Infinite-Dimensional Representations of 2-Groups

Inﬁnite-Dimensional Represen tations of 2-Groups John C. Baez 1 , Aristide Baratin 2 , Lauren t F reidel 3 , 4 , Derek K. Wise 5 1 Departmen t of Mathematics, Univ ersity of California Riv erside, CA 92521, USA 2 Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am M ¨ uhlen b erg 1, 14467 Golm, German y 3 Lab oratoire de Physique, ´ Ecole Normale Sup ´ erieure de Ly on 46 All´ ee d’Italie, 69364 Lyon Cedex 07, F rance 4 P erimeter Institute for Theoretical Physics W aterlo o ON, N2L 2Y5, Canada 5 Institute for Theoretical Physics I II, Univ ersity of Erlangen–N¨ urnberg Staudtstraße 7 / B2, 91058 Erlangen, German y Abstract A ‘2-group’ is a category equipp ed with a m ultiplication satisfying laws lik e those of a group. Just as groups ha ve represen tations on vector spaces, 2-groups hav e representations on ‘2-v ector spaces’, whic h are categories analogous to vector spaces. Unfortunately , Lie 2- groups typically hav e few represen tations on the ﬁnite-dimensional 2-vector spaces in tro duced b y Kaprano v and V o ev o dsky . F or this reason, Crane, Sheppeard and Y etter in tro duced certain inﬁnite-dimensional 2-vector spaces called ‘measurable categories’ (since they are closely related to measurable ﬁelds of Hilbert spaces), and used these to study inﬁnite-dimensional represen- tations of certain Lie 2-groups. Here we contin ue this w ork. W e begin with a detailed study of measurable categories. Then we giv e a geometrical description of the measurable represen- tations, intert winers and 2-intert winers for any skeletal measurable 2-group. W e study tensor pro ducts and direct sums for representations, and v arious concepts of subrepresentation. W e describ e direct sums of intert winers, and sub-in tertwiners—features not seen in ordinary group represen tation theory . W e study irreducible and indecomp osable representations and in tertwin- ers. W e also study ‘irretractable’ representations—another feature not seen in ordinary group represen tation theory . Finally , we argue that measurable categories equipp ed with some extra structure deserve to be considered ‘separable 2-Hilb ert spaces’, and compare this idea to a ten- tativ e deﬁnition of 2-Hilbert spaces as represen tation categories of comm utative von Neumann algebras. 1 Con ten ts 1 In tro duction 3 1.1 2-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 2-V ector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Represen tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Plan of the pap er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Represen tations of 2-groups 16 2.1 F rom groups to 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 2-groups as 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.2 Crossed mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 F rom group representations to 2-group representations . . . . . . . . . . . . . . . . . 20 2.2.1 Represen ting groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Represen ting 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.3 The 2-category of representations . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Measurable categories 28 3.1 F rom vector spaces to 2-vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Categorical p erspective on 2-v ector spaces . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 F rom 2-vector spaces to measurable categories . . . . . . . . . . . . . . . . . . . . . . 33 3.3.1 Measurable ﬁelds and direct integrals . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 The 2-category of measurable categories: Meas . . . . . . . . . . . . . . . . . 39 3.3.3 Construction of Meas as a 2-category . . . . . . . . . . . . . . . . . . . . . . 53 4 Represen tations on measurable categories 54 4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 In vertible morphisms and 2-morphisms in Meas . . . . . . . . . . . . . . . . . . . . 58 4.3 Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.1 Structure of representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.2 Structure of intert winers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3.3 Structure of 2-intert winers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Equiv alence of represen tations and of intert winers . . . . . . . . . . . . . . . . . . . . 77 4.5 Op erations on represen tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.5.1 Direct sums and tensor pro ducts in Meas . . . . . . . . . . . . . . . . . . . . 80 4.5.2 Direct sums and tensor pro ducts in 2Rep ( G ) . . . . . . . . . . . . . . . . . . 85 4.6 Reduction, retraction, and decomp osition . . . . . . . . . . . . . . . . . . . . . . . . 87 4.6.1 Represen tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.6.2 In tertwiners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5 Conclusion 97 A T o ols from measure theory 99 A.1 Leb esgue decomp osition and Radon-Nik o dym deriv ativ es . . . . . . . . . . . . . . . 100 A.2 Geometric mean measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 A.3 Measurable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 A.4 Measurable G -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2 1 In tro duction The goal of ‘categoriﬁcation’ is to dev elop a richer version of existing mathematics b y replacing sets with categories. This lets us exploit the follo wing analogy: set theory category theory elemen ts ob jects equations isomorphisms b et w een elemen ts b et w een ob jects sets categories functions functors equations natural isomorphisms b et w een functions b et w een functors Just as sets hav e elemen ts, categories hav e ob jects. Just as there are functions b et ween sets, there are functors b et ween categories. The correct analogue of an equation b et ween elemen ts is not an equation b et ween ob jects, but an isomorphism. More generally , the analog of an equation b etw een functions is a natural isomorphism b et w een functors. The word ‘categoriﬁcation’ was ﬁrst coined b y Louis Crane [ 23 ] in the context of mathematical ph ysics. Applications to this sub ject hav e alwa ys b een among the most exciting [ 9 ], since categori- ﬁcation holds the promise of generalizing some of the sp ecial features of lo w-dimensional physics to higher dimensions. The reason is that categoriﬁcation b o osts the dimension by one . T o see this in the simplest p ossible wa y , note that we can draw sets as 0-dimensional dots and functions b et ween sets as 1-dimensional arrows: S • f * * • S 0 If we could draw all the sets in the world this w ay , and all the functions b et w een them, w e would ha ve a picture of the category of all sets. But there are man y categories b eside the category of sets, and when we study categories en masse w e see an additional lay er of structure. W e can draw categories as dots, and functors b et ween categories as arrows. But what ab out natural isomorphisms b et ween functors, or more general natural transformations b et ween functors? W e can draw these as 2-dimensional surfaces: C • f * * f 0 4 4 • C 0 h   So, the dimension of our picture has b een b oosted by one! Instead of merely a category of all categories, we say we hav e a ‘2-category’. If w e could dra w all the categories in the world this w ay , and all functors b et w een them, and all natural transformations b et ween those, w e w ould hav e a picture of the 2-category of all categories. This story contin ues indeﬁnitely to higher and higher dimensions: categoriﬁcation is a process than can b e iterated. But our goal here lies in a diﬀerent direction: we wish to take a sp eciﬁc branc h of mathematics, the theory of inﬁnite-dimensional group representations, and categorify 3 that just once. This migh t seem like a purely formal exercise, but we shall see otherwise. In fact, the resulting theory has fascinating relations b oth to w ell-known topics within mathematics (ﬁelds of Hilbert spaces and Mack ey’s theory of induced group representations) and to interesting ideas in ph ysics (spin foam mo dels of quan tum gra vity , most notably the Crane–Shepp eard mo del). 1.1 2-Groups T o categorify group represenation theory , w e m ust ﬁrst choose a wa y to categorify the basic notions in volv ed: the notions of ‘group’ and ‘v ector space’. A t presen t, categorifying mathematical deﬁni- tions is not a completely straightforw ard exercise: it requires a bit of creativity and go o d taste. So, there is work to be done here. By no w, how ever, there is a fairly uncon trov ersial wa y to categorify the concept of ‘group’. The resulting notion of ‘2-group’ can be deﬁned in v arious equiv alent wa ys [ 8 ]. F or example, we can think of a 2-group as a category equipped with a multiplication satisfying the usual axioms for a group. Since categoriﬁcation inv olves replacing equations by natural isomorphisms, w e s hould demand that the group axioms hold up to natur al isomorphism . Then w e should demand that these isomorphisms ob ey some laws of their o wn, called ‘coherence laws’. This is where the creativity comes into pla y . Luc kily , every one agrees on the correct coherence laws for 2-groups. Ho wev er, to simplify our task in this pap er, we shall only consider ‘strict’ 2-groups, where the axioms for a group hold as e quations —not just up to natural isomorphisms. This lets us ignore the issue of coherence laws. Another adv antage of strict 2-groups is that they are essen tially the same as ‘crossed mo dules’ [ 34 ], which are structures already familiar in algebra. So, henceforth w e shall alw ays use the term ‘2-group’ to mean a 2-group of this kind. Supp ose G is a 2-group of this kind. Since G is a category , it has ob jects and morphisms. The ob jects form a group under multiplication, so we can use them to describ e symmetries. The new feature, where w e go b eyond traditional group theory , is the morphisms. F or most of our more substan tial results, we shall make a drastic simplifying assumption: we shall assume G is not only strict but also ‘skeletal’. This means that there only exists a morphism from one ob ject of G to another if these ob jects are actually equal. In other w ords, all the morphisms b et w een ob ject of G are actually automorphisms. Since the ob jects of G describ e symmetries, their automorphisms describ e symmetries of symmetries . The reader should not be fooled b y the somewhat in timidating language. A skeletal 2-group is really a very simple thing. Using the theory of crossed modules, explained in Section 2.1.2 , w e shall see that a skeletal 2-group G consists of: • a group G (the group of ob jects of G ), • an ab elian group H (the group of automorphisms of any ob ject), • a left action B of G as automorphisms of H . A nice example is the ‘P oincar´ e 2-group’, ﬁrst discov ered b y one of the authors [ 4 ]. But to understand this, and to prepare ourselv es for the discussion of physics applications later in this in tro duction, let us ﬁrst recall the ordinary Poincar ´ e group. In special relativity , we think of a p oin t x = ( t, x, y , z ) in R 4 as describing the time and lo cation of an even t. W e equip R 4 with a bilinear form, the so-called ‘Minko wski metric’: x · x 0 = tt 0 − xx 0 − y y 0 − z z 0 4 whic h serves as substitute for the usual dot pro duct on R 3 . With this extra structure, R 4 is called ‘Mink owski spacetime’. The group of all linear transformations T : R 4 → R 4 preserving the Minko wski metric is called O(3 , 1). The connected comp onen t of the identit y in this group is called SO 0 (3 , 1). This smaller group is generated by rotations in space together with transformations that mix time and space co ordinates. Elemen ts of SO 0 (3 , 1) are called ‘Loren tz transformations’. In sp ecial relativity , w e think of Lorentz transformations as symmetries of space- time. How ev er, we also wan t to count translations of R 4 as symmetries. T o include these, we need to take the semidirect pro duct SO 0 (3 , 1) n R 4 , and this is called the P oincar´ e group . The P oincar ´ e 2-group is built from the same ingredients, Lorentz transformation and translations but in a diﬀerent wa y . No w Lorentz transformations are treated as symme tries—that is, ob jects— while the translations are treated as symmetries of symmetries—that is, morphisms. More precisely , the P oincar´ e 2-group is deﬁned to b e the sk eletal 2-group with: • G = SO 0 (3 , 1): the group of Lorentz transformations, • H = R 4 : the group of translations of Minko wski space, • the obvious action of SO 0 (3 , 1) on R 4 . As w e shall see, the representations of this particular 2-group may hav e in teresting applications to ph ysics. F or other examples of 2-groups, see our invitation to ‘higher gauge theory’ [ 7 ]. This is a generalization of gauge theory where 2-groups replace groups. 1.2 2-V ector spaces Just as groups act on sets, 2-groups can act on categories. If a category is equipp ed with structure analogous to that of a vector space, w e ma y call it a ‘2-vector space’, and call a 2-group action preserving this structure a ‘representation’. There is, how ever, quite a bit of experimentation un- derw ay when it comes to axiomatizing the notion of ‘2-vector space’. In this pap er we inv estigate represen tations of 2-groups on inﬁnite-dimensional 2-v ector spaces, follo wing a line of work initiated b y Crane, Shepp eard and Y etter [ 25 , 26 , 71 ]. A quic k review of the history will explain why this is a go od idea. T o b egin with, ﬁnite-dimensional 2-vector spaces were introduced by Kapranov and V o ev o dsky [ 43 ]. Their idea w as to replace the ‘ground ﬁeld’ C by the category V ect of ﬁnite-dimensional complex v ector spaces, and exploit this analogy: ordinary higher linear algebra linear algebra C V ect + ⊕ × ⊗ 0 { 0 } 1 C 5 Just as every ﬁnite-dimensional v ector space is isomorphic to C N for some N , every ﬁnite-dimensional Kaprano v–V o ev o dsky 2-v ector space is equiv alent to V ect N for some N . W e can take this as a deﬁnition of these 2-vector spaces — but just as with ordinary vector spaces, there are also intrinsic c haracterizations which mak e this result into a theorem [ 56 , 70 ]. Similarly , just as every linear map T : C M → C N is equal to one giv en b y a N × M matrix of complex n umbers, every linear map T : V ect M → V ect N is isomorphic to one giv en by an N × M matrix of v ector spaces. Matrix addition and multiplication w ork as usual, but with ⊕ and ⊗ replacing the usual addition and m ultiplication of complex num b ers. The really new feature of higher linear algebra is that w e also ha ve ‘2-maps’ b et w een linear maps. If we ha ve linear maps T , T 0 : V ect M → V ect N giv en b y N × M matrices of vector spaces T n,m and T 0 n,m , then a 2-map α : T ⇒ T 0 is a matrix of linear op erators α n,m : T n,m → T 0 n,m . If we draw linear maps as arrows: V ect M T / / V ect N then we should dra w 2-maps as 2-dimensional surfaces, like this: V ect M T + + T 0 3 3 V ect N α   So, compared to ordinary group represen tation theory , the key nov elty of 2-group representation theory is that b esides in tertwining op erators b et w een representations, we also hav e ‘2-intert winers’, dra wn as surfaces. This bo osts the dimension of our diagrams by one, giving 2-group representation theory an intrinsically 2-dimensional c haracter. The study of represen tations of 2-groups on Kaprano v–V o ev o dsky 2-vector spaces w as initiated by Barrett and Mac k aay [ 18 ], and con tinued b y Elgueta [ 31 ]. They came to some upsetting conclusions. T o understand these, we need to know a bit more ab out 2-v ector spaces. An ob ject of V ect N is an N -tuple of ﬁnite-dimensional v ector spaces ( V 1 , . . . , V N ), so ev ery ob ject is a direct sum of certain sp ecial ob jects e i = (0 , . . . , C |{z} i th place , . . . , 0) . These ob jects e i are analogous to the ‘standard basis’ of C N . How ev er, unlike the case of C N , these ob jects e i are essentially the only basis of V ect N . More precisely , given any other basis e 0 i , we hav e e 0 i ∼ = e σ ( i ) for some p erm utation σ . This fact has serious consequences for representation theory . A 2-group G has a group G of ob jects. Giv en a represen tation of G on V e ct N , eac h g ∈ G maps the standard basis e i to some new basis e 0 i , and th us determines a permutation σ . So, we automatically get an action of G on the ﬁnite set { 1 , . . . , N } . If G is ﬁnite, it will typically ha ve many actions on ﬁnite sets. So, w e can exp ect that ﬁnite 2-groups hav e enough interesting representations on Kapranov–V o ev o dsky 2-vector spaces to yield an interesting theory . But there are many ‘Lie 2-groups’, such as the Poincar ´ e 2-group, where the group of ob jects is a Lie group with few nontrivial actions on ﬁnite sets. Such 2-groups hav e few represen tations on Kaprano v–V o evodsky 2-vector spaces . This prompted the searc h for a ‘less discrete’ v ersion of Kaprano v–V o ev o dsky 2-v ector spaces, where the ﬁnite index set { 1 , . . . , N } is replaced b y something on whic h a Lie group can act in an in teresting wa y . Crane, Shepp eard and Y etter [ 25 , 26 , 71 ] suggested replacing the index set by a 6 measurable space X and replacing N -tuples of ﬁnite-dimensional vector spaces b y ‘measurable ﬁelds of Hilb ert spaces’ on X . Measurable ﬁelds of Hilb ert spaces hav e long b een imp ortan t for studying group represen tations [ 50 ], v on Neumann algebras [ 28 ], and their applications to quantum physics [ 51 , 69 ]. Roughly , a measurable ﬁeld of Hilb ert spaces on a measurable space X can b e thought of as assigning a Hilb ert space to each x ∈ X , in a wa y that v aries measurably with x . There is also a w ell-known concept of ‘measurable ﬁeld of b ounded op erators’ b et w een measurable ﬁelds of Hilb ert spaces ov er a ﬁxed space X . These make measurable ﬁelds of Hilb ert spaces o ver X into the ob jects of a category H X . This is the prototypical example of what Crane, Shepp eard and Y etter call a ‘measurable category’. When X is ﬁnite, H X is essen tially just a Kapranov–V o ev o dsky 2-v ector space. If X is ﬁnite and equipp ed with a measure, H X acquires a kind of inner pro duct, so it b ecomes a ﬁnite-dimensional ‘2-Hilb ert space’ [ 3 ]. When X is inﬁnite, we should think of the measurable category H X as some sort of inﬁnite-dimensional 2-vector space. How ev er, it lacks some features we exp ect from an inﬁnite-dimensional 2-Hilb ert space: in particular, there is no inner pro duct of ob jects. W e discuss this issue further in Section 5 . Most imp ortan tly , since Lie groups hav e many actions on measurable spaces, there is a rich supply of representations of Lie 2-groups on measurable categories. As we shall see, a representation of a 2-group G on the category H X giv es, in particular, an action of the group G of ob jects on the space X , just as representations on V ect N ga ve group actions on N -element sets. These actions lead naturally to a geometric picture of the representation theory . In fact, a measurable category H X already has a considerable geometric ﬂa vor. T o appreciate this, it helps to follow Mac key [ 51 ] and call a measurable ﬁeld of Hilb ert spaces on the measurable space X a ‘measurable Hilb ert space bundle’ o ver X . Indeed, such a ﬁeld H resem bles a v ector bundle in that it assigns a Hilb ert space H x to each p oin t x ∈ X . The diﬀerence is that, since H lives in the w orld of measure theory rather than top ology , we only require that each p oin t x lie in a me asur able subset of X o ver which H can be trivialized, and we only require the existence of me asur able transition functions. As a result, we can alw ays write X as a disjoin t union of countably man y measurable subsets on which H x has constant dimension. In practice, w e demand that this dimension b e ﬁnite or countably inﬁnite. Similarly , measurable ﬁelds of b ounded op erators ma y b e view ed as measurable bundle maps. So, the measurable category H X ma y be viewed as a measurable v ersion of the category of Hilb ert space bundles ov er X . In concrete examples, X is often a manifold or smooth algebraic v ariety , and measurable ﬁelds of Hilb ert spaces often arise from bundles or coheren t sheav es of Hilb ert spaces ov er X . 1.3 Represen tations The study of representations of sk eletal 2-groups on measurable categories w as b egun by Crane and Y etter [ 26 ]. The special case of the Poincar ´ e 2-group w as studied in detail by Crane and Shepp eard [ 25 ]. They noticed in teresting connections to the orbit metho d in geometric quantization, and also to the theory of discrete subgroups of SO(3 , 1), known as ‘Kleinian groups’. These observ ations suggest that Lie 2-group represen tations on measurable categories deserve a thorough and careful treatmen t. This, then, is the goal of the present text. W e give ge ometric descriptions of: • a representation ρ of a skeletal 2-group G on a measurable category H X , • an intert winer b et ween suc h representations: ρ φ / / ρ 0 7 • a 2-intert winer b et ween suc h intert winers: ρ φ ' ' φ 0 7 7 ρ 0 α   . W e use the term ‘in tertwiner’ as short for ‘intert wining operator’. This is a commonly used term for a morphism betw een group representations; here we use it to mean a morphism betw een 2-group represen tations. But in addition to in tertwiners, w e hav e something really new: 2-intert winers b et w een interwiners! This extra la yer of structure arises from categoriﬁcation. W e deﬁne all these concepts in Sections 2 and 3 . Instead of previewing the deﬁnitions here, we prefer to sk etc h the geometric picture that emerges in Section 4 . So, we n ow assume G is a sk eletal 2- group describ ed by the data ( G, H , B ), as ab ov e. W e also assume in what follows that all the spaces and maps inv olv ed are measurable. Under these assumptions we can describ e representations of G , as well as in tertwiners and 2-intert winers, in terms of familiar geometric constructions—but living in the category of measurable spaces, rather than smo oth manifolds. Essentially—ignoring v arious tec hnical issues whic h we discuss later—w e obtain the following dictionary relating represen tation theory to geometry . represen tation theory geometry a representation of G on H X a right action of G on X , and a map X → H ∗ making X a ‘measurable G -equiv arian t bundle’ ov er H ∗ an intert winer b et w een a ‘Hilb ert G -bundle’ o ver the pullbac k of G -equiv arian t bundles represen tations on H X and H Y and a ‘ G -equiv arian t measurable family of measures’ µ y on X a 2-intert winer a map of Hilb ert G -bundles This dictionary requires some explanation! First, H ∗ here is not quite the Pon trjagin dual of H , but rather the group, under p oin t wise m ultiplication, of measurable homomorphisms χ : H → C × where C × is the m ultiplicative group of nonzero complex n umbers. How ev er, this group H ∗ con tains the Pon trjagin dual of H . It turns out that a measurable homomorphism like χ ab o ve, with our deﬁnition of measurable group, is automatically also contin uous. Since C × ∼ = U(1) × R , we hav e H ∗ = b H × hom( H, R ) where b H is the P ontrjagin dual of H . One can consistently restrict to ‘unitary’ represen tations of G , where we replace H ∗ b y b H in the ab o ve table. In most of the pap er, we shall hav e no reason to mak e this restriction, but it is often useful in examples, as w e shall see b elo w. In any case, under some mild conditions on H , H ∗ is again a measurable space, and its group op erations are measurable. The left action B of G on H naturally induces a right action of G on H ∗ , say ( χ, g ) 7→ χ g , given by χ g ( h ) = χ ( g B h ) . 8 This promotes H ∗ to a right G -space. As indicated in the chart, a representation of G is simply a G -equiv ariant map X → H ∗ , where X is a measurable G -space. Because of the measure-theoretic context, we are happ y to call this a ‘bundle’ even with no implied lo cal trivialit y in the top ological sense. Indeed, most of the ﬁb ers ma y even b e empt y . Because of the G -equiv ariance, how ev er, ﬁb ers are isomorphic along any given G -orbit in H ∗ . This geometric pictures helps us understand irreducibility and related notions for 2-group rep- resen tations. Recall that for ordinary groups, a representation is ‘irreducible’ if it has no subrepre- sen tations other than the 0-dimensional represen tation and itself. It is ‘indecomp osable’ if it has no direct summands other than the 0-dimensional represen tation and itself. Since ev ery direct summand is a subrepresentation, every indecomposable representation is irreducible. The conv erse is generally false. Ho w ever, it is true in some cases: for example, ev ery unitary irreducible representation is indecomp osable. The situation with 2-groups is more subtle. The notions of subrepresen tation and direct summand generalize to 2-group represen tations, but there is also an intermediate notion: a ‘ retract’. In fact this notion already exists for group representations. A group represen tation ρ 0 is a ‘retract’ of ρ if ρ 0 is a subrepresen tation and there is also an intert winer pro jecting down from ρ to this subrepresentation. So, we may say a representation is ‘irretractable’ if it has no retracts other than the 0-dimensional represen tation and itself. But for group representations, a retract turns out to b e exactly the same thing as a direct summand, so there is no need for these additional notions. Ho wev er, we can generalize the concept of ‘retract’ to 2-group representations—and now things b ecome more in teresting! No w we ha ve: direct summand = ⇒ retract = ⇒ subrepresentation and thus: irreducible = ⇒ irretractable = ⇒ indecomp osable None of these implications are reversible, except p erhaps every irretractable represen tation is irre- ducible. At presen t this question is unsettled. Indecomp osable and irretractable representations play important roles in our work. Each has a nice geometric picture. Supp ose we ha ve a represen tation of our skeletal 2-group G corresp onding to a G -equiv ariant map X → H ∗ . If the G -space X has more than a single orbit, then w e can write it as a disjoint union of G -spaces X = X 0 ∪ X 00 and split the map X → H ∗ in to a pair of maps. This amoun ts to writing our 2-group represen tation as a direct sum of representations. So, a represen tation on H X is indecomp osable if the G -action on X is transitive. By equiv ariance, this implies that the image of the corresp onding map X → H ∗ is a single orbit of H ∗ , and that the stabilizer of a p oin t in X is a subgroup of the stabilizer of its image in H ∗ . In other words, the orbit in H ∗ is a quotient of X . It follows that indecomp osable represen tations of G are classiﬁed up to equiv alence b y pairs consisting of: • an orbit in H ∗ , and • a subgroup of the stabilizer of a p oin t in that orbit. It turns out that a representation is irretractable if and only if it is indecomposable and the map X → H ∗ is injectiv e. This of course means that X is isomorphic as a G -space to one of the orbits of H ∗ . Thus, irretractable representations are classiﬁed up to equiv alence by G -orbits in H ∗ . In the case of the P oincar ´ e 2-group, this has an in teresting in terpretation. The group H = R 4 has H ∗ ∼ = C 4 . So, a representation in general is given b y a SO 0 (3 , 1)-equiv arian t map p : X → C 4 , where 9 SO 0 (3 , 1) acts indep enden tly on the real and imaginary parts of a vector in C 4 . The representation is irretractable if the image of p is a single orbit. Restricting to the P ontrjagin dual b H amounts to c ho osing the orbit of some r e al vector, an elemen t of R 4 . Th us ‘unitary’ irretractable representations are classiﬁed by the SO 0 (3 , 1) orbits in R 4 , which are familiar ob jects from sp ecial relativit y . If we use p = ( E , p x , p y , p z ) as our name for a p oin t of R 4 , then any orbit is a connected comp onen t of the solution set of an equation of the form p · p = m 2 where the dot denotes the Mink owski metric. In other w ords: E 2 − p 2 x − p 2 y − p 2 y = m 2 . The v ariable names are the traditional ones in relativity: E stands for the energy of a particle, while p x , p y , p z are the three comp onen ts of its momentum, and the constant m is its mass. An orbit corresp onding to a particular mass m describ es the allow ed v alues of energy and momentum for a particle of this mass. These orbits can be dra wn explicitly if we suppress one dimension: m 2 > 0 m 2 =0 m 2 < 0 O O E > 0   E < 0 Though this picture is dimensionally reduced, it faithfully depicts all of the orbits in the 4- dimensional case. There are six t yp es of orbits, thus giving us six t yp es of irretractable representa- tions of the Poincar ´ e 2-group: 1. E = 0, m = 0: the trivial represen tation (orbit is a single p oin t) 2. E > 0, m = 0: the ‘p ositiv e energy massless’ represen tation 3. E < 0, m = 0: the ‘negative energy massless’ representation 4. E > 0, m > 0: ‘p ositiv e energy real mass’ represen tations (one for each m > 0) 5. E < 0, m > 0: ‘negative energy real mass’ representations (one for eac h m > 0) 6. m 2 < 0: ‘imaginary mass’ or ‘tac hy on’ representations (one for eac h − im > 0) On the other hand, there are many more inde c omp osable representations, since these are classiﬁed b y a c hoice of one of the ab o ve orbits together with a subgroup of the corresp onding p oint stabilizer— SO(2), SO(3) or SO 0 (2 , 1) depending on whether m 2 = 0, m 2 > 0, or m 2 < 0. These indecomposable represen tations were studied by Crane and Shepp eard [ 25 ], though they called them ‘irreducible’. 10 T o an y reader familiar with the classiﬁcation of irreducible unitary represen tations of the ordinary P oincar´ e gr oup , the abov e story should seem familiar, but also a bit strange. It should seem familiar b ecause these group representations are p artial ly classiﬁed by SO(3 , 1) orbits in Minko wski spacetime. The strange part is that for these group representations, some extra data is also needed. F or example, a particle with p ositiv e mass and energy is characterized by b oth a mass m > 0 and a spin —an irreducible represen tation of SO(3) (or in a more detailed treatment, the double cov er of this group). By switching to the Poincar ´ e 2-group, we seem to hav e somehow lost the spin information. This is not the case. In fact, as w e no w explain, the ‘spin’ information from the ordinary P oincar ´ e group representation theory has simply b een pushed up one categorical notch—w e will ﬁnd it in the in tertwiners! In other words, the concept of spin shows up not in the classiﬁcation of represen tations of the Poincar ´ e 2-group, but in the classiﬁcation of morphisms b et ween represen tations. The reason, ultimately , is that Lorentz transformations and translations of R 4 sho w up at diﬀeren t levels in the P oincar´ e 2-group: the Lorentz transformations as ob jects, and the translations as morphisms. T o see this in more detail, w e need to understand the geometry of in tertwiners. Supp ose we ha ve t wo represen tations, one on H X and one on H Y , given by equiv ariant bundles χ 1 : X → H ∗ and χ 2 : Y → H ∗ . Lo oking again at the c hart, the k ey geometric ob ject is a Hilb ert bundle o ver the pullbac k of χ 1 and χ 2 . This pullbac k may b e seen as a subspace Z of Y × X : Z X         Y H ∗ χ 2         χ 1   1 1 1 1 1 1   1 1 1 1 1 1 Z = { ( y , x ) ∈ Y × X : χ 2 ( y ) = χ 1 ( x ) } It is easy to see that Z is a G -space under the diagonal action of G on X × Y , and that the pro jections in to X and Y are G -equiv ariant. If H X and H Y are b oth indecomp osable represen tations, then X and Y each lie ov er a single orbit of H ∗ . These orbits must b e the same in order for the pullback Z , and hence the space of intert winers, to b e nontrivial. If H X and H Y are b oth irretractable, this implies that they are equiv alen t. Thus, given an irretractable representation represented b y an orbit X in H ∗ , the self-in tertwiners of this representation are classiﬁed by equiv ariant Hilbert space bundles ov er X . Equiv arian t Hilbert bundles are the sub ject of Mac key’s induced represen tation theory [ 48 , 50 , 51 ]. In general, a wa y to construct an equiv arian t bundle is to pick a p oin t in the base space X and a Hilb ert space that is a representation of the stabilizer of that point, and then use the action of G to ‘translate’ the Hilb ert space along a G -orbit. Con versely , giv en an equiv arian t bundle, the ﬁb er o ver a giv en p oin t is a representation of the stabilizer of that p oint. Indeed, there is an equiv alence of categories:  G -equiv arian t vector bundles o ver a homogeneous space X  '  represen tations of the stabilizer of a p oin t in X  Pro ving this is straightforw ard when we mean ‘vector bundles’ in the in the ordinary topological sense. But in Mack ey’s work, he generalized this correspondence to a measure-theoretic context— precisely the con text that arises in the theory of 2-group represen tations w e are considering here! The upshot for us is that self-intert winers of an irretractable represen tation amount to represen tations of the stabilizer subgroup. T o illustrate this idea, let us return to the example of the Poincar ´ e 2-group. Supp ose w e hav e a unitary irretractable represen tation of this 2-group. As w e ha ve seen, this is given by one of the orbits 11 X ⊂ R 4 of SO 0 (3 , 1). Now, consider any self-intert winer of this represen tation. This is given by a SO 0 (3 , 1)-inv ariant Hilb ert space bundle o ver X . By induced represen tation theory , this amounts to the same thing as a representation of the stabilizer of an y p oin t x ∈ X . F or a ‘p ositiv e energy real mass’ represen tation, for example, corresp onding to an ordinary massiv e particle in sp ecial relativity , this stabilizer is SO(3), so self-in tertwiners are essentially representations of SO(3). In ordinary group represen tation theory , there is no notion of ‘reducibility’ for intert winers. But here, b ecause of the additional level of categorical structure, 2-group intert winers in many w ays more closely resemble group representations than group intert winers. There is a natural concept of ‘direct sum’ of intert winers, and this gives a notion of ‘indecomposable’ intert winer. Similarly , the concept of ‘sub-intert winer’ gives a notion of ‘irreducible’ intert winer. Returning yet again to the P oincar´ e 2-group example, consider the s elf-in tert winers of a p ositiv e energy real mass representation. W e hav e just seen that these corresp ond to represen tations of SO(3). When is suc h a self-in tertwiner irreducible? Unsurprisingly , the answ er is: precisely when the corresp onding represen tation of SO(3) is irreducible. Because of the added lay er of structure, w e can also ask ho w a pair of intert winers with the same source and target represe n tations migh t b e related by 2-intert winer. As w e shall see, intert winers satisfy an analogue of Sc hur’s lemma: a 2-in tertwiner betw een irr e ducible intert winers is either null or an isomorphism, and in the latter case is essen tially unique. So, there is no in teresting information in the self-2-intert winers of an irreducible in tertwiner. W e conclude with a small warning: in the foregoing description of the represen tation theory , w e ha ve for simplicity’s sak e glossed ov er certain subtle measure theoretic issues. Most of these issues make little diﬀerence in the case of the Poincar ´ e 2-group, but may b e imp ortan t for general represen tations of an arbitrary measurable 2-group. F or details, read the rest of the b o ok! 1.4 Applications Next w e describ e some p oten tial applications to ph ysics. Crane and Shepp eard [ 25 ] originally ex- amined representations of the Poincar ´ e 2-group as part of a plan to construct a physical theory of a sp eciﬁc sort. W e b eliev e a v ery similar mo del is implicit in the work of tw o of the current authors on F eynman diagrams in quan tum gravit y [ 10 ]. Since proving this was one of our main motiv ations for studying the representations of Lie 2-groups, we w ould like to recall the ideas here. A ma jor problem in physics today is trying to extend quantum ﬁeld theory , originally formulated for theories that neglect gra vit y , to theories that include gravit y . Quantum ﬁeld theories that neglect gra vity , suc h as the Standard Mo del of particle ph ysics, treat spacetime as ﬂat. More precisely , they treat it as R 4 with its Minko wski metric. The ordinary Poincar ´ e group acts as symmetries here. In quantum ﬁeld theories, physical quantities are often computed with the help of ‘F eynman diagrams’. The details can b e found in an y go o d bo ok on quantum ﬁeld theory—or, for that matter, Borc herds’ review article for mathematicians [ 20 ]. How ev er, from a very abstract persp ective, a F eynman diagram can b e seen as a graph with: • edges lab elled by irreducible represen tations of some group G , and • vertices labelled b y intert winers, where the intert winer at an y vertex go es from the trivial representation to the tensor pro duct of all the representations lab elling edges incident to that vertex. In the simplest theories, the group G is just the P oincar´ e group. In more complicated theories, suc h as the Standard Mo del, w e use a larger group. There is a wa y to ev aluate F eynman diagrams and get complex num b ers, called ‘F eynman am- plitudes’. Ph ysically , w e think of the group representations lab elling F eynman diagram edges as 12 p articles . Indeed, we ha ve already said a bit ab out ho w an irreducible representation of the Poincar ´ e group can describe a particle with a given mass and spin. W e think of the intert winers as inter- actions : w ays for the particles to collide and turn into other particles. So, a F eynman diagram describ es a process inv olving particles. When we tak e the absolute v alue of its amplitude and square it, we obtain the probability for this pro cess to o ccur. F eynman diagrams are essen tially one-dimensional structures, since they ha ve v ertices and edges. On the other hand, there is an approach to quantum gravit y that uses closely analogous two- dimensional structures called ‘spin foams’ [ 5 , 15 , 37 , 65 ]. The 2-dimensional analogue of a graph is called an ‘2-complex’: it is a structure with vertices, edges and fac es . In a spin foam, we lab el the v ertices, edges and faces of a 2-complex with data of some sort. Like F eynman diagrams, spin foams should b e thought of as describing physical pro cesses—but now of a higher-dimensional sort. A spin foam mo del is a recip e for computing complex n umbers from spin foams: their ‘amplitudes’. As b efore, when w e take the absolute v alue of these amplitude and square them, w e obtain probabilities. The ﬁrst spin foam model, only later recognized as suc h, go es back to a famous 1968 pap er by P onzano and Regge [ 59 ]. This describ ed R iemannian quantum gravit y in 3-dimensional spacetime— t wo drastic simpliﬁcations that are worth explaining. First of all, gravit y is muc h easier to deal with in 3d spacetime, since in this case, in the absence of matter, all solutions of Einstein’s equations for general relativity lo ok alike lo cally . More pre- cisely , any spacetime ob eying these equations can be lo cally iden tiﬁed, after a suitable coordinate transformation, with R 3 equipp ed with its Mink owski metric x · x 0 = tt 0 − xx 0 − y y 0 . This is v ery diﬀerent from the physically realistic 4d case, where gravitational w av es can propagate through the v acuum, giving a plethora of lo cally distinct solutions. Ph ysicists say that 3d gravit y lac ks ‘lo cal degrees of freedom’. This makes it muc h easier to study—but it retains some of the conceptual and technical c hallenges of the 4d problem. Second of all, in ‘Riemannian quantum gravit y’, we inv estigate a simpliﬁed w orld where time is just the same as space. In 4d spacetime, this inv olv es replacing Mink owski spacetime with 4d Euclidean space—that is, R 4 with the inner pro duct x · x 0 = tt 0 + xx 0 + y y 0 + z z 0 . While physically quite unrealistic, this switch simpliﬁes some of the math. The reason, ultimately , is that the group of Lorentz transformations, SO 0 (3 , 1), is noncompact, while the rotation group SO(4) is compact. A compact Lie group has a coun table set of irreducible unitary represen tations instead of a contin uum, and this makes some calculations easier. F or example, certain in tegrals b ecome sums. P onzano and Regge found that after making b oth these simpliﬁcations, they could write down an elegant theory of quantum gravit y , no w called the Ponzano–Regge mo del. Their theory is deeply related to represen tations of the 3-dimensional rotation group, SO(3). In mo dern terms, the idea is to start with a 3-manifold equipp ed with a triangulation ∆. Then we form the Poincar ´ e dual of ∆ and lo ok at its 2-skeleton K . In simple terms, K is the 2-complex with: • one vertex for each tetrahedron in ∆, • one edge for each triangle in ∆, • one face for each edge of ∆. 13 W e call suc h a thing a ‘2-complex’. Note that a 2-complex is precisely the sort of structure that, when suitably lab elled, giv es a spin foam! T o obtain a spin foam, w e: • lab el each face of K with an irreducible representation of SO(3), and • lab el each edge of K with an intert winer. There is a wa y to compute an amplitude for such a spin foam, and we can use these amplitudes to answ er physically in teresting questions ab out 3d Riemannian quantum gravit y . The P onzano–Regge mo del served as an inpiration for many further developmen ts. In 1997, Barrett and Crane prop osed a similar mo del for 4-dimensional Riemannian quantum gravit y [ 15 ]. More or less simultaneously , the general concept of ‘spin foam model’ was formulated [ 5 ]. Shortly thereafter, spin foam mo dels of 4d Loren tzian quantum gravit y were prop osed, closely m odelled after the Barrett-Crane model [ 27 , 60 ]. Later, ‘improv ed’ mo dels were developed b y F reidel and Krasno v [ 37 ] and Engle, Pereira, Rov elli and Livine [ 32 ]. These newer mo dels are b eginning to show signs of correctly predicting some phenomena we expect from a realistic theory of quan tum gravit y . Ho wev er, this is work in progress, whose ultimate success is far from certain. One fundamen tal challenge is to incorp orate matter in a spin foam model of quantum gravit y . Indeed, any theory that fails to do this is at b est a warm up for a truly realistic theory . Recen tly , a lot of progress has b een made on incorp orating matter in the P onzano–Regge mo del. Here is where spin foams meet F eynman diagrams! The idea is to compute F eynman amplitudes using a sligh t generalization of the P onzano–Regge mo del which lets us include matter [ 14 ]. This mo del takes the gravitational interactions of particles in to account. As a consistency chec k, we w ant the ‘no-gra vity limit’ of this mo del to reduce to the standard recip e for computing F eynman amplitudes in quantum ﬁeld theory—or more precisely its analogue with Euclidean R 3 replacing 4d Mink owski spacetime. And indeed, this was shown to b e true [ 61 – 63 ]. This raised the hope that the same sort of strategy can w ork in 4-dimensional quantum gravit y . It was natural to start with the ‘no-gravit y limit’, and ask if the usual F eynman amplitudes for quan tum ﬁeld theory in ﬂat 4d spacetime can be computed using a spin foam mo del. If we could do this, the result would not b e a theory of quan tum gravit y , but it w ould provide a radical new form ulation of quantum ﬁeld theory , in which Minko wski spacetime is replaced by an inherently quan tum-mechanical spacetime built from spin foams. If a form ulation exists, it ma y help us dev elop mo dels describing quan tum gravit y and matter in 4 dimensions. Recen t work by [ 10 ] gives precisely such a formulation, at least in the 4-dimensional Riemannian case. In other words, this work gives a spin foam mo del for computing F eynman amplitudes for quan tum ﬁeld theories, not on Minko wski spacetime, but rather on 4-dimensional Euclidean space. F eynman diagrams for such theories are built using representations, not of the Poincar ´ e group, but of the Euclidean group : SO(4) n R 4 . More recently still, it was seen that this new mo del is a close relative of the Crane–Shepp eard mo del [ 11 , 13 ]! The only diﬀerence is that where the Crane–Shepp eard mo del uses the Poincar ´ e 2-group, the new mo del uses the Euclidean 2-group , a skeletal 2-group for which: • G = SO(4): the group of rotations of 4d Euclidean space, • H = R 4 : the group of translations 4d Euclidean space, • the obvious action of SO(4) on R 4 . 14 The representation theory of the Euclidean 2-group is very muc h like that of the P oincar´ e 2-group, but with concentric spheres replacing the hyperb oloids E 2 − p 2 x − p 2 y − p 2 y = m 2 . So, we can no w guess the meaning of the Crane–Shepp eard model: it should giv e a new wa y to compute F eynman integrals for ordinary quantum ﬁeld theories on 4d Minko wski spacetime. T o conclude, let us just say a word about how this mo del actually w orks. It helps to go bac k to the Ponzano–Regge mo del. W e can describ e this directly in terms of a 3-manifold with triangulation ∆, instead of the Poincar ´ e dual picture. In these terms, each spin foam corresp onds to a wa y to: • lab el each edge of ∆ with an irreducible representation of SO(3), and • lab el each triangle of ∆ with an in tertwiner. The Ponzano–Regge mo del gives a w ay to compute an amplitude for any suc h lab elling. The Crane–Shepp eard mo del do es a similar thing one dimension up. Suppose we tak e a 4- manifold with a triangulation ∆. Then we ma y: • lab el each edge of ∆ with an irretractable representation of the Poincar ´ e 2-group, • lab el each triangle of ∆ with an irreducible intert winer, and • lab el each tetrahedron of ∆ with a 2-in tertwiner. The Crane–Shepp eard model giv es a w ay to compute an amplitude for an y such lab elling. 1.5 Plan of the pap er Ab o v e we describe a 2-group as a category equipp ed with a multiplication and in verses. While this is correct, another equiv alen t approach turns out to b e more useful for our purp oses here. Just as a group can b e thought of as a category that has one ob ject and for whic h all morphisms are inv ertible, a 2-group can b e though t of as a 2-category that has one ob ject and for whic h all morphisms and 2-morphisms are in vertible. In Section 2 w e recall the deﬁnition of a 2-category and explain how to think of a 2-group as a 2-category of this sort. W e also describ e how to construct 2-groups from crossed modules, and vice v ersa. W e conclude by deﬁning the 2-category 2Rep ( G ) of represen tations of a ﬁxed 2-group G in a ﬁxed 2-category C . In Section 3 we explain measurable categories. W e ﬁrst recall Kaprano v and V o ev o dsky’s 2- v ector spaces, and then in tro duce the necessary analysis to present Y etter’s results on measurable categories. T o do this, we need to construct the 2-category Meas of measurable categories. The problem is that w e do not yet know an intrinsic characterization of measurable categories. At present, a measurable category is simply deﬁned as one that is ‘ C ∗ -equiv alen t’ to a category of measurable ﬁelds of Hilb ert spaces. So, it is a substan tial task to construct the 2-category Meas . As a warm up, w e carry out a similar construction of the 2-category of Kaprano v–V o ev o dsky 2-vector spaces (for whic h an in trinsic characterization is known, making a simpler approach possible). W orking in this picture, we study the representations of 2-groups on measurable categories in Section 4 . W e present a detailed study of equiv alence, direct sums, tensor products, reducibility , decomp osabilit y , and retractabilit y for represen tations and 1-in tertwiners. While our w ork is h ugely indebted to that of Crane, Shepp eard, and Y etter, w e confron t many issues they did not discuss. Some of these arise from the fact that they implicitly consider represen tations of discrete 2-groups, 15 while w e treat me asur able representations of me asur able 2-groups—for example, Lie 2-groups. The represen tations of a Lie group viewed as a discrete group are v astly more pathological than its measurable represen tations. Indeed, this is already true for R , which has enormous num bers of nonmeasurable 1-dimensional representations if we assume the axiom of c hoice, but none if we assume the axiom of determinacy . The same phenomenon occurs for Lie 2-groups. So, it is important to treat them as measurable 2-groups, and fo cus on their measurable representations. In Section 5 , we conclude by sketc hing some directions for future research. W e argue that a measurable category H X b ecomes a ‘separable 2-Hilbert space’ when the measurable space X is equipp ed with a σ -ﬁnite measure. W e also sketc h how this approach to separable 2-Hilb ert spaces should ﬁt into a more general approach to 2-Hilbert spaces based on von Neumann algebras. Finally , App endix A contains some results from analysis that we need. Nota Bene: in this pap er, we alwa ys use ‘measurable space’ to mean ‘standard Borel space’: that is, a set X with a σ -algebra of subsets generated by the op en subsets for some complete separable metric on X . Similarly , we use ‘measurable group’ to mean ‘lcsc group’: that is, a top ological group for which the top ology is lo cally compact Hausdorﬀ and second countable. W e also assume all our measures are σ -ﬁnite and p ositive. These bac kground assumptions give a fairly conv enient framework for the analysis in this pap er. 2 Represen tations of 2-groups 2.1 F rom groups to 2-groups 2.1.1 2-groups as 2-categories W e hav e said that a 2-group is a category equipp ed with pro duct and inv erse op erations satisfying the usual group axioms. How ev er, a more p o w erful approach is to think of a 2-group as a sp ecial sort of 2-category . T o understand this, ﬁrst note that a group G can be thought of as a category with a single ob ject ? , morphisms lab eled b y elements of G , and comp osition deﬁned by m ultiplication in G : ? g 1 / / ? g 2 / / ? = ? g 2 g 1 / / ? In fact, one can deﬁne a group to b e a category with a single ob ject and all morphisms inv ertible. The ob ject ? can b e though t of as an ob ject whose symmetry group is G . In a 2-group, we add an additional lay er of structure to this picture, to capture the idea of symmetries b etwe en symmetries . So, in addition to ha ving a single ob ject ? and its automorphisms, w e hav e isomorphisms b etwe en automorphisms of ? : ? g ( ( g 0 6 6 ? h   These ‘morphisms b et ween morphisms’ are called 2-morphisms . T o make this precise, we should recall that a 2-category consists of: • ob jects: X , Y , Z , . . . • morphisms: X f / / Y 16 • 2-morphisms: X f ' ' f 0 7 7 Y α   Morphisms can b e comp osed as in a category , and 2-morphisms can b e comp osed in t wo distinct w ays: vertically: X f " " f 0 / / f 00 < < Y α   α 0   = X f % % f 00 9 9 Y α 0 · α   and horizontally: X f 1 ' ' f 0 1 7 7 Y α 1   f 2 ' ' f 0 2 7 7 Z α 2   = X f 2 f 1 % % f 0 2 f 0 1 9 9 Y α 2 ◦ α 1   A few simple axioms must hold for this to b e a 2-category: • Comp osition of morphisms must b e ass ociative, and ev ery ob ject X m ust ha ve a morphism X 1 x / / X serving as an identit y for comp osition, just as in an ordinary category . • V ertical comp osition m ust b e asso ciativ e, and every morphism X f / / Y m ust hav e a 2- morphism X f ' ' f 7 7 Y 1 f   serving as an identit y for vertical comp osition. • Horizontal composition m ust b e asso ciativ e, and the 2-morphism X 1 X ' ' 1 X 7 7 X 1 1 X   m ust serve as an identit y for horizontal composition. • V ertical comp osition and horizontal comp osition of 2-morphisms must satisfy the following exc hange la w : ( α 0 2 · α 2 ) ◦ ( α 0 1 · α 1 ) = ( α 0 2 ◦ α 0 1 ) · ( α 2 ◦ α 1 ) (1) 17 so that diagrams of the form X f 1 " " f 0 1 / / f 00 1 < < Y α 1   α 0 1   f 2 " " f 0 2 / / f 00 2 < < Z α 2   α 0 2   deﬁne unambiguous 2-morphisms. F or more details, see the references [ 44 , 52 ]. W e can now deﬁne a 2-group: Deﬁnition 1 A 2-group is a 2-c ate gory with a unique obje ct such that al l morphisms and 2- morphisms ar e invertible. In fact it is enough for all 2-morphisms to ha ve ‘v ertical’ in v erses; given that morphisms are in vertible it then follo ws that 2-morphisms hav e horizontal in verses. Experts will realize that we are deﬁning a ‘strict’ 2-group [ 8 ]; we will never use an y other sort. The 2-categorical approach to 2-groups is a p o werful conceptual to ol. Ho w ever, for explicit calculations it is often useful to treat 2-groups as ‘crossed mo dules’. 2.1.2 Crossed mo dules Giv en a 2-group G , we can extract from it four pieces of information whic h form something called a ‘crossed mo dule’. Conv ersely , an y crossed module gives a 2-group. In fact, 2-groups and crossed mo dules are just diﬀerent wa ys of describing the same concept. While less elegan t than 2-groups, crossed mo dules are go od for computation, and also go o d for constructing examples. Let G b e a 2-group. F rom this we can extract: • the group G consisting of all morphisms of G : ? g / / ? • the group H consisting of all 2-morphisms whose source is the identit y morphism: ? 1 ( ( g 6 6 ? h   • the homomorphism ∂ : H → G assigning to eac h 2-morphism h ∈ H its target: ? 1 ( ( ∂ ( h ):= g 6 6 ? h   • the action B of G as automorphisms of H giv en by ‘horizon tal conjugation’: ? 1 ( ( g ∂ ( h ) g − 1 6 6 ? g B h   := ? g − 1 & & g − 1 8 8 ? 1 g − 1   1 & & ∂ h 8 8 ? h   g & & g 8 8 ? 1 g   18 It is easy to c heck that the homomorphism ∂ : H → G is compatible with B in the follo wing tw o w ays: ∂ ( g B h ) = g ∂ ( h ) g − 1 (2) ∂ ( h ) B h 0 = hh 0 h − 1 . (3) Suc h a system ( G, H , B , ∂ ) satisfying equations ( 2 ) and ( 3 ) is called a crossed mo dule . W e can reco ver the 2-group G from its crossed module ( G, H, B , ∂ ), using a pro cess we now describ e. In fact, every crossed mo dule giv es a 2-group via this pro cess [ 34 ]. Giv en a crossed mo dule ( G, H , B , ∂ ), we construct a 2-group G with: • one ob ject: ? • elements of G as morphisms: ? g / / ? • pairs u = ( g , h ) ∈ G × H as 2-morphisms, where ( g , h ) is a 2-morphism from g to ∂ ( h ) g . W e dra w such a pair as: u = ? g & & g 0 8 8 ? h   where g 0 = ∂ ( h ) g . Comp osition of morphisms and v ertical comp osition of 2-morphisms are deﬁned using m ultiplication in G and H , resp ectiv ely: ? g 1 / / ? g 2 / / ? = ? g 2 g 1 / / ? and ? g ! ! g 0 / / g 00 = = ? h   h 0   = ? g $ $ g 00 : : ? h 0 h   with g 0 = ∂ ( h ) g and g 00 = ∂ ( h 0 ) ∂ ( h ) g = ∂ ( h 0 h ) g . In other words, supp ose we hav e 2-morphisms u = ( g , h ) and u 0 = ( g 0 , h 0 ). If g 0 = ∂ ( h ) g , they are vertically comp osable, and their vertical comp osite is giv en by: u 0 · u = ( g 0 , h 0 ) · ( g , h ) = ( g , h 0 h ) (4) They are alwa ys horizontally comp osable, and we deﬁne their horizontal composite by: ? g 1 ( ( g 0 1 6 6 ? h 1   g 2 ( ( g 0 2 6 6 ? h 2   = ? g 2 g 1 ' ' g 0 2 g 0 1 7 7 ? h 2 ( g 2 B h 1 )   So, horizon tal composition mak es the set of 2-morphisms into a group, namely the semidirect product G n H with m ultiplication: ( g 2 , h 2 ) ◦ ( g 1 , h 1 ) ≡ ( g 2 g 1 , h 2 ( g 2 B h 1 )) (5) 19 One can chec k that the exchange la w ( u 0 2 · u 2 ) ◦ ( u 0 1 · u 1 ) = ( u 0 2 ◦ u 0 1 ) · ( u 2 ◦ u 1 ) (6) holds for 2-morphisms u i = ( g i , h i ) and u 0 i = ( g 0 i , h 0 i ), so that the diagram ? g 1 ! ! g 0 1 / / g 00 1 = = ? h 1   h 0 1   g 2 ! ! g 0 2 / / g 00 2 = = ? h 2   h 0 2   giv es a w ell-deﬁned 2-morphism. T o see an easy example of a 2-group, start with a group G acting as automorphisms of a group H . If we take B to be this action and let ∂ : H → G be the trivial homomorphism, we can easily c heck that the crossed mo dule axioms ( 2 ) and ( 3 ) hold if H is ab elian . So, if H is ab elian, we obtain a 2-group with G as its group of ob jects and G n H as its group of morphisms, where the semidirect pro duct is deﬁned using the action B . Since ∂ is trivial in this example, any 2-morphism u = ( g, h ) goes from g to itself: ? g & & g 8 8 ? h   So, this type of 2-group has only 2- auto morphisms, and each morphism has precisely one 2-automorphism for each element of H . A 2-group with trivial ∂ is called skeletal , and one can easily see that every sk eletal 2-group is of the form just describ ed. An imp ortan t p oin t is that for a skeletal 2-group, the group H is necessarily ab elian. While w e derived this using ( 3 ) ab ov e, the real reason is the Eckmann–Hilton argumen t [ 29 ]. An imp ortan t example of a skeletal 2-group is the ‘Poincar ´ e 2-group’ coming from the semidirect pro duct S O (3 , 1) n R 4 in precisely the wa y just describ ed [ 4 ]. 2.2 F rom group representations to 2-group represen tations 2.2.1 Represen ting groups In the ordinary theory of groups, a group G may b e represen ted on a vector space. In the language of categories, suc h a represen tation is nothing but a functor ρ : G → V ect, where G is seen as category with one ob ject ∗ , and V ect is the category of vector spaces and linear op erators. T o see this, note that suc h a functor m ust send the ob ject ∗ ∈ G to some vector space ρ ( ∗ ) = V ∈ V ect. It m ust also send each morphism ? g → ? in G —or in other words, eac h element of our group—to a linear map V ρ ( g ) / / V Sa ying that ρ is a functor then means that it preserves iden tities and comp osition: ρ (1) = 1 V ρ ( g h ) = ρ ( g ) ρ ( h ) 20 for all group elements g , h . In this language, an intert wining operator b et ween group represen tations—or ‘intert winer’, for short—is nothing but a natur al tr ansformation . T o see this, supp ose that ρ 1 , ρ 2 : G → V ect are functors and φ : ρ 1 ⇒ ρ 2 is a natural transformation. Suc h a transformation m ust give for each ob ject ? ∈ G a linear operator from ρ 1 ( ∗ ) = V 1 to ρ 2 ( ∗ ) = V 2 . But G is a category with one ob ject, so w e hav e a single op erator φ : V 1 → V 2 . Saying that the transformation is ‘natural’ then means that this square commutes: V 1 ρ 1 ( g ) / / φ   V 1 φ   V 2 ρ 2 ( g ) / / V 2 (7) for each group elemen t g . This says simply that ρ 2 ( g ) φ = φρ 1 ( g ) (8) for all g ∈ G . So, φ is an intert winer in the usual sense. Wh y bother with the categorical viewp oin t on on represen tation theory? One reason is that it lets us generalize the concepts of group representation and in tertwiner: Deﬁnition 2 If G is a gr oup and C is any c ate gory, a representation of G in C is a functor ρ fr om G to C , wher e G is se en as a c ate gory with one obje ct. Given r epr esentations ρ 1 and ρ 2 of G in C , an in tertwiner φ : ρ → ρ 0 is a natur al tr ansformation fr om ρ to ρ 0 . In ordinary representation theory we take C = V ect; but we can also, for example, work with the category of sets C = Set, so that a representation of G in C picks out a set together with an action of G on this set. Quite generally , there is a category Rep ( G ) whose ob jects are representations of G in C , and whose morphisms are the intert winers. Comp osition of in tertwiners is deﬁned b y comp osing natural transformations. W e deﬁne tw o representations ρ 1 , ρ 2 : G → C to b e equiv alent if there exists an in tertwiner betw een them which has an inv erse. In other words, ρ 1 and ρ 2 are equiv alen t if there is a natural isomorphism b et ween them. In the next section w e shall see that the represen tation theory of 2-groups amoun ts to taking all these ideas and ‘b o osting the dimension b y one’, using 2-categories ev erywhere instead of categories. 2.2.2 Represen ting 2-groups Just as groups are t ypically represen ted in the category of v ector spaces, 2-groups ma y be represented in some 2-category of ‘2-v ector spaces’. How ever, just as for group representations, the deﬁnition of a 2-group represen tation do es not dep end on the particular target 2-category we wish to represent our 2-groups in. W e therefore presen t the deﬁnition in its abstract form here, b efore describing precisely what sort of 2-vector spaces we will use, in Section 3 . W e hav e seen that a represen tation of a group G in a category C is a functor ρ : G → C b etw een categories. Similarly , a representation of a 2-group will b e a ‘2-functor’ b etw een 2-categories. As with group representations, w e hav e intert winers b et ween 2-group representations, which in the language of 2-categories are ‘pseudonatural transformations’. But the extra lay er of categorical 21 structure implies that in 2-group representation theory w e also hav e ‘2-intert winers’ going b et ween in tertwiners. These are deﬁned to b e ‘modiﬁcations’ b et w een pseudonatural transformations. The reader can learn the general notions of ‘2-functor’, ‘pseudonatural transformation’ and ‘mo d- iﬁcation’ from the review article by Kelly and Street [ 44 ]. How ev er, to make this paper self-contained, w e describ e these concepts b elow in the sp ecial cases that we actually need. Deﬁnition 3 If G is a 2-gr oup and C is any 2-c ate gory, then a represen tation of G in C is a 2-functor ρ fr om G to C . Let us describ e what such a 2-functor amounts to. Supp ose a 2-group G is given by the crossed mo dule ( G, H , ∂ , B ), so that G is the group of morphisms of G , and G n H is the group of 2-morphisms, as describ ed in section 2.1.2 . Then a representation ρ : G → C is sp eciﬁed b y: • an ob ject V of C , asso ciated to the single ob ject of the 2-group: ρ ( ? ) = V • for each morphism g ∈ G , a morphism in C from V to itself: V ρ ( g ) / / V • for each 2-morphism u = ( g , h ), a 2-morphism in C V ρ ( g ) ) ) ρ ( ∂ hg ) 5 5 V ρ ( u )   That ρ is a 2-functor means these corresp ondences preserve identities and all three composition op erations: comp osition of morphisms, and horizon tal and vertical composition of 2-morphisms. In the case of a 2-group, preserving iden tities follo ws from preserving comp osition. So, we only need require: • for all morphisms g , g 0 : ρ ( g 0 g ) = ρ ( g 0 ) ρ ( g ) (9) • for all v ertically comp osable 2-morphisms u and u 0 : ρ ( u 0 · u ) = ρ ( u 0 ) · ρ ( u ) (10) • for all 2-morphisms u, u 0 : ρ ( u 0 ◦ u ) = ρ ( u 0 ) ◦ ρ ( u ) (11) Here the compositions la ws in G and C hav e b een denoted the same w ay , to av oid an o verabundance of notations. Deﬁnition 4 Given a 2-gr oup G , any 2-c ate gory C , and r epr esentations ρ 1 , ρ 2 of G in C , an in ter- t winer φ : ρ 1 → ρ 2 is a pseudonatur al tr ansformation fr om ρ 1 to ρ 2 . 22 This is analogous to the usual representation theory of groups, where an intert winer is a natural transformation b et ween functors. As b efore, an in tertwiner inv olv es a morphism φ : V 1 → V 2 in C . Ho wev er, as usual when passing from categories to 2-categories, this morphism is only required to satisfy the commutation relations ( 8 ) up to 2-isomorphism . In other words, whereas b efore the diagram ( 7 ) commuted, so that the morphisms ρ 2 ( g ) φ and φρ 1 ( g ) were e qual , here we only require that there is a sp eciﬁed in vertible 2-morphism φ ( g ) from one to the other. (An in v ertible 2-morphism is called a ‘2-isomorphism’.) The commutativ e square ( 7 ) for in tertwiners is th us generalized to: V 1 ρ 1 ( g ) / / φ   V 1 φ   V 2 ρ 2 ( g ) / / V 2 : B φ ( g ) } } } } } } } } } } } } } } } } } } (12) W e say the comm utativity of the diagram ( 7 ) has b een ‘w eakened’. In short, a intert winer from ρ 1 to ρ 2 is really a pair consisting of a morphism φ : V 1 → V 2 together with a family of 2-isomorphisms φ ( g ) : ρ 2 ( g ) φ ∼ − → φ ρ 1 ( g ) (13) one for each g ∈ G . These data m ust satisfy some additional conditions in order to b e ‘pseudonatu- ral’: • φ should b e compatible with the identit y 1 ∈ G : φ (1) = 1 φ (14) where 1 φ : φ → φ is the iden tity 2-morphism. Diagrammatically: V 1 1 V 1 / / φ   V 1 φ   V 2 1 V 2 / / V 2 : B φ (1) } } } } } } } } } } } } } } } } } } } } = V 1 φ   φ 0 0 V 2 : B 1 φ } } } } } } } } } } } } } } } } } } } } • φ should be compatible with composition of morphisms in G . Intuitiv ely , this means w e should b e able to glue φ ( g ) and φ ( g 0 ) together in the most ob vious w ay , and obtain φ ( g 0 g ): V 1 ρ 1 ( g ) / / φ   V 1 ρ 1 ( g 0 ) / / φ   V 1 φ   V 2 ρ 2 ( g ) / / V 2 ρ 2 ( g 0 ) / / V 2 : B φ ( g ) } } } } } } } } } } } } } } } } } } } } : B φ ( g 0 ) } } } } } } } } } } } } } } } } } } } } = V 1 ρ 1 ( g 0 g ) / / φ   V 1 φ   V 2 ρ 2 ( g 0 g ) / / V 2 : B φ ( g 0 g ) } } } } } } } } } } } } } } } } } } } } (15) 23 T o make sense of this equation we need the concept of ‘whisk ering’, which w e no w explain. Supp ose in an y 2-category we hav e morphisms f 1 , f 2 : x → y , a 2-morphism φ : f 1 ⇒ f 2 , and a morphism g : y → z . Then w e can whisker φ b y g b y taking the horizontal composite 1 g ◦ φ , deﬁning: x f 1   f 2 @ @ y g / / z φ   := x f 1   f 2 @ @ y g   g A A z φ   1 g   W e can also whisker on the other side: x f / / y g 1   g 2 A A z φ   := x f   f @ @ y g 1   g 2 A A z 1 f   φ   T o deﬁne the 2-morphism given by the diagram on the left-hand side of ( 15 ), we whisker φ ( g ) on one side b y ρ 2 ( g 0 ), whisk er φ ( g 0 ) on the other side b y ρ 1 ( g ), and then v ertically compose the resulting 2-morphisms. So, the equation in ( 15 ) is a diagrammatic wa y of writing:  φ ( g 0 ) ◦ 1 ρ 1 ( g )  ·  1 ρ 2 ( g 0 ) ◦ φ ( g )  = φ ( g 0 g ) (16) • Finally , the intert winer φ should satisfy a higher-dimensional analogue of diagram ( 7 ), so that it ‘intert wines’ the 2-morphisms ρ 1 ( u ) and ρ 2 ( u ) where u = ( g , h ) is a 2-morphism in the 2- group. So, we demand that the follo wing “pillo w” diagram comm ute for all g ∈ G and h ∈ H : V 1 φ   ρ 1 ( g 0 ) ( ( ρ 1 ( g ) 6 6 V 1 φ   V 2 ρ 2 ( g 0 ) ( ( k g c _ [ W S ρ 2 ( g ) 6 6 V 2 K S ρ 1 ( u ) K S ρ 2 ( u ) > F φ ( g 0 ) ? G φ ( g )               (17) where we hav e introduced g 0 = ∂ ( h ) g . In other words: [ 1 φ ◦ ρ 1 ( u )] · φ ( g ) = φ ( g 0 ) · [ ρ 2 ( u ) ◦ 1 φ ] (18) where we hav e again used whiskering to glue together the 2-morphisms on the front and top, and similarly the b ottom and back. No w a w ord ab out notation is required. While an intert winer from ρ 1 to ρ 2 is really a pair consisting of a morphism φ : V 1 → V 2 and a family of 2-morphisms φ ( g ), for eﬃciency w e refer to an in tertwiner simply as φ , and denote it by φ : ρ 1 → ρ 2 . This should not cause any confusion. 24 So far, we hav e describ ed represen tation of 2-groups as 2-functors and intert winers as pseudo- natur al tr ansformations . As mentioned earlier, there are also things going b et ween pseudonatural transformations, called mo diﬁc ations . The follo wing deﬁnition should thus come as no surprise: Deﬁnition 5 Given a 2-gr oup G , a 2-c ate gory C , r epr esentations ρ 1 and ρ 2 of G in C , and inter- twiners φ, ψ : ρ → ρ 0 , a 2-in tertwiner m : φ ⇒ ψ is a mo diﬁc ation fr om φ to ψ . Let us say what modiﬁcations amoun t to in this case. A mo diﬁcation m : φ ⇒ ψ is a 2-morphism V 1 φ ) ) ψ 5 5 V 2 m   (19) in C such that the following pillow diagram: V 1 ρ 1 ( g ) / / φ   ψ   V 1 φ       # ' , ψ   V 2 ρ 2 ( g ) / / V 2 m + 3 m + 3 ψ ( g ) 7 ? w w w w w w w w w w w w w w φ ( g ) 7 ? (20) comm utes. Equating the front and left with the back and right, this means precisely that: ψ ( g ) ·  1 ρ 2 ( g ) ◦ m  =  m ◦ 1 ρ 1 ( g )  · φ ( g ) (21) where we hav e again used whisk ering to attac h the morphisms ρ i ( g ) to the 2-morphism m . It is helpful to compare this diagram with the condition shown in ( 17 ). One imp ortan t diﬀerence is that in that case, we had a “pillow” for eac h elemen t g ∈ G and h ∈ H , whereas here w e hav e one only for each g ∈ G . F or a intert winer, the pillow in volv es 2-morphisms b et ween the maps given b y represen tations. Here the condition states that we ha ve a ﬁxed 2-morphism m b et ween morphisms I and J b et ween represen tation spaces, making the given diagram commu te for each g . This is what represen tation theory of ordinary groups would lead us to exp ect from an intert winer. 2.2.3 The 2-category of representations Just as any group G gives a category Rep ( G ) with representations as ob jects and intert winers as morphisms, any 2-group G gives a 2-category 2Rep ( G ) with represen tations as ob jects, intert winers as morphisms, 2-intert winers as 2-morphisms. It is w orth describing the structure of this 2-category explicitly . In particular, let us describ e the rules for comp osing intert winers and for vertically and horizon tally comp osing 2-intert winers: • First, given a comp osable pair of intert winers: ρ 1 φ / / ρ 2 ψ / / ρ 3 25 w e wish to deﬁne their comp osite, whic h will b e an intert winer from ρ 1 to ρ 3 . Recall that this in tertwiner is a pair consisting of a morphism ξ : V 1 → V 3 in C together with a family of 2-morphisms ξ ( g ). W e deﬁne ξ to b e the comp osite ψ φ , and for any g ∈ G we deﬁne ξ ( g ) b y gluing together the diagrams ( 12 ) for φ ( g ) and ψ ( g ) in the obvious w ay: V 1 ρ 1 ( g ) / / ξ   V 1 ξ   V 3 ρ 3 ( g ) / / V 3 : B ξ ( g ) } } } } } } } } } } } } } } } } } } } } := V 1 ρ 1 ( g ) / / φ   V 1 φ   V 2 ρ 2 ( g ) / / ψ   V 2 ψ   : B φ ( g ) } } } } } } } } } } } } } } } } } } } } V 3 ρ 3 ( g ) / / V 2 : B ψ ( g ) } } } } } } } } } } } } } } } } } } } } (22) The diagram on the left hand side is once again ev aluated with the help of whiskering: w e whisk er φ ( g ) on one side b y ψ and ψ ( g ) on the other side by φ , then v ertically comp ose the resulting 2-morphisms. In summary: ξ = ψ φ, ξ ( g ) = [ 1 ψ ◦ φ ( g )] · [ ψ ( g ) ◦ 1 φ ] (23) By some calculations b est done using diagrams, one can chec k that these formulas deﬁne an in tertwiner: relations ( 12 ), ( 14 ), ( 15 ) and ( 17 ) follo w from the corresp onding relations for ψ and φ . • Next, supp ose we ha ve a v ertically comp osable pair of 2-intert winers: ρ 1 φ " " ψ / / ξ < < ρ 2 m   n   Then the 2-intert winers m and n can b e vertically comp osed using vertical composition in C . With some further calculations one one chec k that the relation ( 21 ) for n · m : φ ⇒ ξ follows from the corresp onding relations for m and n . • Finally , consider a horizontally composable pair of 2-intert winers: ρ 1 φ ) ) φ 0 5 5 ρ 2 m   ψ ) ) ψ 0 5 5 ρ 3 n   Then m and n can b e comp osed using horizontal comp osition in C . With more calculations, one can c heck that the result n ◦ m deﬁnes a 2-intert winer: it satisﬁes relation ( 21 ) b ecause n and m satisfy the corresp onding relations. 26 All the calculations required ab o ve are w ell-known in 2-category theory [ 44 ]. Quite generally , these calculations show that for any 2-categories X and Y , there is a 2-category with: • 2-functors ρ : X → Y as ob jects, • pseudonatural transformations b et w een these as morphisms, • mo diﬁcations b et ween these as 2-morphisms. W e are just considering the case X = G , Y = C . W e conclude our description of 2Rep ( G ) by discussing inv ertibility for intert winers and 2- in tertwiners; this will allow us to introduce natural equiv alence relations for represen tations and in tertwiners. W e ﬁrst need to ﬁll a small gap in our description of the 2-category 2Rep ( G ): we need to describe the identit y morphisms and 2-morphisms. Every representation ρ , with representation space V , has its identit y in tertwiner giv en b y the identit y morphism 1 V : V → V in C , together with for each g the identit y 2-morphism 1 ρ ( g ) : ρ ( g ) 1 V ∼ − → 1 V ρ ( g ) Also, every intert winer φ has its identit y 2-intert winer , given by the identit y 2-morphism 1 φ in C . W e deﬁne a 2-intert winer m : φ ⇒ ψ to b e inv ertible (for vertical comp osition) if there exists n : ψ ⇒ φ such that n · m = 1 φ and m · n = 1 ψ Similarly , we deﬁne a in tertwiner φ : ρ 1 → ρ 2 to b e strictly inv ertible if there exists an in tertwiner ψ : ρ 2 → ρ 1 with ψ φ = 1 ρ 1 and φψ = 1 ρ 2 (24) Ho wev er, it is better to relax the notion of in vertibilit y for intert winers b y requiring that the equalities ( 24 ) hold only up to invertible 2-intertwiners . In this case w e say that φ is w eakly inv ertible , or simply in vertible . As for ordinary groups, w e often consider equiv alence classes of representations, rather than represen tations themselves: Deﬁnition 6 We say that two r epr esentations ρ 1 and ρ 2 of a 2-gr oup ar e equiv alen t , and write ρ 1 ' ρ 2 , when ther e exists a we akly invertible intertwiner b etwe en them. In the representation theory of 2-groups, how ev er, where an extra la yer of categorical structure is added, it is also natural to consider equiv alence classes of intert winers: Deﬁnition 7 We say two intertwiners ψ , φ : ρ 1 → ρ 2 ar e equiv alent , and write φ ' ψ , when ther e exists an invertible 2-intertwiner b etwe en them. Sometimes it is useful to relax this notion of equiv alence to include pairs of intert winers that are not strictly parallel. Namely , w e call in tertwiners φ : ρ 1 → ρ 2 and ψ : ρ 0 1 → ρ 0 2 ‘equiv alen t’ if there are inv ertible in tertwiners ρ i → ρ 0 i suc h that ρ 1 φ → ρ 2 ∼ → ρ 0 2 and ρ 1 ∼ → ρ 0 1 ψ → ρ 0 2 are equiv alen t, in the sense of the previous deﬁnition. A ma jor task of 2-group represen tation theory is to classify the representations and intert winers up to equiv alence. Of course, one can only do this concretely after c ho osing a 2-category in which to represent a giv en 2-group. W e turn to this task next. 27 3 Measurable categories W e ha ve describ ed the passage from groups to 2-groups, and from representations to 2-representa- tions. Having presen ted these deﬁnitions in a fairly abstract form, our next ob jectiv e is to describ e a suitable target 2-category for represen tations of 2-groups. Just as ordinary groups are t ypically rep- resen ted on vector spaces, 2-groups can b e represen ted on higher analogues called ‘2-v ector spaces’. The idea of a 2-vector space can b e formalized in several w ays. In this section we describ e the general idea of 2-vector spaces, then foc us on a particular formalism: the 2-category Meas deﬁned b y Y etter [ 71 ]. 3.1 F rom v ector spaces to 2-vector spaces T o understand 2-v ector spaces, it is helpful ﬁrst to remember the naive p oin t of view on linear algebra that v ectors are lists of num b ers, op erators are matrices. Namely , any ﬁnite dimensional complex vector space is isomorphic to C N for some natural num b er N , and a linear map T : C M → C N is an N × M matrix of complex num b ers T n,m , where n ∈ { 1 , . . . , N } , m ∈ { 1 , . . . , M } . Comp osition of op erators is accomplished by matrix m ultiplication: ( U T ) k,m = N X n =1 U k,n T n,m for T : C M → C N and U : C N → C K . As a setting for doing linear algebra, we can form a category whose ob jects are just the sets C N and whose morphisms are N × M matrices. This category is smaller than the category V ect of al l ﬁnite dimensional v ector spaces, but it is e quivalent to V ect. This is why one can accomplish the same things with matrices as with abstract linear maps—an oft used fact in practical computations. Kaprano v and V o ev o dsky [ 43 ] observ ed that w e can ‘categorify’ this naive version of the category of vector spaces and deﬁne a 2-category of ‘2-vector spaces’. When we categorify a concept, we replace sets with categories. In this case, w e replace the set C of complex n umbers, along with its usual pro duct and sum op erations, b y the category V ect of complex v ector spaces, with its tensor pro duct and direct sum. Thus a ‘2-vector’ is a list, not of num b ers, but of vector spaces. Since we can deﬁne maps b et ween such lists they form, not just a set, but a category: a ‘2-vector space’. A morphism b etw een 2-vector spaces is a matrix, not of num b ers, but of v ector spaces. W e also get another lay er of structure: 2-morphisms . These are matrices of linear maps. More precisely , there is a 2-category denoted 2V ect deﬁned as follows: Ob jects The ob jects of 2V ect are the categories V ect 0 , V ect 1 , V ect 2 , V ect 3 , . . . where V ect N denotes the N -fold cartesian pro duct. Note in particular that the zero-dimensional 2-v ector space V ect 0 has just one ob ject and one morphism. 28 Morphisms Giv en 2-vector spaces V ect M and V ect N , a morphism T : V ect M → V ect N is given b y an N × M matrix of complex vector spaces T n,m , where n ∈ { 1 , . . . , N } , m ∈ { 1 , . . . , M } . Comp osition is accomplished b y matrix m ultiplication, as in ordinary linear algebra, but using tensor pro duct and direct sum: ( U T ) k,m = N M n =1 U k,n ⊗ T n,m (25) for T : V ect M → V ect N and U : V ect N → V ect K . 2-Morphisms Giv en morphisms T , T 0 : V ect M → V ect N , a 2-morphism α b et w een these: V ect M T + + T 0 3 3 V ect N α   is an N × M matrix of linear maps of vector spaces, with comp onen ts α n,m : T n,m → T 0 n,m . Suc h 2-morphisms can b e composed vertic al ly : V ect M T " " T 0 / / T 00 < < V ect N α   α 0   simply by comp osing comp onen twise the linear maps: ( α 0 · α ) n,m = α 0 n,m α n,m . (26) They can also b e comp osed horizontal ly : V ect N T + + T 0 3 3 V ect M U + + U 0 3 3 V ect K α   β   analogously with ( 25 ), by using ‘matrix multiplication’ with resp ect to tensor product and direct sum of maps: ( β ◦ α ) k,m = N M n =1 β k,n ⊗ α n,m . (27) While simple in spirit, this deﬁnition of 2V ect is problematic for a couple of reasons. First, comp osition of morphisms is not strictly asso ciative, since the direct sum and tensor pro duct of 29 v ector spaces satisfy the associative and distributive laws only up to isomorphism, and these laws are used in pro ving the associativity of matrix multiplication. So, 2V ect as just deﬁned is not a 2-category , but only a ‘weak’ 2-category , or ‘bicategory’. These are a bit more complicated, but luc kily any bicategory is equiv alen t, in a precise sense, to some 2-category . The next section gives a concrete description of a such a 2-category . (See also the work of Elgueta [ 30 ].) The ab o v e deﬁnition of 2V ect is also somewhat naive, since it categoriﬁes a naiv e version of V ect where the only vector spaces are those of the form C N . A more sophisticated approach in v olves ‘abstract’ 2-v ector spaces. One can deﬁne these axiomatically b y listing prop erties of a category that guarantee that it is equiv alent to V ect N (see Def. 2.12 in [ 56 ], and also [ 70 ]). A cruder wa y to accomplish the same eﬀect is to deﬁne an abstract 2-vector space to b e a category equiv alent to V ect N . W e take this approac h in the next section, b ecause w e do not yet know an axiomatic approac h to measurable categories, and we wish to prepare the reader for our discussion of those. 3.2 Categorical p erspective on 2-v ector spaces In this section we give a deﬁnition of 2V ect which in volv es treating it as a sub-2-category of the 2-category Cat , in whic h ob jects, morphisms, and 2-morphisms are categories, functors, and natural transformations, resp ectiv ely . This approach addresses b oth problems men tioned at the end of the last subsection. Similar ideas will be very useful in our study of measurable categories in the sections to come. In this approac h the ob jects of 2V ect are ‘linear categories’ that are ‘linearly equiv alent’ to V ect N for some N . The morphisms are ‘linear functors’ b et w een such categories, and the 2-morphisms are natural transformations. Let us deﬁne the three quoted terms. First, a linear category is a category where for eac h pair of ob jects x and y , the set of morphisms from x to y is equipp ed with the structure of a ﬁnite-dimensional complex v ector space, and comp osition of morphisms is a bilinear op eration. F or example, V ect N is a linear category . Second, a functor F : V → V 0 b et w een linear categories is a linear equiv alence if it is an equiv alence that maps morphisms to morphisms in a linear wa y . W e deﬁne a 2-vector space to b e a linear category that is linearly equiv alen t to V ect N for some N . F or example, given a category V and an equiv alence F : V → V ect N , we can use this equiv alence to equip V with the structure of a linear category; then F b ecomes a linear equiv alence and V becomes a 2-vector space. Third, note that any N × M matrix of vector spaces T n,m giv es a functor T : V ect M → V ect N as follo ws. F or an ob ject V ∈ V ect M , we deﬁne T V ∈ V ect N b y ( T V ) n = M M m =1 T n,m ⊗ V m . F or a morphism φ in V ect M , we deﬁne T φ b y: ( T φ ) n = M M m =1 1 T n,m ⊗ φ m where 1 T n,m denotes the identit y map on the vector space T n,m . It is straigh tforward to c heck that these op erations deﬁne a functor. W e call such a functor from V ect N to V ect M a matrix functor . More generally , giv en 2-vector spaces V and V 0 , w e deﬁne a linear functor from V to V 0 to b e any functor naturally isomorphic to a comp osite V F / / V ect M T / / V ect N G / / V 0 30 where T is a matrix functor and F , G are linear equiv alences. These deﬁnitions ma y seem complicated, but unlike the naiv e deﬁnitions in the previous section, they give a 2-category: Theorem 8 Ther e is a sub-2-c ate gory 2V ect of Cat wher e the obje cts ar e 2-ve ctor sp ac es, the morphisms ar e line ar functors, and the 2-morphisms ar e natur al tr ansformations. The proof of this result will serve as the pattern for a similar argumen t for measurable categories. W e break it in to a series of lemmas. It is easy to see that identit y functors and iden tity natural transformations are linear. It is obvious that natural transformations are closed under vertical and horizon tal comp osition. So, we only need to c heck that linear functors are closed under comp osition. This is Lemma 12 . Lemma 9 A c omp osite of matrix functors is natur al ly isomorphic to a matrix functor. Pro of: Suppose T : V ect M → V ect N and U : V ect N → V ect K are matrix functors. Their comp osite U T applied to an ob ject V ∈ V ect M giv es an ob ject U T V with comp onen ts ( U T V ) k = N M n =1 U k,n ⊗ M M m =1 T n,m ⊗ V m ! but this is naturally isomorphic to M M m =1 N M n =1 U k,n ⊗ T n,m ! ⊗ V m so U T is naturally isomorphic to the matrix functor deﬁned by formula ( 25 ). Lemma 10 If F : V ect N → V ect M is a line ar e quivalenc e, then N = M and F is a line ar functor. Pro of: Let e i b e the standard basis for V ect N : e i = (0 , . . . , C |{z} i th place , . . . , 0) . Since an equiv alence maps indecomp osable ob jects to indecomposable ob jects, we ha v e F ( e i ) ∼ = e σ ( i ) for some function σ . This function must b e a permutation, since F has a weak inv erse. Let ˜ F b e the matrix functor corresp onding to the p erm utation matrix asso ciated to σ . One can c heck that F is naturally isomorphic to ˜ F , hence a linear functor. Checking this mak es crucial use of the fact that F b e a line ar equiv alence: for example, taking the complex conjugate of a vector space deﬁnes an equiv alence K : V ect → V ect that is not a matrix functor. W e lea ve the details to the reader. Lemma 11 If T : V → V 0 is a line ar functor and F : V → V ect M , G : V ect N → V 0 ar e arbitrary line ar e quivalenc es, then T is natur al ly isomorphic to the c omp osite V F / / V ect M ˜ T / / V ect N G / / V 0 for some matrix functor ˜ T . 31 Pro of: Since T is linear we kno w there exist linear equiv alences F 0 : V → V ect M 0 and G 0 : V ect N 0 → V 0 suc h that T is naturally isomorphic to the comp osite V F 0 / / V ect M 0 ˜ T 0 / / V ect N 0 G 0 / / V 0 for some matrix functor ˜ T 0 . W e ha ve M 0 = M and N 0 = N by Lemma 10 . So, let ˜ T b e the composite V ect M ¯ F / / V F 0 / / V ect M ˜ T 0 / / V ect N G 0 / / V 0 ¯ G / / V ect N where ¯ F and ¯ G are weak inv erses for F and G . Since F 0 ¯ F : V ect M → V ect M and ¯ GG 0 : V ect N → V ect N are linear equiv alences, they are naturally isomorphic to matrix functors by Lemma 10 . Since ˜ T is a comp osite of functors that are naturally isomorphic to matrix functors, ˜ T itself is naturally isomorphic to a matrix functor b y Lemma 9 . Note that the comp osite V F / / V ect M ˜ T / / V ect N G / / V 0 is naturally isomorphic to T . Since F and G are linear equiv alences and ˜ T is naturally isomorphic to a matrix functor, it follo ws that T is a linear functor. Lemma 12 A c omp osite of line ar functors is line ar. Pro of: Supp ose we hav e a comp osable pair of linear functors T : V → V 0 and U : V 0 → V 00 . By deﬁnition, T is naturally isomorphic to a comp osite V F / / V ect L ˜ T / / V ect M G / / V 0 where ˜ T is a matrix functor, and F and G are linear equiv alences. By Lemma 11 , U is naturally isomorphic to a comp osite V 0 ¯ G / / V ect M ˜ U / / V ect N H / / V 00 where ˜ U is a matrix functor, ¯ G is a w eak in verse for G , and H is a linear equiv alence. The composite U T is thus naturally isomorphic to V F / / V ect L ˜ U ˜ T / / V ect N H / / V 00 Since ˜ U ˜ T is naturally isomorphic to a matrix functor b y Lemma 9 , it follows that U T is a linear functor. These results justify the naive recip e for comp osing 1-morphisms using matrix m ultiplication, namely equation ( 25 ). First, Lemma 9 shows that the comp osite of matrix functors is naturally isomorphic to their matrix pro duct as given by equation ( 25 ). More generally , given any linear functors T : V ect L → V ect M and U : V ect M → V ect N , we can choose matrix functors naturally isomorphic to these, and the comp osite U T will b e naturally isomorphic to the matrix pro duct of these matrix functors. Finally , w e can reduce the job of comp osing linear functors b et ween arbitr ary 2-v ector spaces to matrix m ultiplication by choosing linear equiv alences b etw een these 2-vector spaces and some of the form V ect N . 32 Similar results hold for natural transformations. Any N × M matrix of linear op erators α n,m : T n,m → T 0 n,m determines a natural transformation betw een the matrix functors T , T 0 : V ect M → V ect N . This natural transformation gives, for each ob ject V ∈ V ect M , a morphism α V : T V → T 0 V with comp o- nen ts ( α V ) n : M M m =1 T n,m ⊗ V m → M M m =1 T 0 n,m ⊗ V m giv en by ( α V ) n = M M m =1 α n,m ⊗ 1 V m . W e call a natural transformation of this sort a matrix natural transformation . How ev er: Theorem 13 Any natur al tr ansformation b etwe en matrix functors is a matrix natur al tr ansforma- tion. Pro of: Giv en matrix functors T , T 0 : V ect M → V ect N , a natural transformation α : T ⇒ T 0 giv es for each basis ob ject e m ∈ V ect M a morphism in V ect N with comp onen ts ( α e m ) n : T n,m ⊗ C → T 0 n,m ⊗ C . Using the natural isomorphism b etw een a vector space and that vector space tensored with C , these can b e rein terpreted as op erators α n,m : T n,m → T 0 n,m . These op erators deﬁne a matrix natural transformation from T to T 0 , and one can chec k using naturalit y that this equals α . One can c heck that vertical comp osition of matrix natural transformations is giv en by th e matrix form ula of the previous section, namely formula ( 26 ). Similarly , the horizontal composite of matrix natural transformations is ‘essentially’ given b y form ula ( 27 ). So, while these matrix formulas are a bit naive, they are useful to ols when prop erly in terpreted. 3.3 F rom 2-v ector spaces to measurable categories In the previous sections, we sa w the 2-category 2V ect of Kapranov–V o ev o dsky 2-vector spaces as a categoriﬁcation of V ect, the category of ﬁnite-dimensional v ector spaces. While one can certainly study representations of 2-groups in 2V ect [ 18 , 31 ], our goal is to describ e represen tations of 2-groups in something more akin to inﬁnite-dimensional 2- Hilb ert spaces. Suc h ob jects should b e roughly lik e ‘Hilb X ’, where Hilb is the category of Hilb ert spaces and X ma y now b e an inﬁnite index set. In fact, for our purposes, X should hav e at least the structure of a measurable space. This allo ws one to categorify Hilb ert spaces L 2 ( X, µ ) in such a w ay that measurable functions are replaced by ‘measurable ﬁelds of Hilb ert spaces’, and in tegrals of functions are replaced by ‘direct in tegrals’ of suc h ﬁelds. W e can construct a c hart like the one in the introduction, outlining the basic strategy for cate- goriﬁcation: 33 ordinary higher L 2 spaces L 2 spaces C Hilb + ⊕ × ⊗ 0 { 0 } 1 C measurable functions measurable ﬁelds of Hilb ert spaces R (in tegral) R ⊕ (direct integral) V arious alternatives spring from this basic idea. In this section and the following one, we pro vide a concrete description of one possible categoriﬁcation of L 2 spaces: ‘measurable categories’ as deﬁned b y Y etter [ 71 ], whic h provide a foundation for earlier w ork by Crane, Shepp eard, and Y etter [ 25 , 26 ]. Measurable categories do not provide a full-ﬂedged categoriﬁcation of the concept of Hilb ert space, so they do not deserv e to b e called ‘2-Hilb ert spaces’. Indeed, ﬁnite-dimensional 2-Hilb ert spaces are well understo o d [ 3 , 17 ], and they hav e a bit more structure than measurable categories with a ﬁnite basis of ob jects. Namely , w e can tak e the ‘inner pro duct’ of tw o ob jects in suc h a 2-Hilb ert space and get a Hilb ert space. W e exp ect something similar in an inﬁnite-dimensional 2-Hilb ert space, and it happ ens in many interesting examples, but the deﬁnition of measurable category lacks this feature. So, our w ork here can b e seen as a stepping-stone tow ards a theory of unitary representations of 2-groups on inﬁnite-dimensional 2-Hilb ert spaces. See Section 5 for a bit more on this issue. The goal of this section is to construct a 2-category of measurable categories, denoted Meas . This requires some work, in part b ecause we do not ha ve an intrinsic characterization of measurable categories. W e also give concrete practical formulas for comp osing morphisms and 2-morphisms in Meas . This will equip the reader with the to ols necessary for calculations in the representation theory dev elop ed in Section 4 . But ﬁrst we need some preliminaries in analysis. F or basic results and standing assumptions the reader ma y also turn to App endix A . 3.3.1 Measurable ﬁelds and direct integrals W e present here some essen tial analytic tools: measurable ﬁelds of Hilb ert spaces and op erators, their measure-classes and direct integrals, and measurable families of measures. W e ha ve explained the categorical motiv ation for generalizing functions on a measurable space to ‘ ﬁelds of Hilb ert sp ac es ’ on a measurable space. But one cannot simply assign an arbitrary Hilb ert space to each p oin t in a measurable space X and exp ect to p erform op erations that make goo d analytic sense. F ortunately , ‘measurable ﬁelds’ of Hilb ert spaces hav e b een studied in detail—see esp ecially the b ook b y Dixmier [ 28 ]. Algebraists ma y view these as represen tations of ab elian von Neumann algebras on Hilb ert spaces, as explained by Dixmier and also Arveson [ 2 , Chap. 2.2]. Geometers may instead prefer to view them as ‘measurable bundles of Hilb ert spaces’, following the treatment of Mack ey [ 51 ]. Measurable ﬁelds of Hilb ert spaces hav e also b een studied from a category-theoretic p erspective b y Y etter [ 71 ]. It will b e conv enien t to imp ose some simplifying assumptions. Our measurable spaces will all b e ‘standard Borel spaces’ and our measures will alw ays be σ -ﬁnite and p ositiv e. Standard Borel spaces can b e c haracterized in sev eral w ays: Lemma 14 L et ( X, B ) b e a me asur able sp ac e, i.e. a set X e quipp e d with a σ -algebr a of subsets B . Then the fol lowing ar e e quivalent: 34 1. X c an b e given the structur e of a sep ar able c omplete metric sp ac e in such a way that B is the σ -algebr a of Bor el subsets of X . 2. X c an b e given the structur e of a se c ond-c ountable, lo c al ly c omp act Hausdorﬀ sp ac e in such a way that B is the σ -algebr a of Bor el subsets of X . 3. ( X , B ) is isomorphic to one of the fol lowing: • a ﬁnite set with its σ -algebr a of al l subsets; • a c ountably inﬁnite set with its σ -algebr a of al l subsets; • [0 , 1] with its σ -algebr a of Bor el subsets. A me asur able sp ac e satisfying any of these e quivalent c onditions is c al le d a standard Borel space . Pro of: It is clear that 3) implies 2). T o see that 2) implies 1), we need to chec k that every second-coun table lo cally compact Hausdorﬀ space X can b e made into a separable complete metric space. F or this, note that the one-p oin t compactiﬁcation of X , say X + , is a second-countable compact Hausdorﬀ space, which admits a metric b y Urysohn’s metrization theorem. Since X + is compact this metric is complete. Finally , any op en subset of separable complete metric space can b e giv en a new metric giving it the same top ology , where the new metric is separable and complete [ 21 , Chap. IX, § 6.1, Prop. 2]. Finally , that 1) implies 3) follo ws from tw o classic results of Kurato wski. Namely: t wo standard Borel spaces (deﬁned using condition 1) are isomorphic if and only if they ha ve the same cardinalit y , and any uncoun table standard Borel space has the cardinality of the contin uum [ 57 , Chap. I, Thms. 2.8 and 2.13]. The following deﬁnitions will b e handy: Deﬁnition 15 By a measurable space we me an a standar d Bor el sp ac e ( X , B ) . We c al l sets in B measurable . Given sp ac es X and Y , a map f : X → Y is measurable if f − 1 ( S ) is me asur able whenever S ⊆ Y is me asur able. Deﬁnition 16 By a measure on a me asur able sp ac e ( X , B ) we me an a σ -ﬁnite me asur e, i.e. a c ountably additive map µ : B → [0 , + ∞ ] for which X is a c ountable union of S i ∈ B with µ ( S i ) < ∞ . A k ey idea is that a measurable ﬁeld of Hilbert spaces should kno w what its ‘measurable sections’ are. That is, there should b e preferred wa ys of selecting one vector from the Hilb ert space at each p oin t; these preferred sections should satisfy some prop erties, given b elow, to guaran tee reasonable measure-theoretic b eha vior: Deﬁnition 17 L et X b e a me asur able sp ac e. A measurable ﬁeld of Hilbert spaces H on X is an assignment of a Hilb ert sp ac e H x to e ach x ∈ X , to gether with a subsp ac e M H ⊆ Q x H x c al le d the measurable sections of H , satisfying the pr op erties: • ∀ ξ ∈ M H , the function x 7→ k ξ x k H x is me asur able. • F or any η ∈ Q x H x such that x 7→ h η x , ξ x i H x is me asur able for al l ξ ∈ M H , we have η ∈ M H . • Ther e is a se quenc e ξ i ∈ M H such that { ( ξ i ) x } ∞ i =1 is dense in H x for al l x ∈ X . Deﬁnition 18 L et H and H 0 b e me asur able ﬁelds of Hilb ert sp ac es on X . A measurable ﬁeld of b ounded linear op erators φ : H → K on X is an X -indexe d family of b ounde d op er ators φ x : H x → H 0 x such that ξ ∈ M H implies φ ( ξ ) ∈ M H 0 , wher e φ ( ξ ) x := φ x ( ξ x ) . 35 Giv en a p ositiv e measure µ on X , measurable ﬁelds can b e integrated. The integral of a function giv es an element of C ; the integral of a ﬁeld of Hilb ert spaces gives an ob ject of Hilb. F ormally , we ha ve the follo wing deﬁnition: Deﬁnition 19 L et H b e a me asur able ﬁeld of Hilb ert sp ac es on a me asur able sp ac e X ; let h· , ·i x denote the inner pr o duct in H x , and k · k x the induc e d norm. The direct integral Z ⊕ X d µ ( x ) H x of H with r esp e ct to the me asur e µ is the Hilb ert sp ac e of al l µ -a.e. e quivalenc e classes of me asur able L 2 se ctions of H , that is, se ctions ψ ∈ M H such that Z X d µ ( x ) k ψ x k 2 x < ∞ , with inner pr o duct given by h ψ , ψ 0 i = Z X d µ ( x ) h ψ x , ψ 0 x i x . for ψ , ψ 0 ∈ R ⊕ X d µ H . That the inner pro duct is well deﬁned for L 2 sections follows b y p olarization. Of course, for R ⊕ X d µ H to b e a Hilbert space as claimed in the deﬁnition, one must also c heck that it is Cauch y-complete with resp ect to the induced norm. This is indeed the case [ 28 , P art I I Ch. 1 Prop. 5]. W e often denote an element of the direct integral of H by Z ⊕ X d µ ( x ) ψ x where ψ x ∈ H x is deﬁned up to µ - a.e. equalit y . W e also hav e a corresp onding notion of direct integral for ﬁelds of linear op erators: Deﬁnition 20 Supp ose φ : H → H 0 is a µ -essential ly b ounde d me asur able ﬁeld of line ar op er ators on X . The direct in tegral of φ is the line ar op er ator acting p ointwise on se ctions: Z ⊕ X d µ ( x ) φ x : Z ⊕ X d µ ( x ) H x → Z ⊕ X d µ ( x ) H 0 x R ⊕ X d µ ( x ) ψ x 7→ R ⊕ X d µ ( x ) φ x ( ψ x ) Note requiring that the ﬁeld b e µ -essen tially b ounded— i.e. that the operator norms k φ x k hav e a common b ound for µ -almost every x —guarantees that the image lies in the direct integral of H 0 , since Z X d µ ( x ) k φ x ( ψ x ) k 2 H 0 x ≤ ess sup x 0 k φ x 0 k 2 Z d µ ( x ) k ψ x k 2 H x < ∞ . Notice that direct integrals indeed generalize direct sums: in the case where X is a ﬁnite set and µ is counting measure, direct integrals of Hilb ert spaces and op erators simply reduce to direct sums. In ordinary in tegration theory , one typically identiﬁes functions that coincide almost everywhere with resp ect to the relev an t measure. This is also useful for the measurable ﬁelds deﬁned ab o v e, for the same reasons. T o make ‘ a.e. -equiv alence of measurable ﬁelds’ precise, we ﬁrst need a notion of ‘restriction’. 36 If A ⊆ X is a measurable set, any measurable ﬁeld H of Hilbert spaces on X induces a ﬁeld H | A on A , called the restriction of H to A . The restricted ﬁeld is constructed in the ob vious wa y: w e let ( H| A ) x = H x for each x ∈ A , and deﬁne the measurable sections to b e the restrictions of measurable sections on X : M H| A = { ψ | A : ψ ∈ M H } . It is straigh tforward to chec k that ( H| A , M H| A ) indeed deﬁnes a measurable ﬁeld. 1 Similarly , if φ : H → K is a ﬁeld of linear op erators, its restriction to a measurable subset A ⊆ X is the obvious A -indexed family of op erators φ | A : H| A → K| A giv en b y ( φ | A ) x = φ x for each x in A . It is easy to c heck that ξ ∈ M H| A implies φ ( ξ ) ∈ M K| A , so φ | A deﬁnes a measurable ﬁeld on A . W e say t wo measurable ﬁelds of Hilb ert spaces on X are µ -almost everwhere equiv alen t if they hav e equal restrictions to some measurable A ⊆ X with µ ( X − A ) = 0. This is obviously an equiv alence relation, and an equiv alence class is called a µ -class of measurable ﬁelds . Two ﬁelds in the same µ -class hav e canonically isomorphic direct in tegrals, so the direct in tegral of a µ -class mak es sense. Equiv alence classes of measurable ﬁelds of linear operators work similarly , but with one subtlet y . First supp ose we hav e tw o measurable ﬁelds of Hilb ert spaces T x and U x on X , and a measurable ﬁeld of operators α x : T x → U x . Given a measure µ , one can clearly identify tw o such α if they coincide outside a set of µ -measure 0, thus deﬁning a notion of µ -class of ﬁelds of op erators from T to U . So far T and U are ﬁxed, but now we wish to take equiv alence classes of them as well. In fact, it is often useful to pass to t -classes of T and u -classes of U , where t and u are in general diﬀer ent measures on X . W e then ask what sort of measure µ m ust b e for the µ -class of α to pass to a well deﬁned map [ α x ] µ : [ T x ] t → [ U x ] u , where brac kets denote the relev an t classes. This w orks if and only if eac h t -null set and eac h u -null set is also µ -null. Thus w e require µ  t and µ  u, (28) where ‘  ’ denotes absolute contin uit y of measures. Given a measure µ satisfying these prop erties, it makes sense to sp eak of the µ -class of ﬁelds of op erators from a t -class of ﬁelds of Hilbert spaces to a u -class of ﬁelds of Hilb ert spaces. In practice , one would lik e to pick µ to b e maximal with respe ct to the required prop erties ( 28 ), so that µ - a.e. equiv alence is the transitive closure of u - a.e. and t - a.e. equiv alences. In fact, if t and u are b oth σ -ﬁnite measures, there is a natural choice for whic h measure µ to tak e in the ab o ve construction: the ‘geometric mean measure’ √ tu of the measures t and u . The notion of geometric mean measure is discussed in App endix A.2 , but the basic idea is as follows. If t is absolutely contin uous with resp ect to u , denoted t  u , then we hav e the Radon–Nikodym deriv ativ e d t d u . More generally , ev en when t is not absolutely contin uous with resp ect to u , w e will use the notation d t d u := d t u d u where t u is the absolutely contin uous part of the Lebesgue decomposition of t with respect to u . An 1 The ﬁrst and third axioms in the deﬁnition are obvious. T o chec k the second, pic k η ∈ Q x ∈ A H x such that x 7→ h η x , ξ x i is a measurable function on A for ev ery ξ ∈ M| H| A . Extend η to ˜ η ∈ Q x ∈ X H x by setting ˜ η x =  η x x ∈ A 0 x 6∈ A. 37 imp ortan t fact, prov ed in App endix A.2 , is that r d t d u d u = r d u d t d t, so we can deﬁne the geometric mean measure , denoted √ d t d u or simply √ tu , using either of these expressions. Ev ery set of t -measure or u -measure z ero also has √ tu -measure zero. That is, √ tu  t and √ tu  u. In fact, every √ tu -n ull set is the union of a t -null set and a u -null set, as we show in App endix A.2 . This means √ tu is a measure that is maximal with resp ect to ( 28 ). Recall that we are assuming our measures are σ -ﬁnite. Using this, one can sho w that d t d u d u d t = 1 √ tu - a.e. (29) This rule, obvious when the t wo measures are equiv alen t, is pro ved in App endix A.2 . W e shall need one more t yp e of ‘ﬁeld’, whic h may be though t of as ‘measurable ﬁelds of me asur es ’. In general, these inv olve tw o measure spaces: they are certain families µ y of measures on a measurable space X , indexed by elements of a measurable space Y . W e ﬁrst introduce the notion of ﬁb ered measure distribution [ 71 ]: Deﬁnition 21 Supp ose X and Y ar e me asur able sp ac es and every one-p oint set of Y is me asur able. Then a Y -ﬁb ered measure distribution on Y × X is a Y -indexe d family of me asur es ¯ µ y on Y × X satisfying the pr op erties: • ¯ µ y is supp orte d on { y } × X : that is, ¯ µ y (( Y − { y } ) × X ) = 0 • F or every me asur able A ⊆ Y × X , the function y 7→ ¯ µ y ( A ) is me asur able • The family is uniformly ﬁnite: that is, ther e exists a c onstant M such that for al l y ∈ Y , ¯ µ y ( X ) < M . An y ﬁb ered measure distributions gives rise to a Y -indexed family of measures on X : Deﬁnition 22 Given me asur able sp ac es X and Y , µ y is a Y -indexed measurable family of measures on X if it is induc e d by a Y -ﬁb er e d me asur e distribution ¯ µ y on Y × X ; that is, if µ y ( A ) = ¯ µ y ( Y × A ) for every me asur able A ⊆ X . Notice that, if ¯ µ y is the ﬁb ered measure distribution associated to the measurable family µ y , we ha ve ¯ µ y = δ y ⊗ µ y (30) as measures on Y × X , where for each y ∈ Y , δ y is the Dirac measure concentrated at y . By itself, a ﬁb ered measure distribution ¯ µ y on Y × X is not a measure on Y × X . How ev er, taken together with a suitable measure ν on Y , it may yield a measure λ on Y × X : λ = Z Y d ν ( δ y ⊗ µ y ) (31) 38 Because this measure λ is obtained from µ y b y inte gr ation with resp ect to ν , the measurable family µ y is also called the disintegration of λ with resp ect to ν . It is often the dis in tegration problem one is interested in: given a measure λ on a product space and a measure ν on one of the factors, can λ b e written as an in tegral of some measurable family of measures on the other factor, as in ( 31 ). Conditions for the disintegration problem to hav e a solution are given by the ‘disintegration theorem’: Theorem 23 (Disintegration Theorem) Supp ose X and Y ar e me asur able sp ac es. Then a me a- sur e λ on Y × X has a disinte gr ation µ y with r esp e ct to the me asur e ν on Y if and only if ν ( U ) = 0 implies λ ( U × X ) = 0 for every me asur able U ⊆ Y . When this is the c ase, the me asur es µ y ar e determine d uniquely for ν -almost every y . Pro of: Graf and Mauldin [ 39 ] state a theorem due to Maharam [ 53 ] that easily implies a stronger v ersion of this result: namely , that the conclusions hold whenev er X and Y are Lusin spaces. Recall that a topological space space homeomorphic to separable complete metric space is called a P olish space , while more generally a Lusin space is a top ological space that is the image of a Polish space under a contin uous bijection. By Lemma 14 , ev ery measurable space we consider — i.e., ev ery standard Borel space—is isomorphic to some P olish space equipp ed with its σ -algebra of Borel sets. 3.3.2 The 2-category of measurable categories: Meas W e are no w in a p osition to giv e a deﬁnition of the 2-category Meas introduced in the work of Crane and Y etter [ 26 , 71 ]. The aim of this section is essentially practical: w e giv e concrete descriptions of the ob jects, morphisms, and 2-morphisms of Meas , and formulae for the comp osition laws. These form ulae will be analogous to those presented in the ﬁnite-dimensional case in Section 3.1 , which the current section parallels. Before diving into the technical details, let us sketc h the basic idea b ehind the 2-category Meas : • The ob jects of Meas are ‘measurable categories’, which are categories somewhat analogous to Hilb ert spaces. The most imp ortant sort of example is the category H X whose ob jects are measurable ﬁelds of Hilb ert spaces on the measurable space X , and whose morphisms are measurable ﬁelds of b ounded op erators. If X is a ﬁnite set with n elements, then H X ∼ = Hilb n . So, H X generalizes Hilb n to situations where X is a measurable space instead of a ﬁnite set. • The morphisms of Meas are ‘measurable functors’. The most imp ortant examples are ‘matrix functors’ T : H X → H Y . Suc h a functor is constructed using a ﬁeld of Hilbert spaces on X × Y , whic h we also denote b y T . When X and Y are ﬁnite sets, suc h ﬁeld is simply a matrix of Hilb ert spaces. But in general, to construct a matrix functor T : H X → H Y w e also need a Y -indexed measure on X . • The 2-morphisms of Meas are ‘measurable natural transformations’. The most imp ortan t ex- amples are ‘matrix natural transformations’ α : T → T 0 b et w een matrix functors T , T 0 : H X → H Y . Suc h a natural transformation is constructed using a uniformly b ounded ﬁeld of linear op erators α y ,x : T y ,x → T 0 y ,x . Here we ha ve sk etchily described the most imp ortan t ob jects, morphisms and 2-morphisms in Meas . How ever, following our treatment of 2V ect in Section 3.2 , w e need to mak e Meas bigger to obtain a 2-category instead of a bicategory . T o do this, w e include as ob jects of Meas certain categories that are e quivalent to categories of the form H X , and include as morphisms certain functors that are natur al ly isomorphic to matrix functors. 39 Ob jects Giv en a measurable space X , there is a category H X with: • measurable ﬁelds of Hilb ert spaces on X as ob jects; • b ounded measurable ﬁelds of linear op erators on X as morphisms. Ob jects of the 2-category Meas are ‘measurable categories’—that is, ‘ C ∗ -categories’ that are ‘ C ∗ - equiv alen t’ to H X for some X . Let us make this precise: Deﬁnition 24 A Banac h category is a c ate gory C enriche d over Banach sp ac es, me aning that for any p air of obje cts x, y ∈ C , the set of morphisms fr om x to y is e quipp e d with the structur e of a Banach sp ac e, c omp osition is biline ar, and k f g k ≤ k f kk g k for every p air of c omp osable morphisms f , g in C . Deﬁnition 25 A Banac h ∗ -category is a Banach c ate gory in which e ach morphism f : x → y has an asso ciate d morphism f ∗ : y → x , such that: • e ach map hom( x, y ) → hom( y , x ) given by f 7→ f ∗ is c onjugate line ar; • ( g f ) ∗ = f ∗ g ∗ , 1 ∗ x = 1 x , and f ∗∗ = f , for every obje ct x and p air of c omp osable morphisms f , g ; • for any morphism f : x → y , ther e exists a morphism g : x → x such that f ∗ f = g ∗ g ; • f ∗ f = 0 if and only if f = 0 . Deﬁnition 26 A C ∗ -category is a Banach ∗ -c ate gory such that for e ach morphism f : x → y , k f ∗ f k = k f k 2 . Note that for each ob ject x in a C ∗ -category , its endomorphisms form a C ∗ -algebra. Note als o that for any measurable space X , H X is a C ∗ -category , where the norm of any b ounded measurable ﬁeld of op erators φ : H → K is k φ k = sup x ∈ X k φ x k and we deﬁne the ∗ op eration p oin t wise: ( φ ∗ ) x = ( φ x ) ∗ where the right-hand side is the Hilb ert space adjoint of the op erator φ x . Deﬁnition 27 A functor F : C → C 0 b etwe en C ∗ -c ate gories is a C ∗ -functor if it maps morphisms to morphisms in a line ar way, and satisﬁes F ( f ∗ ) = F ( f ) ∗ for every morphism f in C . 40 Using the fact that a ∗ -homomorphism betw een unital C ∗ -algebras is automatically norm-decreasing, w e can sho w that an y C ∗ -functor satisﬁes k F ( f ) k ≤ k f k . Deﬁnition 28 Given C ∗ -c ate gories C and C 0 , a natur al tr ansformation α : F ⇒ F 0 b etwe en functors F , F 0 : C → C 0 is b ounded if for some c onstant K we have k α x k ≤ K for al l x ∈ C . If ther e is a b ounde d natur al isomorphism b etwe en functors b etwe en C ∗ -c ate gories, we say they ar e b oundedly naturally isomorphic . Deﬁnition 29 A C ∗ -functor F : C → C 0 is a C ∗ -equiv alence if ther e is a C ∗ -functor ¯ F : C 0 → C such that ¯ F F and F ¯ F ar e b ounde d ly natur al ly isomorphic to identity functors. Deﬁnition 30 A measurable category is a C ∗ -c ate gory that is C ∗ -e quivalent to H X for some me asur able sp ac e X . Morphisms The morphisms of Meas are ‘measurable functors’. The most important measurable functors are the ‘matrix functors’, so we b egin with these. Given t wo ob jects H X and H Y in Meas , w e can construct a functor H X T ,t / / H Y from the following data: • a uniformly ﬁnite Y -indexed measurable family t y of measures on X , • a t -class of measurable ﬁelds of Hilb ert spaces T on Y × X , such that t is concentrated on the supp ort of T ; that is, for each y ∈ Y , t y ( { x ∈ X : T y ,x = 0 } ) = 0 . Here by t -class w e mean a t y -class for each y , as deﬁned in the previous section. F or brevit y , we will sometimes denote the functor constructed from these data simply by T . This functor maps any ob ject H ∈ H X —a measurable ﬁeld of Hilbert spaces on X —to the ob ject T H ∈ H Y giv en by ( T H ) y = Z ⊕ X d t y T y ,x ⊗ H x . Similarly , it maps any morphism φ : H → H 0 to the morphism T φ : T H → T H 0 giv en by the direct in tegral of op erators ( T φ ) y = Z ⊕ X d t y 1 T y,x ⊗ φ x where 1 T y,x denotes the identit y op erator on T y ,x . Note that T is a C ∗ -functor. Deﬁnition 31 Given me asur able sp ac es X and Y , a functor T : H X → H Y of the ab ove sort is c al le d a matrix functor . Starting from matrix functors, we can deﬁne measurable functors in general: 41 Deﬁnition 32 Given obje cts H , H 0 ∈ Meas , a measurable functor fr om H to H 0 is a C ∗ -functor that is b ounde d ly natur al ly isomorphic to a c omp osite H F / / H X T / / H Y G / / H 0 wher e T is a matrix functor and the ﬁrst and last functors ar e C ∗ -e quivalenc es. In Section 3.3.3 we use results of Y etter to show that the composite of measurable functors is measurable. A k ey step is showing that the comp osite of t wo matrix functors: H X T ,t / / H Y U,u / / H Z is b oundedly naturally isomorphic to a matrix functor H X U T ,ut / / H Z . Let us sk etch how this step go es, since we will need explicit formulas for U T and ut . Picking an y ob ject H ∈ H X , we hav e ( U T H ) z = Z ⊕ Y d u z U z ,y ⊗ ( T H ) y = Z ⊕ Y d u z U z ,y ⊗  Z ⊕ X d t y T y ,x ⊗ H x  T o express this in terms of a matrix functor, we will write it as direct integral o ver X with resp ect to a Z -indexed family of measures on X denoted ut , deﬁned by: ( ut ) z = Z Y d u z ( y ) t y . (32) T o do this we use the disintegration theorem, Thm. 23 , to obtain a ﬁeld of measures k z ,x suc h that Z X d( ut ) z ( x ) ( k z ,x ⊗ δ x ) = Z Y d u z ( y ) ( δ y ⊗ t y ) . (33) as measures on Y × X . That is, k z ,x and t y are, resp ectively , the X - and Y -disintegrations of the same measure on X × Y , with respect to the measures ( ut ) z on X and u z on Y . The measures k y ,x are determined uniquely for all z and ( ut ) z -almost every x . With these deﬁnitions, it follo ws that there is a b ounded natural isomorphism ( U T H ) z ∼ = Z ⊕ X d( ut ) z  Z ⊕ Y d k z ,x U z ,y ⊗ T y ,x  ⊗ H x (34) = Z ⊕ X d( ut ) z ( U T ) z ,x ⊗ H x (35) where ( U T ) z ,x = Z ⊕ Y d k z ,x ( y ) U z ,y ⊗ T y ,z , (36) This formula for U T is analogous to ( 25 ). W e refer to Y etter [ 71 ] for pro ofs that the family of measures ut and the ﬁeld of Hilb ert spaces U T are measurable, and hence deﬁne a matrix functor. 42 It is often conv enient to use an alternative form of ( 33 ) in terms of integrals of functions: for ev ery measurable function F on Y × X and for all z ∈ Z , Z X d( ut ) z ( x ) Z Y d k z ,x ( y ) F ( y , x ) = Z Y d u z ( y ) Z X d t y ( x ) F ( y, x ) . (37) This can b e thought of as a sort of ‘F ubini theorem’, since it lets us change the order of integration, but here the measure on one factor in the pro duct is parameterized b y the other factor. Besides comp osition of morphisms in Meas , w e also need identit y morphisms. Given an ob ject H X , to show its identit y functor 1 X : H X → H X is a matrix functor we need an X -indexed family of measures on X , and a ﬁeld of Hilb ert spaces on X × X . Denote the co ordinates of X × X b y ( x 0 , x ). The family of measures assigns to each x 0 ∈ X the unit Dirac measure concen trated at the p oin t x 0 : δ x 0 ( A ) = ( 1 if x 0 ∈ A 0 otherwise for every measurable set A ⊆ X The ﬁeld of Hilb ert spaces on X × X is the constant ﬁeld ( 1 X ) x 0 ,x = C . It is simple to chec k that this acts as b oth left and right identit y for comp osition. Let us chec k that it is a right identit y b y forming this comp osite: H X 1 X ,δ / / H X T ,t / / H Y One can chec k that the comp osite measure is: ( tδ ) y = Z ⊕ X d t y ( x 0 ) δ 0 x = t y , and hence, using ( 37 ), k y ,x = δ x . W e can then calculate the ﬁeld of op erators: ( T 1 X ) y ,x ∼ = Z ⊕ X d δ x ( x 0 ) T y ,x 0 ⊗ C = T y ,x . 2-Morphisms The 2-morphisms in Meas are ‘measurable natural transformations’. The most imp ortan t of these are the ‘matrix natural transformations’. Giv en tw o matrix functors ( T , t ) and ( T 0 , t 0 ), w e can construct a natural transformation b etw een them from a √ tt 0 -class of bounded measurable ﬁelds of linear op erators α y ,x : T y ,x − → T 0 y ,x on Y × X . Here by a √ tt 0 -class , w e mean a p t y t 0 y -class for eac h y , where the p t y t 0 y is the geometric mean of the measures t y and t 0 y . By b ounded , we mean α y ,x ha ve a common b ound for all y and p t y t 0 y -almost every x . W e denote the natural transformation constructed from these data simply by α . This natu- ral transformation assigns to each ob ject H ∈ H X the morphism α H : T H → T 0 H in H Y with comp onen ts: ( α H ) y : Z ⊕ X d t y T y ,x ⊗ H x → Z ⊕ X d t 0 y T 0 y ,x ⊗ H x R ⊕ X d t y ψ y ,x 7→ R ⊕ X d t 0 y [ ˜ α y ,x ⊗ 1 H x ]( ψ y ,x ) 43 where ˜ α is the rescaled ﬁeld ˜ α = s d t y d t 0 y α. (38) T o chec k that α H is well deﬁned, pic k ψ ∈ T H and compute Z X d t 0 y k [ ˜ α y ,x ⊗ 1 H x ]( ψ y ,x ) k 2 = Z X d t c y k [ α y ,x ⊗ 1 H x ]( ψ y ,x ) k 2 ≤ ess sup x 0 k α y ,x 0 k 2 Z ⊕ X d t y k ψ y ,x k 2 < ∞ where t c y is the absolutely con tinuous part of the Leb esgue decomp osition of t y with resp ect to t 0 y ; note that, since t c y is equiv alent to p t y t 0 y , the ﬁeld α is essentially b ounded with resp ect to t c y . This inequalit y shows that the image ( α H ) y ( ψ ) b elongs to ( T 0 H ) y , and that α H is a ﬁeld of b ounded linear maps, as required. Note also that the direct integral deﬁning the image do es not dep end on the chosen representativ e of α . T o chec k that α is natural it suﬃces to choose a morphism φ : H → H 0 in H X and show that the naturalit y square T H T φ / / α H   T H 0 α H 0   T 0 H T 0 φ / / T 0 H 0 comm utes; that is, α H 0 ( T φ ) = ( T 0 φ ) α H . T o chec k this, apply the op erator on the left to ψ ∈ T H and calculate: ( α H 0 ) y ( T φ ) y ( ψ y ) = ( α H 0 ) y  Z ⊕ X d t y ( x )[ 1 T y,x ⊗ φ x ]( ψ y )  = Z ⊕ X d t 0 y ( x )[ ˜ α y ,x ⊗ 1 H 0 x ][ 1 T y,x ⊗ φ x ]( ψ y ) = Z ⊕ X d t 0 y ( x )[ 1 T 0 y,x ⊗ φ x ][ α y ,x ⊗ 1 H x ]( ψ y ) = ( T 0 φ ) y ( α H ) y ( ψ y ) . Deﬁnition 33 Given me asur able sp ac es X and Y and matrix functors T , T 0 : H X → H Y , a natur al tr ansformation α : T ⇒ T 0 of the ab ove sort is c al le d a matrix natural transformation . Ho wev er, in analogy to Thm. 13 , we ha ve: Theorem 34 Given me asur able sp ac es X and Y and matrix functors T , T 0 : H X → H Y , every b ounde d natur al tr ansformation α : T ⇒ T 0 is a matrix natur al tr ansformation, and c onversely. Pro of: The conv erse is easy . So, supp ose T , T 0 : H X → H Y are matrix natural transformations and α : T ⇒ T 0 is a b ounded natural transformation. Denote by t and t 0 the families of measures of the tw o matrix functors. W e will show that α is a matrix natural transformation in three steps. 44 W e b egin b y assuming that for each y ∈ Y , t y = t 0 y ; w e then extend the result to the case where the measures are only equiv alent t y ∼ t 0 y ; then ﬁnally we treat the general case. Assume ﬁrst t = t 0 . Let J b e the measurable ﬁeld of Hilb ert spaces on X with J x = C for all x ∈ X. Then T J and T 0 J are measurable ﬁelds of Hilb ert spaces on Y with canonical isomorphisms ( T J ) y ∼ = Z ⊕ X d t y T x,y , ( T 0 J ) y ∼ = Z ⊕ X d t y T 0 x,y (39) Using these, we ma y think of α J as a measurable ﬁeld of op erators on Y with ( α J ) y : Z ⊕ X d t y T x,y → Z ⊕ X d t y T 0 x,y . W e now show that for an y ﬁxed y ∈ Y there is a b ounded measurable ﬁeld of op erators on X , sa y α y ,x : T x,y → T 0 x,y , with the prop ert y that ( α J ) y : Z ⊕ X d t y ψ y ,x 7→ Z ⊕ X d t y α y ,x ( ψ y ,x ) (40) for an y measurable ﬁeld of vectors ψ y ,x ∈ T y ,x . F or this, note that any measurable bounded function f on X deﬁnes a morphism f : J → J in H X , mapping a vector ﬁeld ψ x to f ( x ) ψ x . The functors T and T 0 map f to the some morphisms T f : T J → T J and T 0 f : T 0 J → T 0 J in H Y . Using the canonical isomorphisms ( 39 ), w e may think of T f as a measurable ﬁeld of multi- plication operators on Y with ( T f ) y : R ⊕ X d t y T y ,x → Z ⊕ X d t y T y ,x R ⊕ X d t y ψ y ,x 7→ R ⊕ X d t y f ( x ) ψ y ,x and similarly for T 0 f . The naturalit y of α implies that the square T J T f / / α J   T J α J   T 0 J T 0 f / / T 0 J comm utes; unrav eling this condition it follows that, for each y ∈ Y , ( α J ) y ( T f ) y = ( T 0 f ) y ( α J ) y . No w we use this result: 45 Lemma 35 Supp ose X is a me asur able sp ac e and µ is a me asur e on X Supp ose T and T 0 ar e me asur able ﬁelds of Hilb ert sp ac es on X and α : Z ⊕ X d µ T x → Z ⊕ X d µ T 0 x is a b ounde d line ar op er ator such that α T f = T 0 f β for every f ∈ L ∞ ( X, µ ) , wher e T f and T 0 f ar e multiplic ation op er ators as ab ove. Then ther e exists a uniformly b ounde d me asur able ﬁeld of op er ators α x : T x → T 0 x such that α : Z ⊕ X d µ ψ x 7→ Z ⊕ X d µ α x ( ψ x ) . Pr o of: This can b e found in Dixmier’s b ook [ 28 , Part I I Chap. 2 Thm. 1].  It follo ws that for any y ∈ Y there is a uniformly b ounded measurable ﬁeld of op erators on X , sa y α y ,x : T x,y → T 0 x,y , satisfying Eq. 40 . Next note that as we let y v ary , α y ,x deﬁnes a uniformly b ounded measurable ﬁeld of op erators on X × Y . The uniform b oundedness follows from the fact that for all y , ess sup x k α y ,x k = k ( α J ) y k ≤ K since α is a b ounded natural transformation. The measurability follo ws from the fact that ( α J ) y is a measurable ﬁeld of b ounded op erators on Y . T o conclude, w e use this measurable ﬁeld α y ,x to pro v e that α is a matrix natural transformation. F or this, we m ust show that for any measurable ﬁeld H of Hilb ert spaces on X , we ha ve ( α H ) y : Z ⊕ X d t y ψ y ,x 7→ Z ⊕ X d t y [ α y ,x ⊗ 1 H x ]( ψ y ,x ) T o prov e this, ﬁrst we consider the case where K is a constant ﬁeld of Hilb ert spaces: K x = K for all x ∈ X, for some Hilb ert space K of countably inﬁnite dimension. W e handle this case b y choosing an orthonormal basis e j ∈ K and using this to deﬁne inclusions i j : J → K , ψ x 7→ ψ x e j The naturality of α implies that the square T J T i j / / α J   T K α K   T 0 J T 0 i j / / T 0 K 46 comm utes; it follo ws that ( α K ) y ( T i j ) y = ( T 0 i j ) y ( α J ) y . Since w e already kno w α J is giv en b y Eq. 40 , writing an y v ector ﬁeld in K in terms of the orthonormal basis e j , we obtain that ( α K ) y : Z ⊕ X d t y ψ y ,x 7→ Z ⊕ X d t y [ α y ,x ⊗ 1 K ]( ψ y ,x ) (41) Next, we use the fact that every measurable ﬁeld H of Hilb ert spaces is isomorphic to a direct summand of K [ 28 , Part II, Chap. 1, Prop. 1]. So, we hav e a pro jection p : K → H . The naturality of α implies that the square T K T p / / α K   T H α H   T 0 J T 0 p / / T 0 H comm utes; it follo ws that ( α H ) y ( T p ) y = ( T 0 p ) y ( α K ) y . Since w e already know α K is giv en by Eq. 41 , using the fact that an y vector ﬁeld in H is the image b y p of a vector ﬁeld in K , we obtain that ( α H ) y : Z ⊕ X d t y ψ y ,x 7→ Z ⊕ X d t y [ α y ,x ⊗ 1 H x ]( ψ y ,x ) W e ha ve assumed so far that the matrix functors T , T 0 are constructed from the same family of measures t = t 0 . Next, let us relax this hypothesis and supp ose that for each y ∈ Y , we hav e t y ∼ t 0 y . Let ˜ T 0 b e the matrix functor constructed from the family of measures t and the ﬁeld of Hilb ert space T 0 . The bounded measurable ﬁeld of identit y operators 1 T 0 y,x deﬁnes a matrix natural transformation r t,t 0 : T ⇒ ˜ T 0 . This natural transformation ass igns to any ob ject H ∈ H X a morphism r t,t 0 H : T H → ˜ T 0 H with comp onen ts: ( r t,t 0 H ) y : Z ⊕ d t 0 y ψ y ,x 7→ Z ⊕ d t y s d t 0 y d t y ψ y ,x Moreo ver, b y equiv alence of the measures, r t,t 0 is a natural isomorphism and r − 1 t,t 0 = r t 0 ,t . Supp ose α : T ⇒ T 0 is a b ounded natural transformation. The comp osite r t,t 0 α : T → ˜ T 0 is a b ounded natural transformation b etw een matrix functors constructed from the same families of measures t . According to the result shown ab o ve, w e kno w that this comp osite is a matrix measurable transformation, deﬁned by some measurable ﬁeld of op erators α y ,x : T y ,x → T 0 y ,x 47 W riting α = r t 0 ,t ( r t,t 0 α ), we conclude that α acts on each ob ject H ∈ H X as ( α H ) y : Z ⊕ X d t y ψ y ,x 7→ Z ⊕ X d t y [ ˜ α y ,x ⊗ 1 H x ]( ψ y ,x ) where ˜ α is the rescaled ﬁeld ˜ α = s d t y d t 0 y α This shows that α is a matrix natural transformation. Finally , to prov e the theorem in its full generality , w e consider the Leb esgue decomp osition of the measures t y and t 0 y with resp ect to eac h other (see App endix A.1 ): t = t t 0 + t t 0 , t t 0  t 0 t t 0 ⊥ t 0 and likewise, t 0 = t 0 t + t 0 t , t 0 t  t t 0 t ⊥ t where the subscript y indexing the measures is dropp ed for clarit y . Prop. 107 shows that t t 0 y ⊥ t t 0 y and t 0 t y ⊥ t 0 t y . Moreov er, Prop. 108 shows that t t 0 y ∼ t 0 t y . Consequently , for eac h y ∈ Y , there are disjoint measurable sets A y , B y and B 0 y suc h that t t 0 y and t 0 t y are supp orted on A y , that is, t t 0 y ( S ) = t t 0 y ( S ∩ A y ) t 0 t y ( S ) = t 0 t y ( S ∩ A y ) , for all measurable sets S ; and such that t t 0 y is supp orted on B y , and t 0 t y is supp orted on B 0 y . Let ˜ T b e the matrix functor constructed from the family of measures t t 0 and the ﬁeld of Hilb ert spaces T y ,x ; let ˜ T 0 b e the matrix functor constructed from the the family of measures t 0 t and the ﬁeld of Hilb ert spaces T 0 y ,x . The b ounded measurable ﬁeld of identit y op erators 1 T y,x deﬁne matrix natural transformations: i : ˜ T ⇒ T , p : T ⇒ ˜ T Giv en an y ob ject H ∈ H X , we get a morphism i H : ˜ T H → T H , whose components act as inclusions: ( i H ) y : Z ⊕ d t t 0 y ψ y ,x 7→ Z ⊕ d t y χ A y ( x ) ψ y ,x where χ A is the characteristic function of the set A ⊂ X : χ A ( x ) =  1 x ∈ A 0 x 6∈ A. W e also get a morphism p H : T H → ˜ T H , whose comp onen ts act as pro jections: ( p H ) y : Z ⊕ d t 0 y ψ y ,x 7→ Z ⊕ d t 0 t y ψ y ,x . Lik ewise, the b ounded measurable ﬁeld of iden tity op erators 1 T 0 y,x deﬁne an inclusion and a pro jec- tion: i 0 : ˜ T 0 ⇒ T 0 , p 0 : T 0 ⇒ ˜ T 0 Supp ose α : T ⇒ T 0 is a b ounded natural transformation. The comp osite p 0 αi : ˜ T ⇒ ˜ T 0 is then a b ounded natural transformation b et ween matrix functors constructed from equiv alent families of 48 measures. According to the result shown abov e, we know that this comp osite is a matrix natural tranformation, deﬁned by some measurable ﬁeld of op erators α y ,x : T y ,x → T 0 y ,x W e will show below the equality of natural transformations: α = i 0 [ p 0 αi ] p (42) This equality leads to our ﬁnal result. Indeed, for any H ∈ H X and each y ∈ Y , it yields: ( α H ) y : Z ⊕ d t y ψ y ,x 7→ Z ⊕ d t 0 y χ A y ( x ) s d t t 0 y d t 0 t y [ α y ,x ⊗ 1 H x ]( ψ y ,x ) and we conclude using the fact that, for all y and t 0 y -almost all x , χ A y ( x ) s d t t 0 y d t 0 t y = s d t t 0 y d t 0 y . The equality ( 42 ) follo ws from naturality of α . In fact, naturality implies that, for any morphism φ : H → H , the square T H T φ / / α H   T H α H   T 0 H T 0 φ / / T 0 H comm utes. It follows that, for eac h y ∈ Y , ( α H ) y ( T φ ) y = ( T 0 φ ) y ( α H ) y Let us ﬁx y ∈ Y . W e apply naturality to the morphism χ B y : H → H mapping any vector ﬁeld ψ x to the vector ﬁeld χ B y ( x ) ψ x . Its image by the functor T 0 deﬁnes a pro jection op erator ( T 0 χ B y ) y ≡ T 0 B y = Z ⊕ B y d t 0 y 1 T 0 y,x ⊗ 1 H x Since B y is a t 0 y -n ull set, this op erator acts trivially on T 0 H . It then follows from naturalit y that ( α H ) y T B y = T 0 B y ( α H ) y = 0 . (43) Lik ewise, applying naturalit y to the morphism χ B 0 y : H → H leads to 0 = ( α H ) y T B 0 y = T 0 B 0 y ( α H ) y (44) 49 W e now use the following decompositions of the iden tities op erators on the Hilb ert spaces ( T H ) y and ( T 0 H ) y in to direct sums of pro jections: 1 ( T H ) y = T A y ⊕ T B y , 1 ( T 0 H ) y = T 0 A y ⊕ T 0 B 0 y to write: ( α H ) y = [ T 0 A y ⊕ T 0 B 0 y ]( α H ) y [ T A y ⊕ T B y ] T ogether with ( 43 ) and ( 44 ), it yields: ( α H ) y = T 0 A y ( α H ) y T A y T o conclude, observe that T A y = ( ip H ) y , T 0 A y = ( i 0 p 0 H ) y W e ﬁnally obtain: ( α H ) y = ( i 0 p 0 H ) y ( α H ) y ( ip H ) y whic h shows our equality ( 42 ). This completes the pro of of the theorem. This allows an easy deﬁnition for the 2-morphisms in Meas : Deﬁnition 36 A measurable natural transformation is a b ounde d natur al tr ansformation b e- twe en me asur able functors. F or our work it will b e useful to hav e explicit formulas for comp osition of matrix natural trans- formations. So, let us compute the vertical composite of t wo matrix natural transformations α and α 0 : H X T ,t T 0 ,t 0 / / T 00 ,t 00 > > H Y α   α 0   F or any ob ject H ∈ H X , we get morphisms α H and α 0 H in H Y . Their comp osite is easy to calculate: ( α H 0 )( α H ) y : Z ⊕ X d t y T y ,x ⊗ H x → Z ⊕ X d t 00 y T 00 y ,x ⊗ H x R ⊕ X d t y ψ y ,x 7→ R ⊕ X d t 00 y [( ˜ α 0 y ,x ˜ α y ,x ) ⊗ 1 H x ]( ψ y ,x ) So, the comp osite is a measurable natural transformation α 0 · α with: ( ] α 0 · α ) y ,x = ˜ α 0 y ,x ˜ α y ,x . (45) F or some calculations it will be useful to hav e this equation written explicitly in terms of the original ﬁelds α and α 0 , rather than their rescalings: ( α 0 · α ) y ,x = s d t 00 y d t y s d t 0 y d t 00 y s d t y d t 0 y α 0 y ,x α y ,x (46) This equality deﬁnes the comp osite ﬁeld almost ev erywhere for the geometric mean measure p t y t 00 y . 50 Next, let us compute the horizon tal composite of tw o matrix natural transformations: H X T ,t * * T 0 ,t 0 4 4 H Y U,u * * U 0 ,u 0 4 4 H Z α   β   Recall that the horizontal composite β ◦ α is deﬁned so that U T H U α H / / β T H   ( β ◦ α ) H F F F F F F F F # # F F F F F F F F U T 0 H β T 0 H   U 0 T H U 0 α H / / U 0 T 0 H comm utes. Let us pick an elemen t ψ ∈ U T H , which can b e written in the form ψ z = Z ⊕ X d( ut ) z ψ z ,x , with ψ z ,x = Z ⊕ Y d k z ,x ψ z ,y,x b y deﬁnition of the comp osite ﬁeld U T . Note that, thanks to Eq. ( 37 ) whic h deﬁnes the family of measures k z ,x , the section ψ z can also b e written as ψ z = Z ⊕ Y d u z ψ z ,y , with ψ z ,y = Z ⊕ X d t y ψ z ,y,x Ha ving introduced all these notations, w e now ev aluate the image of ψ under the morphism ( β ◦ α ) H : (( β ◦ α ) H ) z ( ψ z ) = ( U 0 α H ) z ◦ ( β T H ) z ( ψ z ) =  Z ⊕ Y d u 0 z 1 U 0 z,y ⊗ ( α H ) y   Z ⊕ Y d u 0 z [ ˜ β z ,y ⊗ 1 ( T H ) y ]( ψ z ,y )  = Z ⊕ Y d u 0 z [ ˜ β z ,y ⊗ ( α H ) y ]( ψ z ,y ) = Z ⊕ Y d u 0 z Z ⊕ X d t 0 y [ ˜ β z ,y ⊗ ˜ α y ,x ⊗ 1 H x ]( ψ z ,y,x ) Applying the disintegration theorem, we can rewrite this last direct integral as an integral ov er X with resp ect to the measure ( u 0 t 0 ) z = Z Y d u 0 z ( y ) t 0 y W e obtain (( β ◦ α ) H ) z ( ψ z ) = Z ⊕ X d( u 0 t 0 ) z Z ⊕ Y d k 0 z ,x [ ˜ β z ,y ⊗ ˜ α y ,x ⊗ 1 H x ]( ψ z ,y,x ) = Z ⊕ X d( u 0 t 0 ) z [( ] β ◦ α ) z ,x ⊗ 1 H x ]( ψ z ,x ) 51 where ( ] β ◦ α ) z ,x ( ψ z ,x ) = Z ⊕ Y d k 0 z ,x [ ˜ β z ,y ⊗ ˜ α y ,x ]( ψ z ,y,x ) . (47) Equiv alen tly , in terms of the original ﬁelds α and β : ( β ◦ α ) z ,x ( ψ z ,x ) = s d( u 0 t 0 ) z d( ut ) z Z ⊕ Y d k 0 z ,x [ s d u z d u 0 z s d t y d t 0 y β z ,y ⊗ α y ,x ]( ψ z ,y,x ) (48) A sp ecial case is worth mentioning. When the source and target morphisms of α and β coincide, w e hav e k = k 0 , and the horizontal comp osition form ula ab o ve simply says ( β ◦ α ) z ,x is a direct in tegral of the ﬁelds of op erators β z ,y ⊗ α y ,x . Besides comp osition of 2-morphisms in Meas we also need iden tity 2-morphsms. Given a matrix functor T : H X → H Y , its iden tity 2-morphism 1 T : T ⇒ T is, up t - a.e. –equiv alence, given by the ﬁeld of identit y op erators: ( 1 T ) y ,x = 1 T y,x : T y ,x − → T y ,x . This acts as an identit y for the v ertical comp osition; the identit y 2-morphism of an identit y mor- phism, 1 1 X , acts as an identit y for horizon tal comp osition as well. In calculations, it is often con venien t to be able to describe a 2-morphism either by α or its rescaling ˜ α . The relationship b et ween these t wo descriptions is given by the following: Lemma 37 The ﬁelds α y ,x and α 0 y ,x ar e p t y t 0 y -e quivalent if and only if their r esc alings ˜ α y ,x and ˜ α 0 y ,x ar e t 0 y -e quivalent. Pro of: F or each y , let A y and ˜ A y b e the subsets of X on which α 6 = α 0 , and ˜ α 6 = ˜ α 0 , resp ectiv ely . Observ e that ˜ A y is the intersection of A y with the set of x for which the rescaling factor is non-zero: ˜ A y = A y ∩ n x : q d t y d t 0 y ( x ) 6 = 0 o . Supp osing ﬁrst that α y ,x and α 0 y ,x are p t y t 0 y -equiv alen t, w e ha v e p t y t 0 y ( A y ) = 0, so b y the deﬁnition of the geometric mean measure q t y t 0 y ( A y ) = Z A y d t 0 y s d t y d t 0 y = 0 . Th us the rescaling factor v anishes for t 0 y -almost every x ∈ A y ; that is, ˜ A y has t 0 y -measure zero. Con versely , if t 0 y ( ˜ A y ) = 0, we hav e: q t y t 0 y ( A y ) = q t y t 0 y ( ˜ A y ) + q t y t 0 y ( A y − ˜ A y ) . The ﬁrst term on the righ t v anishes because p t y t 0 y  t 0 y , while the second v anishes since q d t y d t 0 y = 0 on A y − ˜ A y . So, the rescaling α 7→ ˜ α induces a one-to-one corresp ondence b et w een √ tt 0 -classes of ﬁelds α and t 0 -classes of rescaled ﬁelds ˜ α . 52 3.3.3 Construction of Meas as a 2-category Theorem 38 Ther e is a sub-2-c ate gory Meas of Cat wher e the obje cts ar e me asur able c ate gories, the morphisms ar e me asur able functors, and the 2-morphisms ar e me asur able natur al tr ansforma- tions. In Section 3.3.2 w e sho wed that for any measurable space X , the iden tity 1 X : H X → H X is a matrix functor. It follows that the identit y on an y measurable category is a measurable functor. Similarly , in Section 3.3.2 we sho wed that for any matrix functor T , the iden tity 1 T : T ⇒ T is a matrix natural transformation. This implies that the iden tity on an y measurable functor is a measur- able natural transformation. T o prov e that the composite of measurable functors is measurable, we will use the sequence of lemmas b elo w. Since measurable natural transformations are just b ounded natural transformations b et ween measurable functors, by Thm. 34 , it will then easily follow that measurable natural transformations are closed under vertical and horizon tal comp osition. Lemma 39 A c omp osite of matrix functors is b ounde d ly natur al ly isomorphic to a matrix functor. Pro of: This was prov ed by Y etter [ 71 , Thm. 45], and w e hav e sk etched his argument in Section 3.3.2 . Y etter did not emphasize that the natural isomorphism is b ounded, but one can see from equation ( 34 ) that it is. Lemma 40 If F : H X → H Y is a C ∗ -e quivalenc e, then ther e is a me asur able bije ction b etwe en X and Y , and F is a me asur able functor. Pro of: This was prov ed by Y etter [ 71 , Thm. 40]. In fact, Y etter failed to require that F b e linear on morphisms, whic h is necessary for this result. Careful examination of his pro of shows that it can b e repaired if w e include this extra condition, which holds automatically for a C ∗ -equiv alence. Lemma 41 If T : H → H 0 is a me asur able functor and F : H → H X , G : H Y → H 0 ar e arbitrary C ∗ -e quivalenc es, then T is natur al ly isomorphic to the c omp osite H F / / H X ˜ T / / H Y G / / H 0 for some matrix functor ˜ T . Pro of: The pro of is analogous to the pro of of Lemma 11 . Since T is measurable we kno w there exist C ∗ -equiv alences F 0 : H → H X 0 , G 0 : H Y 0 → H 0 suc h that T is b oundedly naturally isomorphic to the comp osite H F 0 / / H X 0 ˜ T 0 / / H Y 0 G 0 / / H 0 for some matrix functor ˜ T 0 . By Lemma 40 we ma y assume X 0 = X and Y 0 = Y . So, let ˜ T b e the comp osite H X ¯ F / / H F 0 / / H X ˜ T 0 / / H Y G 0 / / H 0 ¯ G / / H Y where the w eak inv erses ¯ F and ¯ G are chosen using the fact that F and G are C ∗ -equiv alences. Since F 0 ¯ F : H X → H X and ¯ GG 0 : H Y → H Y are C ∗ -equiv alences, they are matrix functors b y Lemma 40 . It follows that ˜ T is a comp osite of three matrix functors, hence b oundedly naturally isomorphic to a matrix functor by Lemma 39 . Moreov er, the comp osite H F / / H X ˜ T / / H Y G / / H 0 53 is b oundedly naturally isomorphic to T . Since F and G are C ∗ -equiv alences and ˜ T is boundedly naturally isomorphic to a matrix functor, it follows that T is a measurable functor. Lemma 42 A c omp osite of me asur able functors is me asur able. Pro of: The pro of is analogous to the pro of of Lemma 12 . Suppose we ha ve a comp osable pair of measurable functors T : H → H 0 and U : H 0 → H 00 . By deﬁnition, T is b oundedly naturally isomorphic to a comp osite H F / / H X ˜ T / / H Y G / / H 0 where ˜ T is a matrix functor and F and G are C ∗ -equiv alences. By Lemma 41 , U is naturally isomorphic to a comp osite H 0 ¯ G / / H Y ˜ U / / H X H / / H 00 where ˜ U is a matrix functor, ¯ G is the chosen weak inv erse for G , and H is a C ∗ -equiv alence. The comp osite U T is thus boundedly naturally isomorphic to H F / / H X ˜ U ˜ T / / H Z H / / H 00 Since ˜ U ˜ T is a matrix functor by Lemma 39 , it follows that U T is a measurable functor. 4 Represen tations on measurable categories With the material presen ted in the previous sections, we no w ha ve a general framework to study represen tations of 2-groups on measurable categories—that is, representations in the 2-category Meas . Unpacking this representation theory and seeing what it amoun ts to concretely is no w an essen tially computational matter, which we turn to in this section. W e b egin b y summarizing the main results. 4.1 Main results Let us summarize our main results. W e no w assume that G is a skeletal 2-group. In the crossed mo dule description, since the homomorphism ∂ : H → G is trivial, G simply amounts to an ab elian group H and an action B of a group G as automorphisms of H . W e also assume that all the spaces and maps inv olv ed are measurable. Under these assumptions we can describ e representations of G , as well as in tertwiners and 2-intert winers, in terms of familiar geometric constructions—but living in the category of measurable spaces, rather than smo oth manifolds. T o understand these constructions, we ﬁrst deﬁne H ∗ to be the set of measurable homomorphisms χ : H → C × where C × is the multiplicativ e group of nonzero complex num bers. The set H ∗ b ecomes a group under p oin twise multiplication: ( χξ )( h ) = χ ( h ) ξ ( h ) . Under some mild conditions on H , H ∗ is again a measurable space, and its group op erations are measurable. The left action B of G on H naturally induces a right action of G on H ∗ , say ( χ, g ) 7→ χ g , giv en by χ g [ h ] = χ [ g B h ] . 54 This promotes H ∗ to a right G -space. Essen tially—ignoring technical conditions on measures, and issues of a.e. -equiv alence and cate- gorical equiv alence—we then hav e the following dictionary relating represen tation theory to geome- try: represen tation theory of a sk eletal 2-group G = ( G, H , B ) geometry a representation of G on H X a right action of G on X , and a map X → H ∗ making X a ‘measurable G -equiv arian t bundle’ ov er H ∗ an intert winer b et w een a ‘ G -equiv arian t measurable family of measures’ µ y on X , represen tations on H X and H Y and a ‘ G -equiv arian t Hilb ert space bundle’ o ver Y × X a 2-intert winer a map of G -equiv ariant Hilb ert space bundles Let us now explain this corresp ondence in more detail. Represen tations Consider a representation ρ : G → Meas on a measurable category H X . An essential step in understanding suc h a representation is understanding what the measurable automorphisms of the category H X lo ok lik e. In Section 4.2 , w e sho w that any automorphism of H X is 2-isomorphic to one induced by pullback along some measurable automorphism f : X → X . Suc h an automorphism, whic h we denote H f : H X → H X , acts on ﬁelds of Hilb ert spaces and linear maps on X , simply by pulling them back along f . In Thm. 49 , we show that if ρ is a representation on H X suc h that for each g ∈ G , ρ ( g ) = H f g for some f g , then ρ is determined, up to equiv alence of representations, b y: • a right action C of G as measurable transformations of the measurable space X , • a map χ : X → H ∗ that is G -equiv arian t, i.e.: χ ( x C g ) = χ ( x ) g (49) for all x ∈ X and g ∈ G . Geometrically , this states that the map χ : X → H ∗ is an equiv arian t ﬁb er bundle o ver the ‘c haracter group’ H ∗ = hom( H , C × ): X χ   H ∗ W e deﬁne ‘measurable represen tations’ of G to b e ones of this form for which b oth the map χ and the actions of G on X and H ∗ are measurable, where H ∗ inherits a measurable structure from that of H . In the rest of this summary of results we consider only measurable representations. 55 Tw o represen tations on H X are equiv alen t, b y deﬁnition, if they are related by a pair of inter- t winers that are weak inv erses of eac h other. W e discuss general intert winers and their geometry b elo w; for no w we merely men tion that invertible intert winers b etw een measurable representations corresp ond to in vertible measurable bundle maps: X Y ∼ / / H ∗ χ 2         χ 1   1 1 1 1 1 1 So, equiv alence of representations corresp onds geometrically to isomorphism of bundles. W e say that a represen tation is ‘indecomp osable’ if it is not equiv alen t to a ‘2-sum’ of nontriv- ial representations, where a ‘2-sum’ is a categoriﬁed version of the direct sum of ordinary group represen tations. W e sa y a represen tation is ‘irreducible’ if, roughly sp eaking, it do es not con tain an y subrepresen tations other than itself and the trivial representation. Irreducible represen tations are automatically indecomp osable, but not necessarily vice versa. An ( a priori ) intermediate no- tion is that of an ‘irretractable representation’—a representation ρ such that if any comp osite of in tertwiners of the form ρ 0 / / ρ / / ρ 0 is equiv alent to the identit y intert winer on ρ 0 , then ρ 0 is either trivial or equiv alent to ρ . While for ordinary group represen tations irretractable representations are the same as indecomp osable ones, this is not true for 2-group represen tations in Meas . W e thus classify b oth the irretractable and indecomp osable 2-group represen tations in Meas . The irreducible ones remain more challenging: in particular, we do not know if ev ery irretractable represen tation is irreducible. In Thm. 85 we sho w that a measurable representation of G on H X is indecomp osable if and only if G acts transitively on X . The study of indecomposable representations, and hence irreducible and irretractable representations as special cases, is thus rooted in Klein’s geometry of homogeneous spaces. Recall that for an y p oin t x o ∈ X , the stabilizer of x o is the subgroup S ⊆ G consisting of group elements g with x o C g = x o . By a standard argument, w e hav e X ∼ = G/S. Then, let χ o = χ ( x o ). By equation ( 49 ), the image of χ : X → H ∗ is a single G -orbit in H ∗ , and S is con tained in the stabilizer S ∗ of χ o . This shows that an indecomp osable represen tation essentially amoun ts to an equiv ariant map of homogeneous spaces χ : G/S → G/S ∗ , where S ∗ is the stabilizer of some p oin t in H ∗ , and S ⊆ S ∗ . In other words, indecomposable representations are classiﬁed up to equiv alence b y a choice of G -orbit in H ∗ , along with a subgroup S of the stabilizer of a p oin t χ o in the orbit. In Thm. 87 , we show an indecomp osable representation ρ is irretractable if and only if S is e qual to the stabilizer of χ o ; irretractable representations are th us classiﬁed up to equiv alence b y G -orbits in H ∗ . In tertwiners Next w e turn to the main results concerning in tert winers. T o state these, w e ﬁrst need some concepts from m easure theory . Let X b e a measurable space. Recall that t wo measures µ and ν on X are equiv alent , or in the same measure class , if they ha v e the same n ull sets. Next, supp ose G acts on 56 X as measurable transformations. Given a measure µ on X , for eac h g we deﬁne the ‘transformed’ measure µ g b y setting µ g ( A ) := µ ( A C g − 1 ) . (50) The measure is inv arian t if µ g = µ for every g . If µ g and µ are only equiv alen t, we sa y that µ is quasi-inv arian t . It is w ell-known that if G is a separable, lo cally compact top ological group, acting measurably and transitively on X , then there exist nontrivial quasi-inv ariant measures on X , and moreov er, all suc h measures belong to the same measure class (see App endix A.4 for further details). Next, let X and Y be tw o G -spaces. W e may consider Y -indexed families µ y of measures on X . Suc h a family is equiv arian t 1 under the action of G if for all g , µ y C g is e quivalent to µ g y . With these deﬁnitions we can no w giv e a concrete description of intert winers. Supp ose ρ 1 and ρ 2 are measurable representations of a sk eletal 2-group G on measurable categories H X and H Y , resp ectiv ely , with corresp onding equiv arian t bundles χ 1 and χ 2 : X Y H ∗ χ 2         χ 1   1 1 1 1 1 1 Then an intert winer φ : ρ 1 → ρ 2 is sp eciﬁed, up to equiv alence, b y: • an equiv arian t Y -indexed family of measures µ y on X , with each µ y supp orted on χ − 1 1 ( χ 2 ( y )). • an assignment, for each g ∈ G and all y , of a µ y -class of Hilb ert spaces φ y ,x and linear maps Φ g y ,x : φ y ,x → φ ( y ,x ) C g − 1 satisfying the co cycle conditions Φ g 0 g y ,x = Φ g 0 ( y ,x ) C g − 1 Φ g y ,x and Φ 1 y ,x = 1 φ y,x µ - a.e. for each pair g , g 0 ∈ G , where ( y , x ) C g is short for ( y C g , x C g ). There is a more geometric wa y to think of these in tertwiners. F or simplicit y , assume that, among the measure class of ﬁelds of linear op erators Φ g y ,x : φ y ,x → φ ( y ,x ) C g − 1 w e may c ho ose a represen tative suc h that the co cycle conditions hold ev erywhere in Y × X and for all g ∈ G . W e then think of the union of all the Hilb ert spaces: φ = a ( y ,x ) φ y ,x as a bundle of Hilbert spaces ov er the pro duct space Y × X . The group G acts on b oth the total space and the base space of this bundle. Indeed, the maps Φ g y ,x giv e a map Φ g : φ → φ ; the co cycle conditions then b ecome Φ g 0 g = Φ g 0 Φ g and Φ 1 y ,x = 1 φ y,x 1 Since we do not require equality , a more descriptiv e term w ould be ‘quasi-equivariance’; we stick to ‘equivariance’ for simplicit y . 57 whic h are simply the conditions that φ 7→ Φ g φ deﬁne a left action of G on φ . If we turn this into a righ t action b y deﬁning φ g = Φ g − 1 ( φ ) w e ﬁnd that the bundle map is equiv arian t with resp ect to this action of G on φ and the diagonal action of G on Y × X . It is thus helpful to think of an intert winer φ : ρ 1 → ρ 2 as b eing given b y an equiv ariant family of measures µ y and a µ -class of G -equiv ariant bundles of Hilbert spaces φ y ,x o ver Y × X . W e emphasize that it is not clear this picture is completely accurate for arbitrary in tertwiners, particularly when G is an uncoun table group, since there are separate co cycle equations for each pair g , g 0 ∈ G , each holding only almost everywhere. How ever, it is a useful heuristic picture, and can b e made precise at least in imp ortan t sp ecial cases. As with representations, we in tro duce and discuss the notions of reducibility , retractabilit y , and decomp osabilit y for intert winers. 2-In tertwiners Finally , the main results concerning the 2-intert winers are as follows. Consider a pair of represen- tations ρ 1 and ρ 2 of the skeletal 2-group G on the measurable categories H X and H Y , and tw o in tertwiners φ, ψ : ρ 1 ⇒ ρ 2 . Supp ose φ = ( µ, φ, Φ) and ψ = ( ν , ψ , Ψ). F or any y , we denote by √ µ y ν y the geometric mean of the measures µ y and ν y . A 2-in tertwiner turns out to consist of: • an assignment, for each y , of a √ µ y ν y -class of linear maps m y ,x : φ y ,x → ψ y ,x , whic h satisﬁes the intert wining rule Ψ g y ,x m y ,x = m ( y ,x ) g − 1 Φ g y ,x √ µν a.e. In the geometric picture of intert winers as equiv arian t bundles of Hilb ert spaces, this charac- terization of a 2-in tertwiner simply amoun ts to a morphism of equiv arian t bundles , up to almost-ev erywhere equality . The in tertwiners satisfy an analogue of Schur’s lemma . Namely , in Prop. 105 w e sho w that under some mild tec hnical conditions, any 2-intert winer b et ween irreducible in tertwiners is either n ull or an isomorphism. 4.2 In v ertible morphisms and 2-morphisms in Meas A 2-group represen tation ρ gives invertible morphisms ρ ( g ) and invertible 2-morphisms ρ ( g , h ) in the target 2-category . T o understand 2-group represen tations in Meas , it is thus a useful preliminary step to characterize in vertible measurable functors and inv ertible measurable natural transforma- tions. W e address these in this section, b eginning with the 2-morphisms. Consider tw o parallel measurable functors T and T 0 . A measurable natural transformation α : T ⇒ T 0 is in vertible if it has a vertical in verse, namely a measurable natural transformation α 0 : T 0 ⇒ T such that α 0 · α = 1 T and α · α 0 = 1 T 0 . W e often call the inv ertible 2-morphism α in Meas a 2-isomorphism , for short; we also sa y T and T 0 are 2-isomorphic . The following theorem classiﬁes 2-isomorphisms in the case where T and T 0 are matrix functors. Theorem 43 L et ( T , t ) , ( T 0 , t 0 ) : H X → H Y b e matrix functors. Then ( T , t ) and ( T 0 , t 0 ) ar e b ound- e d ly natur al ly isomorphic if and only if the me asur es t y and t 0 y ar e e quivalent, for every y , and ther e is a me asur able ﬁeld of b ounde d line ar op er ators α y ,x : T y ,x → T 0 y ,x such that α y ,x is an isomorphism 58 for e ach y and t y -a.e. in x . In this c ase, ther e is one 2-isomorphism T ⇒ T 0 for e ach t -class of ﬁelds α y ,x . Pro of: Supp ose α : T ⇒ T 0 is a b ounded natural isomorphism, with inv erse α 0 : T 0 ⇒ T . By Lemma 35 , α and α 0 are b oth matrix natural transformations, hence deﬁned b y ﬁelds of b ounded linear operators α y ,x and α 0 y ,x on Y × X . By the composition form ula ( 46 ), the comp osite α 0 · α = 1 T is given by ( α 0 · α ) y ,x = s d t 0 y d t y s d t y d t 0 y α 0 y ,x α y ,x = 1 T y,x t y - a.e. W e know by the chain rule ( 29 ) that the pro duct of Radon-Nikodym deriv atives in this form ula equals one p t y t 0 y - a.e. , but not yet that equals one t y - a.e. How ev er, by deﬁnition of the morphism ( T , t ), the Hilb ert spaces T y ,x are non-trivial t y - a.e. ; hence 1 T y,x 6 = 0. This shows that the pro duct of Radon-Nikodym deriv ativ es ab ov e is t y - a.e. nonzero; in particular, d t 0 t y d t y ( x ) 6 = 0 t y - a.e. where t 0 t y denotes the absolutely con tinuous part of t 0 y in its Leb esgue decomp osition t 0 y = t 0 t y + t 0 t y with resp ect to t y . But this prop ert y is equiv alen t to the statement that the measure t y is absolutely con tinuous with resp ect to t 0 y . T o c heck this, pic k a measurable set A and write t 0 y ( A ) = Z A d t y ( x ) d t 0 t y d t y ( x ) + t 0 t y ( A ) No w if t 0 y ( A ) = 0, both terms of the righ t-hand-side of this equalit y v anish—in particular the in tegral term. But since the Radon-Nik o dym deriv ative is a strictly p ositiv e function t y - a.e. , this requires the t y -measure of A to b e zero. So we hav e sho wn that t 0 y ( A ) = 0 implies t y ( A ) = 0 for an y measurable set A , i.e. t y  t 0 y . Starting with α · α 0 = 1 T 0 , the same analysis leads to the conclusion t y  t 0 y . Hence the tw o measures are equiv alent. F rom this it is immediate that ( α 0 · α ) y ,x = α 0 y ,x α y ,x = 1 T y,x t y - a.e. and thus α 0 y ,x = α − 1 y ,x . In particular, the op erators α y ,x are inv ertible t y - a.e. Con versely , supp ose the measures t y and t 0 y are equiv alen t and we are giv en a measurable ﬁeld α : T → T 0 suc h that for all y , the op erators α y ,x are inv ertible for almost ev ery x . It is easy to c heck, using the formula for vertical comp osition, that the matrix natural transformation deﬁned by α y ,x has inv erse deﬁned by α − 1 y ,x . A morphism T : H X → H Y is strictly in vertible if it has a strict inv erse, namely a 2-morphism U : H Y → H X suc h that U T = 1 X and T U = 1 Y . In 2-category theory , ho wev er, it is more natural to weak en the notion of in vertibilit y , so these equations hold only up to 2-isomorphism . In this case w e say that T is w eakly in vertible or an equiv alence . W e shall give tw o related c haracterizations of w eakly in vertible morphisms in Meas . F or the ﬁrst one, recall that if f : Y → X is a measurable function, then any measure µ on Y pushes forward to a measure f ∗ µ on X, by f ∗ µ ( A ) = µ ( f − 1 A ) for each measurable set A ⊆ X . In the case where µ = δ y , we hav e f ∗ δ y = δ f ( y ) 59 Denoting by δ the Y -indexed family of measures y 7→ δ y on Y , the follo wing theorem shows that ev ery in vertible matrix functor T : H X → H Y is essentially ( C , f ∗ δ ) for some inv ertible measurable map f : Y → X . As shown b y the following theorem, the condition for a morphism to b e an equiv alence is very restrictiv e [ 71 ]: Theorem 44 A matrix functor ( T , t ) : H X → H Y is a me asur able e quivalenc e if and only if ther e is an invertible me asur able function f : Y → X b etwe en the underlying sp ac es such that, for al l y , the me asur e t y is e quivalent to δ f ( y ) , and a me asur able ﬁeld of line ar op er ators fr om T y ,x to the c onstant ﬁeld C that is t y -a.e. invertible. Pro of: If ( T , t ) is an equiv alence, it has w eak inv erse that is also a matrix functor, say ( U, u ). The comp osite U T is 2-isomorphic to the identit y morphism 1 X , and T U is 2-isomorphic to 1 Y . Since 1 X : H X → H X is 2-isomorphic to the matrix functor ( C , δ x ), and similarly for 1 Y , Thm. 43 implies that the comp osite measures ut and tu are equiv alen t to Dirac measures: ( ut ) x = Z Y d u x ( y ) t y ∼ δ x ( tu ) y = Z X d t y ( x ) u x ∼ δ y An immediate consequence is that the measures u x and t y m ust b e non-trivial, for all x and y . Also, for all x , the subset X − { x } has zero ( ut ) x -measure Z Y d u x ( y ) t y ( X − { x } ) = 0 As a result the nonnegative function y 7→ t y ( X − { x } ) v anishes u x -almost everywhere. This means that, for all x and u x -almost all y , the measure t y is equiv alent to δ x . Likewise, we ﬁnd that, for all y and t y -almost all x , the measure u x is equiv alen t to δ y . Let us consider further the consequences of these t wo prop erties, by ﬁxing a p oin t y 0 ∈ Y . F or t y 0 -almost every x , we kno w, on one hand, that t y ∼ δ x for u x -almost all y (since this actually holds for all x ), and on the other hand, that u x ∼ δ y 0 . It follo ws that for t y 0 -almost every x , w e ha ve t y 0 ∼ δ x . The measure t y 0 b eing non-trivial, this requires t y 0 ∼ δ f ( y 0 ) for at least one p oin t f ( y 0 ) ∈ X ; moreov er this p oint is unique, b ecause tw o Dirac measures are equiv alen t only if they c harge the same p oin t. This deﬁnes a function f : Y → X such that t y is equiv alen t to δ f ( y ) . Likewise, we can deﬁne a function g : X → Y such that u x is equiv alen t to δ g ( x ) . Finally , b y expressing the comp osite measures in terms of Dirac measures, we get f g = 1 X and g f = 1 Y , establishing the inv ertibilit y of the function f . The measurability of the function f can b e shown as follo ws. Consider a measurable set A ⊆ X . Since the family of measures t y is measurable, w e know the function y 7→ t y ( A ) is measurable. Since t y ( A ) = δ f ( y ) , so this function is giv en b y: y 7→ t y ( A ) = ( 1 if y ∈ f − 1 ( A ) 0 if not This coincides with the characteristic function of the set f − 1 ( A ) ⊆ Y , whic h is measurable precisely when f − 1 ( A ) is measurable. Hence, f is measurable. Finally , w e can use ( 36 ) to compose the ﬁelds U x,y and T y ,x . Since ( tu ) y ∼ δ y , the only essential comp onen ts of the comp osite ﬁeld are the diagonal ones: ( T U ) y ,y = Z ⊕ X d k y ,y ( x ) T y ,x ⊗ U x,y . 60 Applying ( 37 ) in this case, w e ﬁnd that the measures k y ,y are deﬁned by the property Z X d k y ,y ( x ) F ( x, y ) = Z X d δ f ( y ) ( x ) Z Y δ g ( x ) ( y ) F ( x, y ) for any meas urable function F on X × Y . F rom this w e obtain k y ,y = δ f ( y ) and ( T U ) y ,y = T y ,f ( y ) ⊗ U f ( y ) ,y for all y ∈ Y . Since w e know T U is 2-isomorphic to the matrix functor ( C , δ y ), we therefore obtain ( T U ) y ,y = T y ,f ( y ) ⊗ U f ( y ) ,y ∼ = C ∀ y ∈ Y . where the isomorphism of ﬁelds is measurable. This can only happ en if eac h factor in the tensor pro duct is measurably isomorphic to the constant ﬁeld C . Con versely , if the measures t y are equiv alent to δ f ( y ) for an inv ertible measurable function f , and if T y ,f ( y ) ∼ = C , construct a matrix functor U : H Y → H X from the family of measures δ f − 1 ( x ) and the constant ﬁeld U x,y = C . One can immediately chec k that U is a weak in verse for T . T ak en together, these theorems hav e the following corollary: Corollary 45 If T : H X → H Y is a we akly invertible me asur able functor, ther e is a unique me a- sur able isomorphism f : Y → X such that T is b ounde d ly natur al ly isomorphic t o the matrix functor ( C , δ f ( y ) ) . Pro of: An y measurable functor is b oundedly naturally isomorphic to a matrix functor, say T ∼ = ( T y ,x , t y ). By Thm. 44 , we may in fact tak e T y ,x = C and t y = δ f ( y ) for some measurable isomorphism f : Y → X . By Thm. 43 , t wo such matrix functors, say ( C , δ f ( y ) ) and ( C , δ f 0 ( y ) ) are b oundedly naturally isomorphic if and only if f = f 0 , so the choice of f is unique. W e ha ve classiﬁed measurable equiv alences by giving one represen tative—a speciﬁc matrix equiv- alence—of eac h 2-isomorphism class. These represen tatives are quite handy in calculations, but they do hav e one drawbac k: matrix functors are not strictly closed under comp osition. In particular, the comp osite of tw o of our representativ es ( C , f ∗ δ ) is isomorphic, but not equal, to another of this form. While in general this is the b est w e might exp ect, it is natural to w onder whether these 2-isomorphism classes hav e a set of represen tations that is closed under comp osition. They do. If X and Y are measurable spaces, any measurable function f : Y → X giv es a functor H f called the pullback H f : H X → H Y deﬁned b y pulling bac k measurable ﬁelds of Hilbert spaces and linear op erators along f . Explicitly , giv en a measurable ﬁeld of Hilb ert spaces H ∈ H X , the ﬁeld H f H has comp onents ( H f H ) y = H f ( y ) Similarly , for φ : H → H 0 a measurable ﬁeld of linear op erators on X , ( H f φ ) y = φ f ( y ) . It is easy to see that this is functorial; to chec k that H f is a me asur able functor, we note that it is b oundedly naturally isomorphic to the matrix functor ( C , δ f ( y ) ), whic h sends an ob ject H ∈ H X to Z ⊕ X d δ f ( y ) ( x ) C ⊗ H x ∼ = H f ( y ) = ( H f H ) y 61 and do es the analogous thing to morphisms in H X . The ob vious isomorphism in this equation is natural, and has unit norm, so is b ounded. Prop osition 46 If T : H X → H Y is a we akly invertible me asur able functor, ther e exists a unique me asur able isomorphism f : Y → X such that T is b ounde d ly natur al ly isomorphic to the pul lb ack H f . Pro of: Any measurable functor from H X to H Y is equiv alent to some matrix functor; by Cor. 45 , this matrix functor ma y be tak en to b e ( C , δ f ( x ) ) for a unique isomorphism of measurable spaces f : Y → X . This matrix functor is 2-isomorphic to H f . While the pullbacks H f are closely related to the matrix functors ( C , f ∗ δ ), the former hav e sev eral adv antages, all stemming from the basic equations: H 1 X = 1 H X and H f H g = H g f (51) In particular, comp osition of pullbacks is strictly asso ciative, and eac h pullbac k H f has strict inv erse H f − 1 . In fact, there is a 2-category M with measurable spaces as ob jects, invertible measurable functions as morphisms, and only identit y 2-morphisms. The assignments X 7→ H X and f 7→ H f giv e a contra v ariant 2-functor M → Meas . The forgoing analysis shows this 2-functor is faithful at the level of 1-morphisms. If f , f 0 are distinct measurable isomorphisms, the measurable functors H f and H f 0 are never 2-isomorphic. How ever, eac h H f has many 2-automorphisms: Theorem 47 L et f : Y → X b e an isomorphism of me asur able sp ac es, and H f : H X → H Y b e its pul lb ack. Then the gr oup of 2-automorphisms of H f is isomorphic to the gr oup of me asur able maps Y → C × , with p ointwise multiplic ation. Pro of: Let α b e a 2-automorphism of H f , where f is in vertible. H X H f * * H f 4 4 H Y α   Using the 2-isomorphism β : H f ⇒ ( C , f ∗ δ ), w e can write α as a comp osite α = β − 1 · ˜ α · β By Thm. 43 , ˜ α : ( C , f ∗ δ ) ⇒ ( C , f ∗ δ ) is necessarily a matrix functor given by a measurable ﬁeld of linear op erators ˜ α y ,x : C → C , deﬁned and inv ertible δ f ( y ) - a.e. for all y . Such a measurable ﬁeld is just a measurable function ˜ α : Y × X → C , with ˜ α y ,f ( y ) ∈ C × . F rom the deﬁnition of matrix natural transformations, we can then compute for each ob ject H ∈ H X , the morphism α H : H f H → H f H . Explicitly , H f ( y ) β H / / R ⊕ d δ f ( y ) ( x ) H x ˜ α H / / R ⊕ d δ f ( y ) ( x ) H x β − 1 H / / H f ( y ) ψ f ( y )  / / R ⊕ d δ f ( y ) ( x ) ψ x  / / R ⊕ d δ f ( y ) ( x ) α y ,x ψ x  / / ˜ α y ,f ( y ) ψ f ( y ) So, the natural transformation α acts via multiplication b y α ( y ) := ˜ α y ,f ( y ) ∈ C × . 62 It is easy to show that α ( y ) : Y → C × is measurable, since ˜ α and f are b oth measurable. Con versely , given a measurable map α ( y ), we get a 2-automorphism α of H f b y letting α H : H f H → H f H b e giv en by ( α H ) y : H f ( y ) → H f ( y ) ψ y 7→ α ( y ) ψ y One can easily chec k that the pro cedures just describ ed are inv erses, so we get a one-to-one corre- sp ondence. Moreo ver, comp osition of 2-automorphisms α 1 , α 2 , corresponds to multiplication of the functions α 1 ( y ), α 2 ( y ), so this corresp ondence giv es a group isomorphism. It will also b e useful to know ho w to comp ose pullbac k 2-automorphisms horizontally: Prop osition 48 L et f : Y → X and g : Z → Y b e me asur able isomorphisms, and c onsider the fol lowing diagr am in Meas : H X H f * * H f 4 4 H Y H g * * H g 4 4 H Z α   β   wher e α and β ar e 2-automorphisms c orr esp onding to me asur able maps α : Y → C × and β : Z → C × as in the pr evious the or em. Then the horizontal c omp osite β ◦ α c orr esp onds to the me asur able map fr om Z to C × deﬁne d by ( β ◦ α )( z ) = β ( z ) α ( g ( z )) Pro of: This is a straightforw ard computation from the deﬁnition of horizontal comp osition. 4.3 Structure theorems W e no w b egin the precise description of the represen tation theory , as outlined in Section 4.1 . W e ﬁrst giv e the detailed structure of representations, follo wed b y that of in tertwiners and 2-in tertwiners. 4.3.1 Structure of represen tations Giv en a generic 2-group G = ( G, H, B , ∂ ), w e are interested in the structure of a representation ρ in the target 2-category Meas . Since an y ob ject of Meas is C ∗ -equiv alen t to one of the form H X , w e shall assume that ρ ( ? ) = H X for some measurable space X . The representation ρ also gives, for each g ∈ G , a morphism ρ ( g ) : H X → H X , and we assume for no w that all of these morphisms are pullbacks of measur- able automorphisms of X . Theorem 49 (Representations) L et ρ b e a r epr esentation of G = ( G, H , ∂ , B ) on H X , and as- sume that e ach ρ ( g ) is of the form H f g for some f g : X → X . Then ρ is determine d uniquely by: 63 • a right action C of G as me asur able tr ansformations of X , and • an assignment to e ach x ∈ X of a gr oup homomorphism χ ( x ) : H → C × . satisfying the fol lowing pr op erties: (i) for e ach h ∈ H , the function x 7→ χ ( x )[ h ] is me asur able (ii) any element of the image of ∂ acts trivial ly on X via C . (iii) the ﬁeld of homomorphisms is e quivariant under the actions of G on H and X : χ ( x )[ g B h ] = χ ( x C g )[ h ] . Pro of: Consider a represen tation ρ on H X and supp ose that for each g , ρ ( g ) = H f g , where f g : X → X is a measurable isomorphism. Thanks to the strict comp osition la ws ( 51 ) for such 2-morphisms, the conditions that ρ resp ect comp osition of morphisms and the iden tity morphism, namely ρ ( g 0 g ) = ρ ( g 0 ) ρ ( g ) and ρ (1) = 1 H X can b e expressed as conditions on the functions f g : f g 0 g = f g f g 0 and f 1 = 1 X . (52) In tro ducing the notation x C g = f g ( x ), these equations can b e rewritten x C g 0 g = ( x C g 0 ) C g and x C 1 = x Th us, the mapping ( x, g ) 7→ x C g is a right action of G on X . Next, consider a 2-morphism ρ ( u ), where u = ( g , h ) is a 2-morphism in G . Since u is in vertible, so is ρ ( u ). In particular, applying ρ to the 2-morphism (1 , h ), we get a 2-isomorphism ρ (1 , h ) : 1 H X ⇒ H f ∂ h for each h ∈ H . Such 2-isomorphisms exists only if f ∂ h = 1 X for all h ; that is, x C ∂ ( h ) = x for all x ∈ X and h ∈ H . Th us, the image ∂ ( H ) of the homomorphism ∂ ﬁxes ev ery element x ∈ X under the action C . F or arbitrary , u ∈ G × H , Thm. 47 implies ρ ( u ) is given by a measurable function on X , which w e also denote by ρ ( g , h ): ρ ( g , h ) : X → C . W e can derive conditions on the these functions from the requirement that ρ resp ect b oth kinds of comp osition of 2-morphisms. First, by Thm. 47 , vertical composition corresp onds to p oin twise multiplication of functions, so the condition ( 10 ) that ρ resp ect v ertical comp osition b ecomes: ρ ( g , h 0 h )( x ) = ρ ( ∂ hg , h 0 )( x ) ρ ( g , h )( x ) . (53) Similarly , using the formula for horizontal comp osition provided b y Prop. 48 , we obtain ρ ( g 0 g , h 0 ( g 0 B h ))( x ) = ρ ( g 0 , h 0 )( x ) ρ ( g , h )( x C g 0 ) . (54) 64 Applying this formula in the case g 0 = 1 and h = 1, w e ﬁnd that the functions ρ ( g , h ) are indep enden t of g : ρ ( g , h )( x ) = ρ (1 , h )( x ) This allows a drastic simpliﬁcation of the form ula for vertical comp osition ( 53 ). Indeed, if we deﬁne χ ( x )[ h ] = ρ (1 , h )( x ) , (55) then ( 53 ) is simply the statemen t that h 7→ χ ( x )[ h ] is a homomorphism for each x : χ ( x )[ h 0 h ] = χ ( x )[ h 0 ] χ ( x )[ h ] . T o chec k that the ﬁeld of homomorphisms χ ( x ) satisﬁes the equiv ariance prop ert y χ ( x C g )[ h ] = χ ( x )[ g B h ] , (56) one simply uses ( 54 ) again, this time with g = h 0 = 1. T o complete the pro of, we sho w how to reconstruct the representation ρ : G → Meas , given the measurable space X , right action of G on X , and ﬁeld χ of homomorphisms from H to C × . This is a straigh tforward task. T o the unique ob ject of our 2-group, we assign H X ∈ Meas . If g ∈ G is a morphism in G , we let ρ ( g ) = H f g , where f g ( x ) = x C g ; if u = ( g , h ) ∈ G × H is a 2-morphism in G , we let ρ ( u ) b e the automorphism of H f g deﬁned by the measurable function x 7→ χ ( x )[ h ]. This theorem suggests an in teresting question: is every representation of G on H X equiv alen t to one of the ab o ve t yp e? As a weak piece of evidence that the answer migh t b e ‘yes’, recall from Prop. 46 that any inv ertible morphism from H X to itself is isomorphic to one of the form H f . How ever, this fact alone is not enough. The ab o ve theorem also suggests that we view representations of 2-groups in a more geometric w ay , as equiv ariant bundles. In a representation of a 2-group G on H X , the assignmen t x 7→ χ ( x ) can be viewed as promoting X to the total space of a kind of bundle ov er the set hom( H, C × ) of homomorphisms from H to C × : X χ   hom( H , C × ) Here w e are using ‘bundle’ in a v ery lo ose sense: no top ology is inv olved. The group G acts on b oth the total space and the base of this bundle: the right action C of G on X comes from the represen tation, while its left action B on H induces a righ t action ( χ, g ) 7→ χ g on hom( H, C × ), where χ g [ h ] = χ [ g B h ] . The equiv ariance prop ert y in Thm. 49 means that the map χ satisﬁes χ ( x C g ) = χ ( x ) g . So, we say χ : X → hom( H, C × ) is a ‘ G -equiv ariant bundle’. So far w e ha ve ignored an y measurable structure on the groups G and H , treating them as discrete groups. In practice these groups will come with measurable structures of their o wn, and the maps in volv ed in the 2-group will all b e measurable. F or such 2-groups the in teresting representations will b e the ‘measurable’ ones, meaning roughly that all the maps deﬁning the ab o ve G -equiv arian t bundle are measurable. T o make this line of thought precise, w e need a concept of ‘measurable group’: 65 Deﬁnition 50 We deﬁne a measurable group to b e a top olo gic al gr oup whose top olo gy is lo c al ly c omp act, Hausdorﬀ, and se c ond c ountable. V aradara jan calls these lcsc groups , and his b ook is an excellen t source of information ab out them [ 69 ]. By Lemma 14 , they are a sp ecial case of Polish gr oups : that is, top ological groups G that are homeomorphic to complete separable metric spaces. F or more information on Polish groups, see the b ook by Bec ker and Kec hris [ 19 ]. It ma y seem o dd to deﬁne a ‘measurable group’ to b e a sp ecial sort of top olo gic al group. The ﬁrst reason is that ev ery measurable group has an underlying measurable space, by Lemma 14 . The second is that by Lemma 114 , an y measurable homomorphism betw een measurable groups is automatically con tinuous. This implies that the top ology on a measurable group can be uniquely reconstructed from its group structure together with its σ -algebra of measurable subsets. Next, instead of working with the set hom( H , C × ) of al l homomorphisms from H to C × , w e restrict attention to the me asur able ones: Deﬁnition 51 If H is a me asur able gr oup, let H ∗ denote the set of me asur able (henc e c ontinuous) homomorphisms χ : H → C × . W e make H ∗ in to a group with p oin twise m ultiplication as the group op eration: ( χχ 0 )[ h ] = χ [ h ] χ 0 [ h ] . H ∗ then b ecomes a top ological group with the compact-op en top ology . This is the same as the top ology where χ α → χ when χ α ( h ) → χ ( h ) uniformly for h in any ﬁxed compact subset of H . Unfortunately , H ∗ ma y not b e a measurable group! An example is the free ab elian group on coun tably man y generators, for which H ∗ fails to b e lo cally compact. Ho w ever, H ∗ is measurable when H is a measurable group with ﬁnitely man y connected components. F or more details, including a necessary and suﬃcient condition for H ∗ to b e measurable, see App endix A.3 . In our deﬁnition of a ‘measurable 2-group’, we will demand that H and H ∗ b e measurable groups. The left action of G on H giv es a righ t action of G on H ∗ : C : H ∗ × G → H ∗ ( χ, g ) 7→ χ g where χ g [ h ] = χ [ g B h ] . W e will demand that both these actions b e measurable. W e do not know if these are independent conditions. How ev er, in Lemma 119 we show that if the action of G on H is contin uous, its action on H ∗ is contin uous and thus measurable. This handles most of the examples we care ab out. With these preliminaries out of the wa y , here are the main deﬁnitions: Deﬁnition 52 A measurable 2-group G = ( G, H , B , ∂ ) is a 2-gr oup for which G , H and H ∗ ar e me asur able gr oups and the maps B : G × H → H, C : H ∗ × G → H ∗ , ∂ : H → G ar e me asur able. Deﬁnition 53 L et G = ( G, H , B , ∂ ) b e a me asur able 2-gr oup and supp ose the r epr esentation ρ of G on H X is sp e ciﬁe d by the maps C : X × G → X , χ : X → H ∗ as in Thm. 49 . Then ρ is a measurable representation if b oth these maps ar e me asur able. 66 F rom now on, w e will alwa ys b e interested in me asur able representations of me asur able 2-groups. F or suc h a representation, Lemma 120 guarantees that we can choose a top ology for X , compatible with its structure as a measurable space, suc h that the action of G on X is con tinuous. This may not mak e χ : X → H ∗ con tinuous. How ever, Lemma 114 implies that each χ ( x ) : H → C × is contin uous. Before concluding this section, w e p oin t out a corollary of Thm. 49 that reveals an in teresting feature of the represen tation theory in the 2-category Meas . This corollary inv olves a certain sk eletal 2-group constructed from G (recall that a 2-group is ‘sk eletal’ when its corresponding crossed module has ∂ = 0). Let G b e a 2-group, not ne c essarily me asur able , with corresp onding crossed mo dule ( G, H , ∂ , B ). Then, let ¯ G = G/∂ ( H ) , ¯ H = H / [ H, H ] Note that the image ∂ ( H ) is a normal subgroup of G by ( 2 ), and the commutator subgroup [ H , H ] is a normal subgroup of H . One can chec k that the action B naturally induces an action ¯ B of ¯ G on ¯ H . If we also deﬁne ¯ ∂ : ¯ H → ¯ G to b e the trivial homomorphism, it is straightforw ard to c heck that these data deﬁne a new crossed mo dule, from which we get a new 2-group: Deﬁnition 54 L et G b e a 2-gr oup with c orr esp onding cr osse d mo dule ( G, H, ∂ , B ) . Then the 2-gr oup ¯ G c onstructe d fr om the cr osse d mo dule ( ¯ G, ¯ H , ¯ ∂ , ¯ B ) is c al le d the sk eletization of G . No w consider a representation ρ of the 2-group G . First, by Thm. 49 , ∂ ( H ) acts trivially on X , so ¯ G acts on X . Second, the group C × b eing ab elian, [ H, H ] is con tained in the kernel of the homomorphisms χ ( x ) : H → C × for all x . In light of Thm. 49 , these remarks lead to the following corollary: Corollary 55 F or any 2-gr oup, its r epr esentations of the form describ e d in Thm. 49 ar e in natur al one-to-one c orr esp ondenc e with r epr esentations of the same form of its skeletization. This corollary means measurable representations in Meas fail to detect the ‘non-skeletal part’ of a 2-group. Ho wev er, the representation theory of G as a whole is generally richer than the represen tation theory of its sk eletization ¯ G . One can indeed sho w that, while G and ¯ G can not b e distinguished b y lo oking at their r epr esentations , they generally do not hav e the same intertwiners . In what follows, w e will nevertheless restrict our study to the case of skeletal 2-groups. Th us, from now on, we suppose the group homomorphism ∂ : H → G to b e trivial, and hence the group H to b e ab elian. Considering Thm. 49 in light of the preceding discussion, w e easily obtain the following geometric c haracterization of measurable representations of sk eletal 2-groups. Theorem 56 A me asur able r epr esentation ρ of a me asur able skeletal 2-gr oup G = ( G, H , B ) on H X is determine d uniquely by a me asur able right G -action on X , to gether with a G -e quivariant me asur able map χ : X → H ∗ . Since w e consider only sk eletal 2-groups and measurable represen tations in the rest of the paper, this is the description of 2-group represen tations to k eep in mind. It is helpful to think of this description as giving a ‘measurable G -equiv ariant bundle’ X χ   H ∗ 67 4.3.2 Structure of in tert winers In this section we study intert winers b etw een tw o ﬁxed measurable representations ρ 1 and ρ 2 of a sk eletal 2-group G . Supp ose ρ 1 and ρ 2 are sp eciﬁed, resp ectiv ely , by the measurable G -equiv ariant bundles χ 1 and χ 2 , as in Thm. 56 : X Y H ∗ χ 2         χ 1   1 1 1 1 1 1 T o state our main structure theorem for in tert winers, it is con v enient to ﬁrst deﬁne t w o properties that a Y -indexed measurable of measures µ y on X might satisfy . First, w e sa y the family µ y is ﬁb erwise if each µ y is supp orted on the ﬁb er, in X , ov er the point χ 2 ( y ). That is, µ y is ﬁb erwise if µ y ( X ) = µ y ( χ − 1 1 ( χ 2 ( y ))) for all y . W e also recall from Section 4.1 that w e sa y a measurable family of measures is equiv ariant if for every g ∈ G and y ∈ Y , µ y C g is e quivalent to the transformed measure µ g y deﬁned by: µ g y ( A ) := µ y ( A C g − 1 ) . (57) Note that, to chec k that a giv en e quivariant family of measures is ﬁb erwise, it is enough to chec k that, for a set of representativ es y o of the G -orbits in Y , the measure µ y o concen trates on the ﬁb er o ver χ 2 ( y o ). W e are no w ready to give a concrete c haracterization of in tertwiners φ : ρ 1 → ρ 2 b et w een measur- able representations. F or notational simplicity we now omit the symbol ‘ C ’ for the right G -actions on X and Y deﬁned by the representations, using simple concatenation instead. Theorem 57 (Intert winers) L et ρ 1 , ρ 2 b e me asur able r epr esentations of G = ( G, H, B ) , sp e ciﬁe d r esp e ctively by the G -e quivariant bund les χ 1 : X → H ∗ and χ 2 : Y → H ∗ , as in Thm. 56 . Given an intertwiner φ : ρ 1 → ρ 2 , we c an extr act the fol lowing data: (i) an e quivariant and ﬁb erwise Y -indexe d me asur able family of me asur es µ y on X ; (ii) a µ -class of ﬁelds of Hilb ert sp ac es φ y ,x on Y × X ; (iii) for e ach g ∈ G , a µ -class of ﬁelds of invertible line ar maps Φ g y ,x : φ y ,x → φ ( y ,x ) g − 1 such that, for al l g , g 0 ∈ G , the c o cycle c ondition Φ g 0 g y ,x = Φ g 0 ( y ,x ) g − 1 Φ g y ,x holds for al l y and µ y -almost every x . Conversely, such data c an b e use d to c onstruct an intertwiner. Before commencing with the pro of, note what this theorem do es not state. It do es not state that the data extracted from an in tertwiner are unique, nor that starting with these data and constructing an in tertwiner gives ‘the same’ in tertwiner. This do es turn out to b e essentially true, at least for an certain broad class of in tertwiners. The sense in which this result classiﬁes intert winers will b e clariﬁed in Prop ositions 71 and 72 . 68 Pro of: Recall that an intert winer provides a morphism φ : H X → H Y in Meas , together with a family φ ( g ) : ρ 2 ( g ) φ ⇒ φρ 1 ( g ) g ∈ G of inv ertible 2-morphisms, sub ject to the compatibility conditions ( 14 ), ( 16 ) and ( 18 ), namely φ (1) = 1 φ (58) and  φ ( g 0 ) ◦ 1 ρ 1 ( g )  ·  1 ρ 2 ( g 0 ) ◦ φ ( g )  = φ ( g 0 g ) (59) and [ 1 φ ◦ ρ 1 ( u )] · φ ( g ) = φ ( g ) · [ ρ 2 ( u ) ◦ 1 φ ] (60) where u = ( g , h ). Let us show ﬁrst that we may assume φ is a matrix functor. Since φ is a measurable functor, w e can pick a b ounded natural isomorphism m : φ ⇒ ˜ φ where ˜ φ is a matrix functor. W e then deﬁne, for each g ∈ G , a measurable natural transformation ˜ φ ( g ) =  m ◦ 1 ρ 1 ( g )  · φ ( g ) ·  1 ρ 2 ( g ) ◦ m − 1  c hosen to mak e the follo wing diagram commute: H X ρ 1 ( g ) / / φ   ˜ φ   H X φ       # ' , ˜ φ   H Y ρ 2 ( g ) / / H Y m + 3 m + 3 ˜ φ ( g ) 6 > u u u u u u u u u u u u u u φ ( g ) 6 > The matrix functor ˜ φ , together with the family of measurable natural transformations ˜ φ ( g ), gives an in tertwiner, which we also denote ˜ φ . The natural isomorphism m gives an inv ertible 2-in tertwiner m : φ → ˜ φ . So, ev ery intert winer is equiv alent to one for which φ : H X → H Y is a matrix functor. Hence, we no w assume φ = ( φ, µ ) is a matrix functor, and work out what equations ( 59 ) and ( 60 ) amount to in this case. W e use the following result, which simply collects in one place several useful comp osition form ulas: Lemma 58 L et ρ 1 and ρ 2 b e r epr esentations c orr esp onding to G -e quivariant bund les X and Y over H ∗ , as in the the or em. 1. Given any matrix functor ( T , t ) : H X → H Y : • The c omp osite T ρ 1 ( g ) is a matrix functor; in p articular, it is deﬁne d by the ﬁeld of Hilb ert sp ac es T y ,xg − 1 and the family of me asur es t g y . 69 • The c omp osite ρ 2 ( g ) T is a matrix functor; in p articular, it is deﬁne d by the ﬁeld of Hilb ert sp ac es T y g,x and the family of me asur es t y g . 2. Given a p air of such matrix functors ( T , t ) , ( T 0 , t 0 ) , and any matrix natur al tr ansformation α : T ⇒ T 0 : • Whiskering by ρ 1 ( g ) pr o duc es a matrix natur al tr ansformation whose ﬁeld of line ar op er- ators is α y g,x . • Whiskering by ρ 2 ( g ) pr o duc es a matrix natur al tr ansformation whose ﬁeld of line ar op er- ators is α y g,x . That is: H X ρ 1 ( g ) / / H X T y,x , t y ' ' T 0 y,x , t 0 y 7 7 H Y α y,x   = H X T g,xg − 1 , t g y ' ' T 0 y,xg − 1 , t 0 y g 7 7 H Y α y,xg − 1   and H X T y,x , t y ' ' T 0 y,x , t 0 y 7 7 H Y ρ 2 ( g ) / / H Y α y,x   = H X T yg ,x , t yg ' ' T 0 yg ,x , t 0 yg 7 7 H Y α yg ,x   Pr o of: This is a direct computation from the deﬁnitions of comp osition for functors and natural transformations.  W e return to the pro of of the theorem. Using this lemma, we immediately obtain explicit descriptions of the source and target of each φ ( g ): we ﬁnd that comp osites ρ 2 ( g ) φ and φρ 1 ( g ) are the matrix functors whose families of measures are given b y µ y g and µ g y resp ectiv ely , and whose ﬁelds of Hilb ert spaces read [ ρ 2 ( g ) φ ] y ,x = φ y g,x and [ φρ 1 ( g )] y ,x = φ y ,xg − 1 An immediate consequence is that the family µ y is equiv arian t. Indeed, since each φ ( g ) is a matrix natural isomorphism, Thm. 43 implies the source and target measures µ y g and µ g y are equiv alent for all g . Thus, for all g , the 2-morphism φ ( g ) deﬁnes a ﬁeld of inv ertible op erators φ ( g ) y ,x : φ y g,x − → φ y ,xg − 1 , (61) determined for each y and p µ g y µ y g - a.e. in x , or equiv alently µ y g - a.e. in x , by equiv ariance. The lemma also helps mak e the compatibility condition ( 59 ) explicit. The composites φ ( g 0 ) ◦ 1 ρ 1 ( g ) and 1 ρ 2 ( g 0 ) ◦ φ ( g ) are matrix natural transformations whose ﬁelds of op erators read  φ ( g 0 ) ◦ 1 ρ 1 ( g )  y ,x = φ ( g 0 ) y ,xg − 1 and  1 ρ 2 ( g 0 ) ◦ φ ( g )  y ,x = φ ( g ) y g 0 ,x Hence, ( 59 ) can b e rewritten as φ ( g 0 ) y ,xg − 1 φ ( g ) y g 0 ,x = φ ( g 0 g ) y ,x . 70 Deﬁning a ﬁeld of linear op erators Φ g y ,x ≡ φ ( g ) y g − 1 ,x , (62) the condition ( 59 ) ﬁnally b ecomes: Φ g 0 g y ,x = Φ g 0 ( y ,x ) g − 1 Φ g y ,x . (63) W e note that since φ ( g ) y ,x is deﬁned and inv ertible µ y g - a.e. , Φ g y ,x is deﬁned and inv ertible µ y - a.e. Finally , we must work out the consequences of the “pillow condition” ( 60 ). W e start by ev alu- ating the “whisk ered” comp ositions ρ 2 ( u ) ◦ 1 φ and 1 φ ◦ ρ 1 ( u ), using the form ula ( 48 ) for horizon tal comp osisiton. By the lemma ab ov e, the comp osites ρ 2 ( g ) φ and φρ 1 ( g ) are matrix functors. Hence the 2-isomorphisms [ ρ 2 ( u ) ◦ 1 φ ] and [ 1 φ ◦ ρ 1 ( u )] are necessarily matrix natural transformations. W e can work out their matrix comp onen ts using the deﬁnition of horizontal composition, [ ρ 2 ( u ) ◦ 1 φ ] y ,x = χ 2 ( y )[ h ] and [ 1 φ ◦ ρ 1 ( u )] y ,x = χ 1 ( xg − 1 )[ h ] . The v ertical compositions with φ ( g ) can then b e p erformed with ( 46 ); since all the measures in v olved are equiv alent to eac h other, these compositions reduce to point wise compositions of operators—here m ultiplication of complex num bers. Thus the condition ( 60 ) yields the equation ( χ 2 ( y )[ h ] − χ 1 ( xg − 1 )[ h ]) φ ( g ) y ,x = 0 whic h holds for all h , all y and µ y g -almost every x . Thanks to the cov ariance of the ﬁelds of c haracters, this equation can equiv alen tly b e written as ( χ 2 ( y ) − χ 1 ( x )) Φ g y ,x = 0 (64) for all y and µ y -almost every x . This last equation actually expresses a condition for the family of measures µ y . Indeed, it requires that, for every y , the subset of the x ∈ X suc h that χ 1 ( x ) 6 = χ 2 ( y ) as well as Φ g y ,x 6 = 0 is a null set for the measure µ y . But w e kno w that, for µ y -almost every x , Φ g y ,x is an inv ertible op erator with a non-trivial source space φ y ,x , so that it do es not v anish. Therefore the condition expressed by ( 64 ) is that for each y , the measure µ y is supp orted within the set { x ∈ X | χ 1 ( x ) = χ 2 ( y ) } = χ − 1 1 ( χ 2 ( y )). So, the family µ y is ﬁb erwise. Con versely , given an equiv arian t and ﬁb erwise Y -indexed measurable family µ y of measures on X , a measurable ﬁeld of Hilb ert spaces φ y ,x on Y × X , and a measurable ﬁeld of in vertible linear maps Φ g y ,x satisfying the cocycle condition ( 63 ) µ - a.e. for eac h g , g 0 ∈ G , we can easily construct an intert winer. The pair ( µ y , φ y ,x ) gives a morphism φ : H X → H Y . F or eac h g ∈ G , y ∈ Y , and x ∈ X , w e let φ ( g ) y ,x = Φ g y g,x : [ ρ 2 ( g ) φ ] y ,x → [ φρ 1 ( g )] y ,x This gives a 2-morphism in Meas for each morphism in G . The co cycle condition, and the ﬁberwise prop ert y of µ y , ensure that the equations ( 59 ) and ( 60 ) hold. Giv en that the maps Φ g y ,x are inv ertible, the co cycle condition ( 63 ) implies that g = 1 giv es the iden tity: Φ 1 y ,x = 1 φ y,x µ - a.e. (65) 71 In fact, given given the cocycle condition, this equation is clearly equiv alen t to the statemen t that the maps Φ g y ,x are a.e. -inv ertible. W e also easily get a useful form ula for in verses:  Φ g y ,x  − 1 = Φ g − 1 ( y ,x ) g − 1 µ - a.e. for each g . (66) F ollo wing our classiﬁcation of representations, we noted that only some of them deserve to be called ‘measurable representations’ of a measurable 2-group. Similarly , here we introduce a notion of ‘measurable intert winer’. First, in the theorem, there is no statement to the eﬀect that the linear maps Φ g y ,x : φ y ,x → φ ( y ,x ) g − 1 are ‘measurably indexed’ by g ∈ G . T o correct this, for an intert winer to b e ‘measurable’ w e will demand that Φ g y ,x giv e a measurable ﬁeld of linear op erators on G × Y × X , where the ﬁeld φ y ,x can be thought of as a measurable ﬁeld of Hilb ert spaces on G × Y × X that is independent of its g -co ordinate. Second, in the theorem, for each pair of group elemen ts g , g 0 ∈ G , w e hav e a sep ar ate co cycle condition Φ g 0 g y ,x = Φ g 0 ( y ,x ) g − 1 Φ g y ,x µ - a.e. In other w ords, for each choice of g , g 0 , there is a set U g ,g 0 with µ y ( X − U g ,g 0 ) = 0 for all y , such that the co cycle condition holds on U g ,g 0 . Unless the group G is countable, the union of the sets X − U g ,g 0 ma y hav e p ositive measure. This se ems to cause serious problems for c haracterization of such intert winers, unless we imp ose further conditions. F or an intert winer to b e ‘measurable’, w e will th us demand that the co cycle condition hold outside some null set, indep enden tly of g , g 0 . Similarly , the theorem implies Φ g y ,x is in vertible µ - a.e. , but sep ar ately for each g ; for measurable in tertwiners w e demand inv ertibility outside a ﬁxed null set, independently of g . Let us now formalize these concepts: Deﬁnition 59 L et Φ g y ,x : φ y ,x → φ ( y ,x ) g − 1 b e a me asur able ﬁeld of line ar op er ators on G × Y × X , with Y , X me asur able G -sp ac es. We say Φ is inv ertible at ( y , x ) if Φ g y ,x is invertible for al l g ∈ G ; we say Φ is co cyclic at ( y , x ) if Φ g 0 g y ,x = Φ g 0 ( y ,x ) g − 1 Φ g y ,x for al l g , g 0 ∈ G . Deﬁnition 60 An intertwiner ( φ, Φ , µ ) , of the form describ e d in Thm. 57 , is measurable if: • The ﬁelds Φ g ar e obtaine d by r estriction of a me asur able ﬁeld of line ar op er ators Φ on G × Y × X ; • Φ has a r epr esentat ive (fr om within its µ -class) that is invertible and c o cyclic at al l p oints in some ﬁxe d subset U ⊆ Y × X with ¯ µ y ( Y × X ) = ¯ µ y ( U ) for al l y . Mor e gener al ly a measurable intert winer is an intertwiner that is isomorphic to one like this. The generalization in the last sen tence of this deﬁnition is needed for tw o comp osable measurable in tertwiners to ha ve measurable composite. F rom no w on, w e will alw ays be in terested in measurable in tertwiners b etw een measurable representations; w e sometimes omit the word “measurable” for brevit y , but it is alwa ys implicit. The measurable ﬁeld Φ g y ,x in an intert winer is very similar to a kind of co cycle used in the theory of induced representations on lo cally compact groups (see, for example, the discussion in V aradara jan’s b ook [ 69 , Sec. V.5]). How ev er, one ma jor diﬀerence is that our co cycles here are m uch b etter b ehav ed with resp ect to n ull sets. In particular, we easily ﬁnd that Φ is co cyclic and in vertible at a p oin t, it is satisﬁes the same prop erties everywhere on an G -orbit: 72 Lemma 61 L et Φ g y ,x : φ y ,x → φ ( y ,x ) g − 1 b e a me asur able ﬁeld of line ar op er ators on G × Y × X . If Φ is invertible and c o cyclic at ( y o , x o ) , then it is invertible and c o cyclic at every p oint on the G -orbit of ( y o , x o ) . Pro of: If Φ is inv ertible and co cyclic at ( y o , x o ), then for any g , g 0 w e hav e Φ g 0 ( y o ,x o ) g − 1 = Φ g 0 g y o ,x o  Φ g y o ,x o  − 1 so Φ g 0 at ( y o , x o ) g − 1 is the composite of tw o in vertible maps, hence is in vertible. Since g , g 0 w ere arbitrary , this sho ws Φ is inv ertible everywhere on the orbit. Replacing g 0 in the previous equation with a product g 00 g 0 , and using only the cocycle condition at ( y o , x o ), we easily ﬁnd that Φ is co cyclic at ( y o , x o ) g − 1 for arbitrary g . This lemma immediately implies, for an y measurable intert winer ( φ, Φ , µ ), that a representativ e of Φ may b e c hosen to be inv ertible and co cyclic not only on some set with null compliment, but actually everywhere on an y orbit that meets this set. This fact simpliﬁes man y calculations. Deﬁnition 62 The me asur able ﬁeld of line ar op er ators Φ g y ,x : φ y ,x → φ ( y ,x ) g − 1 on G × Y × X is c al le d a strict G -co cycle if the e quations Φ 1 y ,x = 1 φ y,x and Φ g 0 g y ,x = Φ g 0 ( y ,x ) g − 1 Φ g y ,x hold for all ( g , y , x ) ∈ G × Y × X . An intertwiner (Φ , φ, µ ) for which the me asur e-class of Φ g y ,x has such a strict r epr esentative is a measurably strict intertwiner. An in teresting question is which measurable intert winers are measurably strict. This may b e a diﬃcult problem in general. How ev er, there is one case in whic h it is completely ob vious from Lemma 61 : when the action of G on Y × X is tr ansitive . In fact, it is enough for the G -action on Y × X to b e ‘essen tially transitive’, with resp ect to the family of measures µ . W e introduce a sp ecial case of intert winers for which this is true: Deﬁnition 63 A Y -indexe d me asur able family of me asur es µ y on X is transitiv e if ther e is a single G -orbit o in Y × X such that, for every y ∈ Y , ¯ µ y = δ y ⊗ µ y is supp orte d on o . A transitive in tertwiner is a me asur able intertwiner (Φ , φ, µ ) such that the family µ is tr ansitive. It is often con v enient to ha ve a description of transitiv e families of measures using the measurable ﬁeld µ y of measures on X directly , rather than the associated ﬁb ered measure distribution ¯ µ y . It is easy to chec k that µ y is transitive if and only if there is a G -orbit o ⊆ Y × X such that whenever ( { y } × A ) ∩ o = ∅ , we hav e µ y ( A ) = 0. T ransitive intert winers will play an imp ortan t role in our study of intert winers. Theorem 64 A tr ansitive intertwiner is me asur ably strict. Pro of: The orbit o ⊆ Y × X , on which the measures ¯ µ y are supp orted, is a measurable set (see Lemma 121 ). W e ma y therefore take a representativ e of Φ g y ,x : φ y ,x → φ ( y ,x ) g − 1 for which φ y ,x is trivial on the null set ( Y × X ) − o . The co cycle condition then automatically holds not only on o , b y Lemma 61 , but also on its compliment. In fact, it is clear that a transitive intert winer has an essentially unique ﬁeld representativ e. Indeed, an y tw o representativ es of Φ must b e equal at almost every p oin t on the supp orting orbit, but then Lemma 61 implies they must be equal everywher e on the orbit. 73 Let us turn to the geometric description of in tertwiners. F or simplicity , we restrict our attention to measurably strict intert winers, for which the geometric corresp ondence is clearest. F ollowing Mac key , we can view the measurable ﬁeld φ as a measurable bundle of Hilb ert spaces ov er Y × X : φ   Y × X whose ﬁb er ov er ( y , x ) is the Hilb ert space φ y ,x . As p oin ted out in Section 4.1 , the strict co cycle Φ y ,x can b e viewed as a left action of G on the ‘total space’ φ of this bundle since, b y ( 63 ) and ( 65 ), Φ g : φ → φ satisﬁes Φ g 0 g = Φ g 0 Φ g and Φ 1 = 1 φ . The corresp onding right action g 7→ Φ g − 1 of G on φ is then an action of G ov er the diagonal action on Y × X : φ Φ g − 1 / /   φ   Y × X · C g / / Y × X So, loosely sp eaking, an intert winer can b e viewed as providing a ‘measurable G -equiv ariant bundle of Hilb ert spaces’ ov er Y × X . The asso ciated equiv ariant family of measures µ serv es to indicate, via µ - a.e. equiv alence, when t wo such Hilb ert space bundles actually describ e the same in tertwiner. While these ‘Hilb ert space bundles’ are determined only up to measure-equiv alence, in general, they do share many of the essential features of their counterparts in the top ological category . In particular, the ‘ﬁb er’ φ y ,x is a linear representation of the stabilizer group S y ,x ⊆ G , since the co cyle condition reduces to: Φ s 0 s y ,x = Φ s 0 y ,x Φ s y ,x : φ y ,x → φ y ,x for s, s 0 ∈ S y ,x . Deﬁnition 65 Given any me asur able intertwiner φ = ( φ, Φ , µ ) , we deﬁne the stabilizer repre- sen tation at ( y , x ) ∈ Y × X to b e the line ar r epr esentation of S y ,x = { s ∈ G : ( y , x ) s = ( y , x ) } on φ y ,x deﬁne d by R φ y ,x ( s ) = Φ s y ,x . These r epr esentations ar e deﬁne d µ y -a.e. for e ach y . Along a given G -orbit o in Y × X , the stabilizer groups are all conjugate in G , so if we choose ( y o , x o ) ∈ o with stabilizer S o = S y o ,x o , then the stabilizer representations elsewhere on o can b e view ed as represen tations of S o . Explicitly , s 7→ R φ y ,x ( g − 1 sg ) deﬁnes a linear representation of S o on φ y ,x , where ( y , x ) = ( y o , x o ) g . Moreov er, the cocycle condition 74 implies φ y ,x R φ y ,x ( g − 1 sg ) / / Φ g y ,x   φ y ,x Φ g y ,x   φ y o ,x o R φ y o ,x o ( s ) / / φ y o ,x o comm utes for all s ∈ S o , and all g suc h that ( y , x ) = ( y o , x o ) g . In other words, the maps Φ g are in tertwiners b et ween stabilizer representations. W e thus see that the assignment φ y ,x , Φ g y ,x deﬁnes, for eac h orbit in Y × X , a representation of the stabilizer group as well as a consistent wa y to ‘transp ort’ it along the orbit with inv ertible intert winers. In the case of a transitive intert winer, the only relev ant Hilb ert spaces are the ones ov er the sp ecial orbit o , so w e ma y think of a transitive intert winer as a Hilb ert space bundle ov er a single orbit in Y × X : φ   o W e hav e also observ ed that the Hilbert spaces on the orbit o are uniquely determined, so there is no need to mo d out by µ -equiv alence. W e therefore obtain: Theorem 66 A tr ansitive intertwiner is uniquely determine d by: • A tr ansitive family of me asur es µ y on X , with ¯ µ y supp orte d on the G -orbit o , • A me asur able ﬁeld of line ar op er ators Φ g y ,x : φ y ,x → φ ( y ,x ) g − 1 on G × o that is c o cyclic and invertible at some (and henc e every) p oint. 4.3.3 Structure of 2-in tert winers W e now turn to the problem of classifying the all 2-intert winers b etw een a ﬁxed pair of parallel in tertwiners φ, ψ : ρ 1 → ρ 2 . If ρ 1 and ρ 2 are represen tations of the t yp e describ ed in Thm. 49 , then φ and ψ are, up to equiv alence, of the t yp e describ ed in Thm. 57 . Thus, we let φ and ψ b e giv en resp ectively by the equiv arian t and ﬁb erwise families of measures µ y and ν y , and the (classes of ) ﬁelds of Hilb ert spaces and inv ertible maps φ y ,x , Φ g y ,x and ψ y ,x , Ψ g y ,x . A characterization of 2-in tertwiners betw een such in tertwiners is giv en by the following theorem: Theorem 67 (2-Intert winers) L et ρ 1 , ρ 2 b e r epr esentations on H X and H Y , and let intertwiners φ, ψ : ρ 1 → ρ 2 b e sp e ciﬁe d by the data ( φ, Φ , µ ) and ( ψ , Φ , ν ) as in Thm. 57 . A 2-intertwiner m : φ → ψ is sp e ciﬁe d uniquely by a √ µν -class of ﬁelds of line ar maps m y ,x : φ y ,x → ψ y ,x satisfying Ψ g y ,x m y ,x = m ( y ,x ) g − 1 Φ g y ,x √ µν -a.e. As usual, by √ µν -class of ﬁelds we mean equiv alence class of ﬁelds modulo identiﬁcation of the ﬁelds whic h coincide for all y and √ µ y ν y -almost every x . 75 Pro of: By deﬁnition, a 2-intert winer m b etw een the given intert winers deﬁnes a 2-morphism in Meas b et ween the morphisms ( φ, µ ) and ( ψ , ν ), which satisﬁes the pillow condition ( 21 ), namely ψ ( g ) ·  1 ρ 2 ( g ) ◦ m  =  m ◦ 1 ρ 1 ( g )  · φ ( g ) (67) By Thm. 34 , since m is a measurable natural transformation betw een matrix functors, it is auto- matically a matrix natural transformation. W e th us ha ve merely to sho w that the conditions ( 67 ) imp oses on its matrix comp onen ts m y ,x are precisely those stated in the theorem. First, using Lemma 58 , the t wo whisk ered comp osites 1 ρ 2 ( g ) ◦ m and m ◦ 1 ρ 1 ( g ) in ( 67 ) are matrix natural transformations whose ﬁelds of op erators read [ 1 ρ 2 ( g ) ◦ m ] y ,x = m y g,x and [ m ◦ 1 ρ 1 ( g ) ] y ,x = m y ,xg − 1 resp ectiv ely . Next, we need to p erform the vertical compositions on b oth sides of the equalit y ( 67 ). F or this, w e use the general form ula ( 46 ) for vertical comp osition of matrix natural transformations, whic h in volv es the square ro ot of a pro duct of three Radon-Nykodym deriv ativ es. These deriv atives are, in the present con text: d ν g y d µ y g d ν y g d ν g y d µ y g d ν y g and d ν g y d µ y g d µ g y d ν g y d µ y g d µ g y (68) for the left and right sides of ( 67 ), resp ectively . No w the equiv ariance of the families µ y , ν y yields d µ y g d ν y g = d µ y g d ν g y d ν g y d ν y g , d µ g y d ν g y = d µ y g d ν g y d µ g y d ν g y so that b oth pro ducts in ( 68 ) reduce to d ν g y d µ y g d µ y g d ν g y Thanks to the chain rule ( 29 ), namely d µ d ν d ν d µ = 1 √ µν − a.e. , this last term equals 1 almost everywhere for the geometric mean of the source and target meas ures for the 2-morphism describ ed b y either side of ( 67 ). This sho ws that the v ertical composition reduces to the point wise comp osition of the ﬁelds of op erators. Performing this comp osition and reindexing, ( 67 ) takes the form Ψ g y ,x m y ,x = m ( y ,x ) g − 1 Φ g y ,x (69) as we wish to show. This equation holds for all y and √ µ y ν y -almost every x . Th us, a 2-intert winer m : φ ⇒ ψ essentially assigns linear maps m y ,x : φ y ,x → ψ y ,x to elements ( y , x ) ∈ Y × X , in such a w ay that ( 69 ) is satisﬁed. Diagrammatically , this equation can b e written: φ y ,x Φ g y,x / / m y,x   φ ( y ,x ) g − 1 m ( y,x ) g − 1   ψ y ,x Ψ g y,x / / ψ ( y ,x ) g − 1 76 whic h commutes √ µν - a.e. for each g . It is helpful to think of this as a generalization of the equation for an in tertwiner b et ween ordinary group representations. Indeed, when restricted to elements of the stabilizer S y ,x ⊆ G of ( y , x ) under the diagonal action on Y × X , it b ecomes: R ψ y ,x ( s ) m y ,x = m y ,x R φ y ,x ( s ) s ∈ S y ,x This states that m y ,x is an intert wining op erator, in the ordinary group-theoretic sense, b etw een the stabilizer representations of φ and ψ . If equation ( 69 ) is satisﬁed everywhere along some G -orbit o in Y × X , the maps m y ,x of such an assignmen t are determined by the one m o : φ o → ψ o assigned to a ﬁxed p oin t ( y o , x o ), since for ( y , x ) = ( y o , x o ) g − 1 , we hav e m y ,x = Ψ g o m o (Φ g o ) − 1 If the measure class of m y ,x has a representativ e for whic h equation ( 69 ) is satisﬁed everywhere, m y ,x is determined by its v alues at one representativ e of each G -orbit. In the previous tw o sections, we introduced ‘measurable’ v ersions of representations and inter- t winers. F or 2-intert winers, there are no new data indexed by morphisms or 2-morphisms in our 2-group. Since a 2-group has a unique ob ject, there are no new measurability conditions to imp ose. W e thus make the following simple deﬁnition. Deﬁnition 68 A measurable 2-intert winer is a 2-intertwiner b etwe en me asur able intertwiners, as classiﬁe d in Thm. 67 . 4.4 Equiv alence of represen tations and of intert winers In the previous sections we ha ve c haracterized representations of a 2-group G on measurable cat- egories, as w ell as intert winers and 2-intert winers. In this section we w ould like to describ e the e quivalenc e classes of representations and in tertwiners. The general notions of equiv alence for repre- sen tations and intert winers was in tro duced, for a general target 2-category , in Section 2.2.3 . Recall from that section that tw o representations are equiv alen t when there is a (weakly) inv ertible in- tert winer b etw een them. In the case of representations in Meas , it is natural to sp ecialize to ‘measurable equiv alence’ of representations: Deﬁnition 69 Two me asur able r epr esentations of a 2-gr oup ar e measurably equiv alent if they ar e r elate d by a p air of me asur able intertwiners that ar e we ak inverses of e ach other. In what follows, b y ‘equiv alence’ of representations w e alwa ys mean measurable equiv alence. Similarly , recall that t wo parallel intert winers are equiv alent when there is an inv ertible 2- in tertwiner b et ween them. Since measurable 2-intert winers are simply 2-in tertwiners with mea- surable source and target, there are no extra conditions necessary for equiv alent in tertwiners to b e ‘measurably’ equiv alen t. Let ρ 1 and ρ 2 b e me asurable representations of G = ( G, H , B ) on the measurable categories H X and H Y deﬁned by G -equiv ariant bundles χ 1 : X → H ∗ and χ 2 : Y → H ∗ . W e use the same sym b ol “ C ” for the action of G on b oth X and Y . The following theorem explains the geometric meaning of equiv alence of representations. Theorem 70 (Equiv alen t represen tations) Two me asur able r epr esentations ρ 1 and ρ 2 ar e e quiv- alent if and only if the c orr esp onding G -e quivariant bund les χ 1 : X → H ∗ and χ 2 : Y → H ∗ ar e isomorphic. That is, ρ 1 ∼ ρ 2 if and only if ther e is an invertible me asur able function f : Y → X that is G -e quivariant: f ( y C g ) = f ( y ) C g 77 and ﬁb er-pr eserving: χ 1 ( f ( y )) = χ 2 ( y ) . Pro of: Suppose ﬁrst the representations are equiv alen t, and let φ b e an in vertible in tertwiner b et w een them. Recall that eac h intert winer deﬁnes a morphism in Meas ; moreov er, as sho wn by the la w ( 23 ), the morphism deﬁned b y the comp osition of tw o in tertwiners in the 2-category of represen tations 2Rep ( G ) coincides with the comp osition of the tw o morphisms in Meas . As a consequence, the inv ertibilit y of φ yields the inv ertibilit y of its asso ciated morphism ( φ, µ ). By Theorem 44 , this means the measures µ y are equiv alen t to Dirac measures δ f ( y ) for some inv ertible (measurable) function f : Y → X . On the other hand, by deﬁnition of an intert winer, the family µ y is equiv arian t. This means here that the measure δ f ( y C g ) is equiv alen t to the measure δ g f ( y ) = δ f ( y ) C g . Thus, the t wo Dirac measures c harge the same point, so f ( y C g ) = f ( y ) C g . W e also know that the supp ort of µ y , that is, the singlet { f ( y ) } , is included in the set { x ∈ X | χ 1 ( x ) = χ 2 ( y ) } . This yields χ 1 ( f ( y )) = χ 2 ( y ). Con versely , supp ose there is a function f which satisﬁes the conditions of the theorem. One can immediately construct from it an inv ertible in tertwiner b et ween the t wo representations, by considering the family of measures δ f ( y ) , the constant ﬁeld of one-dimensional spaces C and the constan t ﬁeld of identit y maps 1 . W e now consider tw o in tertwiners φ and ψ betw een the same pair of representations ρ 1 and ρ 2 , sp eciﬁed by equiv ariant and ﬁberwise families of measures µ y and ν y , and classes of ﬁelds φ y ,x , Φ g y ,x and ψ y ,x , Ψ g y ,x . As we kno w, these carry standard linear representations R φ y ,x and R ψ y ,x of the stabilizer S y ,x of ( y , x ) under the diagonal action of G , resp ectiv ely in the Hilb ert spaces φ y ,x and ψ y ,x . The following prop osition gives necessary conditions for intert winers to b e equiv alen t: Prop osition 71 If the intertwiners φ and ψ ar e e quivalent, then for al l y ∈ Y , µ y and ν y ar e in the same me asur e class and the stabilizer r epr esentations R φ y ,x and R ψ y ,x ar e e quivalent for µ y -almost every x ∈ X . Pro of: Assume φ ∼ ψ , and let m : φ ⇒ ψ b e an inv ertible 2-intert winer. Recall that any 2- in tertwiner deﬁnes a 2-morphism in Meas ; moreov er, the morphism deﬁned b y the comp osition of t wo 2-intert winers in the 2-category of representations 2Rep ( G ) coincides with the comp osition of the tw o 2-morphisms in Meas . As a consequence, the in vertibilit y of m yields the inv ertibility of its asso ciated 2-morphism. By Thm. 43 , this means that the measures of the source and the target of m are equiv alen t. Thus, for all y , µ y and ν y are in the same measure class. W e kno w that m deﬁnes a µ -class of ﬁelds of linear maps m y ,x : φ y ,x → ψ y ,x , such that for all y and µ y -almost every x , m y ,x in tertwines the stabilizer representations R φ y ,x and R ψ y ,x . Moreov er, since m is inv ertible as a 2-morphism in Meas , we know b y Thm. 43 that the maps m y ,x are in vertible. Th us, for all y and almost every x , the t wo group representations R φ y ,x and R ψ y ,x are equiv alen t. This prop osition admits a partial conv erse, if one restricts to transitive intert winers: Prop osition 72 (Equiv alen t transitiv e in tertwiners) Supp ose the intertwiners φ and ψ ar e tr ansitive. If for al l y , µ y and ν y ar e in the same me asur e class and the stabilizer r epr esentations R φ y ,x and R ψ y ,x ar e e quivalent for µ y -almost every x ∈ X , then φ and ψ ar e e quivalent. Pro of: Let o b e an orbit of Y × X such that µ y ( A ) = 0 for each { y } × A in ( Y × X ) − o . First of all, if the family µ y is trivial, so is ν y ; and in that case the intert winers are obviously equiv alent. 78 Otherwise, there is a p oin t u o = ( y o , x o ) in o at whic h the representations R φ o and R ψ o of the stabilizer S o are equiv alent. Now, assume the t wo intert winers are sp eciﬁed by the assignments of Hilb ert spaces φ u , ψ u and in vertible maps Φ g u : φ u → φ ug − 1 and Ψ g u : ψ u → ψ ug − 1 to the points of the orbit, satisfying co cycle conditions. These yield, for u = u o k − 1 , Φ g u = Φ g k o  Φ k o  − 1 , Ψ g u = Ψ g k o  Ψ k o  − 1 (70) where φ o , Φ g o denote the v alue of the ﬁelds at the p oin t u o . Now, let m o : φ o → ψ o b e an inv ertible in tertwiner betw een the representations R φ o and R ψ o . Then for u = u o k − 1 , the formula m u = Ψ k o m o  Φ k o  − 1 deﬁnes inv ertible maps m u : φ u → ψ u . It is then straightforw ard to show that ( 70 ) yields the in tertwining equation Ψ g u m u = m ug − 1 Φ g u . Th us, the maps m u deﬁne a 2-intert winer m : φ ⇒ ψ . W e furthermore deduce from the Thm. 43 that m is inv ertible. Thus, the intert winers φ and ψ are equiv alen t. In fact, an y transitive intert winer is equiv alent to one for whic h the ﬁeld of Hilb ert spaces φ y ,x is c onstant , φ y ,x ≡ φ o . More generally , this is true, for an y in tert winer, on an y single G -orbit o ⊆ Y × X on whic h the co cycle is strict. T o see this, pick u o = ( y o , x o ) in o and let S o = S y o ,x o b e its stabilizer. Since o ∼ = G/S o is a homogeneous space of G , there is a measurable section (see Lemma 123 ) σ : o → G deﬁned by the prop erties σ ( y o , x o ) = 1 ∈ G and ( y o , x o ) σ ( y , x ) = ( y , x ) If we deﬁne φ o = φ y o ,x o , then for each ( y , x ) ∈ o , w e get a speciﬁc isomorphism of φ y ,x with φ o : α y ,x = Φ σ ( y,x ) y ,x : φ y ,x → φ o If we then deﬁne P g y ,x = α ( y ,x ) g − 1 Φ g y ,x ( α y ,x ) − 1 a straightforw ard calculation shows that P is co cyclic: P g 0 g y ,x = α ( y ,x )( g 0 g ) − 1 Φ g 0 g y ,x ( α y ,x ) − 1 = α ( y ,x ) g − 1 g 0− 1 Φ g 0 ( y ,x ) g − 1  α ( y ,x ) g − 1  − 1 α ( y ,x ) g − 1 Φ g y ,x ( α y ,x ) − 1 = P g 0 ( y ,x ) g − 1 P g y ,x W e thus get a new measurable intert winer ( φ o , P , µ ), which is equiv alen t to the original intert winer ( φ, Φ , µ ) via an inv ertible 2-in tertwiner deﬁned by α y ,x . In geometric language, this sho ws that an y ‘measurable G -equiv ariant bundle’ can be trivialized b y via a ‘measurable bundle isomorphism’, while maintaining G -equiv ariance. So there are no global ‘t wists’ in suc h ‘bundles’, as there are in top ological or smooth categories. 79 4.5 Op erations on representations Some of the most in teresting features of ordinary group represen tation theory arise because there are natural notions of ‘direct sum’ and ‘tensor pro duct’, which w e can use to build new representations from old. The same is true of 2-group represen tation theory . In the group case, these sums and pro ducts of representations are built from the corresponding op erations in V ect. Lik ewise, for sums and pro ducts in our representation theory , we ﬁrst need to develop such notions in the 2-category Meas . Th us, in this section, w e ﬁrst consider direct sums and tensor pro ducts of measurable cate- gories and measurable functors. W e then use these to describ e direct sums and tensor pro ducts of measurable representations, and measurable intert winers. 4.5.1 Direct sums and tensor pro ducts in Meas W e now introduce imp ortant op erations on ‘higher v ector spaces’, analogous to taking ‘tensor prod- ucts’ and ‘direct sums’ of ordinary vector spaces. These op erations are well understo od in the case of 2V ect [ 18 , 43 ]; here we discuss their generalization to Meas . W e b egin with ‘direct sums’. As emphasized by Barrett and Mack aa y [ 18 ] in the case of 2V ect , there are sev eral levels of ‘linear structure’ in a 2-category of higher v ector spaces. In ordinary linear algebra, the set V ect( V , V 0 ) of all linear maps b et w een ﬁxed vector spaces V , V 0 is itself a ve ctor sp ac e . But the c ate gory V ect has a similar structure: we can take direct sums of both vector spaces and linear maps, making V ect in to a (symmetric) monoidal c ate gory . In c ate goriﬁe d linear algebra, this ‘micro cosm’ of linearity go es one lay er deeper. Here we can add 2-maps b et ween ﬁxed maps, so the top-dimensional hom sets form vector spaces. But there are no w tw o distinct wa ys of taking ‘direct sums’ of maps . Namely , since w e can think of a map b etw een 2-v ector spaces as a ‘matrix of vector spaces’, we can either take the ‘matrix of direct sums’, when the matrices hav e the same size, or, more generally , we can take the ‘direct sum of matrices’. These ideas lead to tw o distinct operations whic h w e call the ‘direct sum’ and the ‘2-sum’. The direct sum leads to the idea that the hom c ate gories , consisting of all maps b et ween ﬁxed 2-v ector spaces, as w ell as 2-maps b et ween those, should b e monoidal categories; the second leads to the idea that a 2-category of 2-vector spaces should itself b e a ‘monoidal 2-category’. Let us make these ideas more precise, in the case of Meas . The most ob vious lev el of linear structure in Meas applies only at 2-morphism level. Since sums and constan t m ultiples of bounded natural transformations are b ounded, the set of measurable natural transformations b et w een ﬁxed measurable functors is a complex v ector space. Next, ﬁxing tw o measurable spaces X and Y , let Mat( X , Y ) b e the category with: • matrix functors ( T , t ) : H X → H Y as ob jects • matrix natural transformations as morphisms Mat( X, Y ) is c learly a linear category , since composition is bilinear with resp ect to the vector space structure on each hom set. Next, there is a notion of dir e ct sum in Mat( X , Y ), which corresponds to the intuitiv e idea of a ‘matrix of direct sums’. In tuitively , given tw o matrix functors ( T , t ) , ( T 0 , t 0 ) ∈ Mat( X, Y ), w e would lik e to form a new matrix functor with matrix comp onen ts T y ,x ⊕ T 0 y ,x . This makes sense as long as the families of measures t y and t 0 y are equiv alen t, but in general we must be a bit more careful. W e ﬁrst deﬁne a y -indexed measurable family of measures t ⊕ t 0 on X b y ( t ⊕ t 0 ) y = t y + t 0 y (71) 80 This will b e the family of measures for a matrix functor we will call the direct sum of T and T 0 . T o obtain the corresponding ﬁeld of Hilbert spaces, w e use the Lebesgue decompositions of the measures with resp ect to eac h other: t = t t 0 + t t 0 , t 0 = t 0 t + t 0 t with t t 0  t 0 and t t 0 ⊥ t 0 , and similarly t 0 t  t and t 0 t ⊥ t . The subscript y indexing the measures has b een dropp ed for simplicity . The measures t t 0 and t 0 t are equiv alen t, and these are singular with resp ect to b oth t t 0 and t 0 t ; moreov er, these latter tw o measures are m utually singular. F or each y ∈ Y , w e can th us write X as a disjoint union X = A y q B y q C y with t t 0 supp orted on A y , t 0 t supp orted on B y , and t t 0 , t 0 t supp orted on C y . (In particular, t y is supp orted on A y q C y and t 0 y is supported on B y q C y .) W e then deﬁne a new ( t ⊕ t 0 )-class of ﬁelds of Hilb ert spaces T ⊕ T 0 b y setting [ T ⊕ T 0 ] y ,x =    T y ,x x ∈ A y T 0 y ,x x ∈ B y T y ,x ⊕ T 0 y ,x x ∈ C y (72) The ( t ⊕ t 0 )-class do es not dep end on the c hoice of sets A y , B y , C y , so the data ( T ⊕ T 0 , t ⊕ t 0 ) giv e a well deﬁned matrix functor H X → H Y , an ob ject of Mat( X , Y ). W e call this the direct sum of ( T , t ) and ( T 0 , t 0 ), and denote it b y ( T , t ) ⊕ ( T 0 , t 0 ), or simply T ⊕ T 0 for short. Note that this direct sum is b oundedly naturally isomorphic to the functor mapping H ∈ H X to the H Y -ob ject with comp onen ts ( T H ) y ⊕ ( T 0 H ) y . There is an ob vious unit ob ject 0 ∈ Mat( X , Y ) for the tensor pro duct, deﬁned by the trivial Y -indexed family of measures on X , µ y ≡ 0. In fact, this is a strict unit ob ject, meaning that we ha ve the equations: ( T , t ) ⊕ 0 = ( T , t ) = 0 ⊕ ( T , t ) for any ob ject ( T , t ) ∈ Mat( X , Y ). W e migh t exp ect these to hold only up to isomorphism, but since t + 0 = t , and T is deﬁned up to measure-class, the equations hold strictly . Also, given any pair of 2-morphisms in Mat( X , Y ), say matrix natural transformations α and α 0 : H X T ,t * * U,u 4 4 H Y α   and H X T 0 ,t 0 * * U 0 ,u 0 4 4 H Y α 0   w e can construct their direct sum, a matrix natural transformation H X T ⊕ T 0 ,t ⊕ t 0 + + U ⊕ U 0 ,u ⊕ u 0 3 3 H Y α ⊕ α 0   as follows. Again, dealing with measure-classes is the tricky part. This time, let us decomp ose X in t wo w ays, for eac h y : X = A y q B y q C y = A 0 y q B 0 y q C 0 y 81 with t y supp orted on A y q C y , u y on B y q C y , and t y and u y equiv alen t on C y , and similarly , t 0 y supp orted on A 0 y q C 0 y , u 0 y on B 0 y q C 0 y , and t 0 y and u 0 y equiv alen t on C 0 y . W e then deﬁne [ α ⊕ α 0 ] y ,x =        α y ,x ⊕ α 0 y ,x x ∈ C y ∩ C 0 y α y ,x x ∈ C y − C 0 y α 0 y ,x x ∈ C 0 y − C y 0 otherwise (73) F or this to determine a matrix natural transformation betw een the indicated matrix functors, w e m ust show that our form ula determines the ﬁeld of linear op erators for eac h y and µ y -almost every x , where µ y = q ( t y + t 0 y )( u y + u 0 y ) On the set C y ∩ C 0 y , the measures t y , u y , t 0 y , u 0 y are all equiv alen t, hence are also equiv alen t to µ , so α is clearly determined on this set. On the set C y − C 0 y , w e hav e t y ∼ u y , while t 0 y ⊥ u 0 y . Using these facts, we show that µ y ∼ q ( t y + t 0 y )( t y + u 0 y ) ∼ t y + q t 0 y u 0 y = t y ∼ p t y u y on C y − C 0 y . But the matrix comp onen ts of α ⊕ α given in ( 73 ) are determined precisely √ t y u y - a.e. , hence µ y - a.e. on C y − C 0 y . By an identical argumen t with primed and un-primed symbols reversing roles, w e ﬁnd µ y ∼ q t 0 y u 0 y on C 0 y − C y . So the comp onents of α ⊕ α 0 are determined µ y - a.e. for each y , hence give a matrix natural trans- formation. W e ha v e deﬁned the ‘direct sum’ in Mat( X, Y ) as a binary op eration on ob jects (matrix functors) and a binary operation on morphisms (matrix natural transformations). One can c heck that the direct sum is functorial, i.e. it resp ects comp osition and identities: ( β · α ) ⊕ ( β 0 · α 0 ) = ( β ⊕ β 0 ) · ( α ⊕ α 0 ) and 1 T ⊕ 1 T 0 = 1 T ⊕ T 0 . Deﬁnition 73 The direct sum in Mat( X , Y ) is the functor: ⊕ : Mat( X, Y ) × Mat( X , Y ) → Mat( X , Y ) . deﬁne d by • The dir e ct sum of obje cts T , T 0 ∈ Mat( X , Y ) is the obje ct T ⊕ T 0 sp e ciﬁe d by the family of me asur es t ⊕ t 0 given in ( 71 ), and by the t ⊕ t 0 -class of ﬁelds [ T ⊕ T 0 ] y ,x given in ( 72 ); • The dir e ct sum morphisms α : T → U and α 0 : T 0 → U 0 is the morphism α ⊕ α 0 : T ⊕ T 0 → U ⊕ U 0 sp e ciﬁe d by the p ( t + u )( t 0 + u 0 ) -class of ﬁelds of line ar maps given in ( 73 ). 82 The direct sum can be used to promote Mat( X , Y ) to a monoidal category . There is an obvious ‘asso ciator’ natural transformation; namely , giv en ob jects T , T 0 , T 00 ∈ Mat( X, Y ), w e get a morphism A T ,T 0 ,T 00 : ( T ⊕ T 0 ) ⊕ T 00 → T ⊕ ( T 0 ⊕ T 00 ) obtained by using the usual asso ciator for direct sums of Hilb ert spaces, on the common supp ort of the resp ectiv e measures t , t 0 , and t 00 . The left and right ‘unit la ws’, as mentioned already , are iden tity morphisms. A straightforw ard exercise sho ws that that Mat( X , Y ) b ecomes a monoidal category under direct sum. There is also an obvious ‘symmetry’ natural transformation in Mat( X, Y ), S T ,T 0 : T ⊕ T 0 → T 0 ⊕ T making Mat( X , Y ) into a symmetric monoidal category . W e can go one step further. Given any measurable categories H and H 0 , the ‘hom-category’ Meas ( H , H 0 ) has • measurable functors T : H → H as ob jects • measurable natural transformations as morphisms An imp ortan t corollary of Thm. 34 is that this category is equiv alen t to some Mat( X , Y ). Picking an adjoint pair of equiv alences: Meas ( H , H 0 ) F / / Mat( X, Y ) F o o w e can transp ort the (symmetric) monoidal structure on Mat( X , Y ) to one on Meas ( H , H 0 ) by a standard pro cedure. F or example, we deﬁne a tensor pro duct of T , T 0 ∈ Meas ( H , H 0 ) by T ⊕ T 0 = F ( F ( T ) ⊕ F ( T 0 )) . This provi des a wa y to take direct sums of arbitrary parallel measurable functors, and arbitrary measurable natural transformations b et ween them. W e no w explain the notion of ‘2-sum’, which is a kind of sum that applies not only to measurable functors and natural transformations, like the direct sum deﬁned ab o ve, but also to measurable categories themselves. First, we deﬁne to 2-sum of measurable categories of the form H X b y the form ula H X  H X 0 = H X q X 0 where q denotes disjoin t union. Th us, an ob ject of H X  H X 0 consists of a measurable ﬁeld of Hilb ert spaces on X , and one on X 0 . Next, for arbitrary matrix functors ( T , t ) : H X → H Y and ( T 0 , t 0 ) : H X 0 → H Y 0 , w e will deﬁne a matrix functor ( T  T 0 , t  t 0 ) called the 2-sum of T and T 0 . Intuitiv ely , whereas the ‘direct sum’ was lik e a ‘matrix of direct sums’, the ‘2-sum’ should b e like a ‘direct sum of matrices’. Th us, we use the ﬁelds of Hilb ert spaces T on Y × X and T 0 on Y 0 × X 0 to deﬁne a ﬁeld T  T 0 on Y q Y 0 × X q X 0 , giv en by [ T  T 0 ] y ,x =    T y ,x ( y , x ) ∈ Y × X T 0 y ,x ( y , x ) ∈ Y 0 × X 0 0 otherwise (74) 83 This is well deﬁned on measure-equiv alence classes, almost everywhere with resp ect to the Y q Y 0 - indexed family t  t 0 of measures on X q X 0 , deﬁned by: [ t  t 0 ] y =  t y y ∈ Y t 0 y y ∈ Y 0 (75) In this deﬁnition we ha ve iden tiﬁed t y with its obvious extension to a measure on X q X 0 . Finally , supp ose we hav e tw o arbitrary matrix natural transformations, deﬁned b y the ﬁelds of linear maps α y ,x : T y ,x → U y ,x and α 0 y 0 ,x 0 : T 0 y 0 ,x 0 → U 0 y 0 ,x 0 . F rom these, we construct a new ﬁeld of maps from [ T  T 0 ] y ,x to [ U  U 0 ] y ,x , given by [ α  α 0 ] y ,x =    α y ,x ( y , x ) ∈ Y × X α 0 y ,x ( y , x ) ∈ Y 0 × X 0 0 otherwise (76) This is determined p ( t  t 0 )( u  u 0 )- a.e. , and hence deﬁnes a matrix natural transformation α  α 0 : T  T 0 ⇒ U  U 0 . Deﬁnition 74 The term 2-sum r efers to any of the fol lowing binary op er ations, deﬁne d on c ertain obje cts, morphisms, and 2-morphisms in Meas : • The 2-sum of me asur able c ate gories H X and H X 0 is the me asur able c ate gory H X  H X 0 = H X q X 0 ; • The 2-sum of matrix functors ( T , t ) : H X → H Y and ( T 0 , t 0 ) : H X 0 → H Y 0 is the matrix functor ( T  T 0 , t  t 0 ) : H X q X 0 → H Y q Y 0 sp e ciﬁe d by the family of me asur es t  t 0 given in ( 75 ) and the class of ﬁelds T  T 0 given in ( 74 ); • The 2-sum of matrix natur al tr ansformations α : ( T , t ) ⇒ ( U, u ) and α 0 : ( T 0 , t 0 ) ⇒ ( U, u 0 ) is the matrix natur al tr ansformation α  β : ( T  T 0 , t  t 0 ) ⇒ ( U  U 0 , u  u 0 ) sp e ciﬁe d by the class of ﬁelds of line ar op er ators given in ( 76 ). It should b e possible to extend the notion of 2-sum to apply to arbitrary ob jects, morphisms, or 2-morphisms in Meas , and deﬁne additional structure so that Meas b ecomes a ‘monoidal 2- category’. While we b eliev e our limited deﬁnition of ‘2-sum’ is a go od starting p oin t for a more thorough treatmen t, we make no such attempts here. F or our immediate purp oses, it suﬃces to kno w how to take 2-sums of ob jects, morphisms, and 2-morphisms of the sp ecial t yp es described. There is an important relationship b et w een the direct sum ⊕ and the 2-sum  . Giv en arbitrary— not necessarily parallel—matrix functors ( T , t ) : H X → H Y and ( T 0 , t 0 ) : H X 0 → H Y 0 , their 2-sum can b e written as a direct sum: T  T 0 ∼ = [ T  0 0 ] ⊕ [0  T 0 ] (77) Here 0 and 0 0 denote the unit ob jects in the monoidal categories Mat( X , Y ) and Mat( X 0 , Y 0 ). A similar relation holds for matrix natural transformations. W e no w brieﬂy discuss ‘tensor pro ducts’. As with the additiv e structures discussed abov e, there ma y b e multiple la y ers of related multiplicativ e structures. In particular, we can presumably use the ordinary tensor pro duct of Hilb ert spaces and linear maps to turn each Mat( X , Y ), and ultimately eac h Meas ( H , H 0 ), into a (symmetric) monoidal category . But, we should also b e able to turn Meas itself into a monoidal 2-category , using a ‘tensor 2-pro duct’ analogous to the ‘direct 2-sum’. W e shall not develop these ideas in detail here, but it is p erhaps worth while outlining the general structure we exp ect. First, the tensor pro duct in Mat( X , Y ) should b e giv en as follows: 84 • Given ob jects ( T , t ), ( T 0 , t 0 ), deﬁne their tensor product ( T ⊗ T 0 , t ⊗ t 0 ) b y the family of measures ( t ⊗ t 0 ) y = q t y t 0 y and the ﬁeld of Hilb ert spaces [ T ⊗ T 0 ] y ,x = T y ,x ⊗ T y ,x • Given morphisms α : T → U and α 0 : T 0 → U 0 , deﬁne their tensor pro duct α ⊗ α 0 : T ⊗ T 0 → U ⊗ U 0 b y the class of ﬁelds deﬁned b y ( α ⊗ α 0 ) y ,x = α y ,x ⊗ α 0 y ,x These are simpler than the corresp onding formulae for the direct sum, as null sets turn out to be easier to handle. As with the direct sum, we expect the tensor pro duct to give Mat( X , Y ) the structure of a symmetric monoidal category , allowing us to transp ort this structure to any hom- category Meas ( H , H 0 ) in Meas . Next, let us describ e the ‘tensor 2-pro duct’. • Given t wo measurable categories of the form H X and H X 0 , we deﬁne their tensor 2-pro duct to b e H X  H X 0 := H X × X 0 • Given matrix functors ( T , t ) : H X → H Y and ( T 0 , t 0 ) : H X 0 → H Y 0 , deﬁne their tensor 2- pro duct to be the matrix functor ( T  T 0 , t  t 0 ) deﬁned by the Y × Y 0 -indexed family of measures on X × X 0 [ t  t 0 ] y ,y 0 = t y ⊗ t 0 y 0 , where ⊗ on the right denotes the ordinary tensor pro duct of measures, and the ﬁeld of Hilbert spaces [ T  T 0 ] ( y ,y 0 ) , ( x,x 0 ) = T y ,x ⊗ T y 0 ,x 0 . • Given matrix natural transformations α : ( T , t ) ⇒ ( U, u ) and α 0 : ( T 0 , t 0 ) ⇒ ( U, u 0 ), deﬁne their tensor 2-product to b e the matrix natural transformation α  β : ( T  T 0 , t  t 0 ) ⇒ ( U  U 0 , u  u 0 ) sp eciﬁed b y: [ α  α 0 ] ( y ,y 0 ) , ( x,x 0 ) = α y ,x ⊗ α y 0 ,x 0 determined almost everywhere with respect to the family of geometric mean measures: p ( t  t 0 )( u  u 0 ) = √ tu  √ t 0 u 0 As with the 2-sum, it should b e p ossible to use this tensor 2-pro duct to make Meas in to a monoidal 2-category . W e lea ve this to further work. 4.5.2 Direct sums and tensor pro ducts in 2Rep ( G ) No w let G b e a skeletal measurable 2-group, and consider the representation 2-category 2Rep ( G ) of (measurable) representations of G in Meas . Monoidal structures in Meas give rise to monoidal structures in this representation category in a natural wa y . 85 Let us consider the v arious notions of ‘sum’ that 2Rep ( G ) inherits from Meas . First, and most ob vious, since the 2-morphisms in Meas b et w een a ﬁxed pair of morphisms form a vector space, so do the 2-intert winers b et ween ﬁxed in tertwiners. Next, ﬁx t wo representations ρ 1 and ρ 2 , on the measurable categories H X and H Y , resp ectively . An intert winer φ : ρ 1 → ρ 2 giv es an ob ject of φ ∈ Meas ( H X , H Y ) and for each g ∈ G a morphism in Meas ( H X , H Y ). Since Meas ( H X , H Y ) is equiv alen t to Mat( X , Y ), the former becomes a symmetric monoidal category with direct sum, and this in turn induces a direct sum of intert winers b et ween ρ 1 and ρ 2 . W e get a direct sum of 2-in tertwiners in an analogous wa y . Deﬁnition 75 L et ρ 1 , ρ 2 b e r epr esentations on H X and H Y . The direct sum of intert winers φ, φ 0 : ρ 1 → ρ 2 is the intertwiner φ ⊕ φ 0 : ρ 1 → ρ 2 given by the morphism φ ⊕ φ 0 in Meas , to gether with the 2-morphisms φ ( g ) ⊕ φ 0 ( g ) in Meas . The direct sum of 2-in tertwiners m : φ → ψ and m 0 : φ 0 → ψ 0 is the 2-intertwiner given by the me asur able natur al tr ansformation m ⊕ m 0 : φ ⊕ φ 0 → ψ ⊕ ψ 0 . The intert winers deﬁne families of measures µ y and µ 0 y , and classes of ﬁelds of Hilb ert spaces φ y ,x and φ 0 y ,x and inv ertible maps Φ g y ,x and Φ 0 g y ,x that are in vertible and co cyclic. It is straightforw ard to deduce the structure of the direct sum of intert winers in terms of these data: Prop osition 76 L et φ = ( φ, Φ , µ ) , φ 0 = ( φ 0 , Φ 0 , µ 0 ) b e me asur able intertwiners with the same sour c e and tar get r epr esentations. Then the intertwiner φ ⊕ φ 0 sp e ciﬁe d by the family of me asur es µ + µ 0 , and the classes of ﬁelds φ y ,x ⊕ φ 0 y ,x and Φ g y ,x ⊕ Φ 0 g y ,x , is a dir e ct sum for φ and φ 0 . The in tertwiner sp eciﬁed by the family of trivial measures, µ y ≡ 0, pla ys the role of unit for the direct sum. This unit is the null intert winer b et ween ρ 1 and ρ 2 . Finally , 2Rep ( G ) inherits a notion of ‘2-sum’. W e b egin with the representations. Deﬁnition 77 The 2-sum of represen tations ρ  ρ 0 is the r epr esentation deﬁne d by ( ρ  ρ 0 )( ς ) = ρ ( ς )  ρ 0 ( ς ) wher e ς denotes the obje ct ? , or any morphism or 2-morphism in G . W e immediately deduce, from the deﬁnition of the 2-sum in Meas , the structure of the 2-sum of represen tations: Prop osition 78 L et ρ , ρ 0 b e me asur able r epr esentations of G = ( G, H , B ) , with c orr esp onding e quiv- ariant maps χ : X → H ∗ , χ 0 : X → H ∗ . The 2-sum of represen tations ρ  ρ 0 is the r epr esentation on the me asur able c ate gory H X q X 0 , sp e ciﬁe d by the action of G induc e d by the actions on X and X 0 , and the obvious e quivariant map χ q χ 0 : X q X 0 → H ∗ . The empty space X = ∅ deﬁnes a represen tation 1 whic h pla ys the role of unit elemen t for the direct sum. This unit element is the n ull representation . There is a notion of 2-sum for in tertwiners, which allows one to deﬁne the sum of in tertwiners that are not necessarily parallel. This notion can essentially b e deduced from that of the direct sum, using ( 77 ). Indeed, if φ = ( φ, Φ , µ ) is a measurable intert winer, a 2-sum of the form φ  0 is simply giv en by the trivial extensions of the ﬁelds φ, Φ , µ to a disjoint union, and lik ewise for 0  φ 0 ; w e then simply write φ  φ 0 as a direct sum via ( 77 ) and the analogous equation for 2-morphisms. There should also b e notions of ‘tensor pro duct’ and ‘tensor 2-pro duct’ in the representation 2-category 2Rep ( G ). Since w e hav e not constructed these pro ducts in detail in Meas , we shall not giv e the details here; the constructions should b e analogous to the ‘direct sum’ and ‘2-sum’ just describ ed. 1 Note that the measurable category H ∅ is the category with just one ob ject and one morphism. 86 4.6 Reduction, retraction, and decomp osition In this section, we introduce notions of reducibility and decomp osabilit y , in analogy with group represen tation theory , as w ell as an a priori in termediate notion, ‘retractability’. These notions mak e sense not only for representations, but also for intert winers. W e classify the indecomp os- able, irretractable and irreducible measurable representations, and intert winers b et ween these, up to equiv alence. 4.6.1 Represen tations Let us start with the basic deﬁnitions. Deﬁnition 79 A r epr esentation ρ 0 is a subrepresen tation of a given r epr esentation ρ if ther e exists a we akly monic intertwiner ρ 0 → ρ . W e remind the reader that an in tertwiner φ : ρ 0 → ρ is (strictly) monic if whenev er ξ , ξ 0 : τ → ρ are in tertwiners such that φ · ξ = φ · ξ 0 , w e hav e ξ = ξ 0 ; we sa y it is w eakly monic if this holds up to in vertible 2-in tertwiners, i.e. φ · ξ ∼ = φ · ξ 0 implies ξ ∼ = ξ 0 . Deﬁnition 80 A r epr esentation ρ 0 is a retract of ρ if ther e exist intertwiners φ : ρ 0 → ρ and ψ : ρ → ρ 0 whose c omp osite ψ φ is e quivalent to the identity intertwiner of ρ 0 ρ 0 φ / / ρ ψ / / ρ 0 ' ρ 0 1 ρ 0 / / ρ 0 Deﬁnition 81 A r epr esentation ρ 0 is a 2-summand of ρ if ρ ' ρ 0  ρ 00 for some r epr esentation ρ 00 . It is straightforw ard to show that an y 2-summand is automatically a retract, since the diagram ρ 0 → ρ 0  ρ 00 → ρ 0 , built from the obvious ‘injection’ and ‘pro jection’ intert winers, is equiv alent to the identit y . On the other hand, we shall see that a representation ρ generally has retracts that are not 2-summands; this is in stark contrast to linear represen tations of ordinary groups, where summands and retracts coincide. Similarly , any retract is automatically a subrepresentation, since ψ φ ' 1 easily implies φ is w eakly monic. An y represen tation ρ has b oth itself and the n ull representation as subrepresentations, as retracts, and as summands. This leads us to the following deﬁnitions: Deﬁnition 82 A r epr esentation ρ is irreducible if it has exactly two subr epr esentations, up to e quivalenc e, namely ρ itself and the nul l r epr esentation. Deﬁnition 83 A r epr esentation ρ is irretractable if it has exactly two r etr acts, up to e quivalenc e, namely ρ itself and the nul l r epr esentation. Deﬁnition 84 A r epr esentation ρ is indecomp osable if it has exactly two 2-summands, up to e quivalenc e, namely ρ itself and the nul l r epr esentation. 87 Note that according to these deﬁnitions, the null representation is neither irreducible, nor inde- comp osable, nor irretractable. An irreducible representation is automatically irretractable, and an irretractable representation is automatically indecomp osable. A priori, neither of these implications is reversible. Indecomp osable represen tations are characterized b y the follo wing theorem: Theorem 85 (Indecomp osable represen tations) L et ρ b e a me asur able r epr esentation on H X , making X into a me asur able G -sp ac e. Then ρ is inde c omp osable if and only if X is nonempty and G acts tr ansitively on X . Pro of: Observ e ﬁrst that, since the null representation is not indecomp osable, the theorem is ob vious for the case X = ∅ . W e may th us assume ρ is not the null represen tation. Assume ﬁrst ρ indecomposable, and let U and V be t wo disjoint G -inv ariant subsets suc h that X = U q V . ρ naturally induces represen tations ρ U in H U and ρ V in H V , and furthermore ρ = ρ U  ρ V . Since by hypothesis ρ is indecomp osable, at least one of these representations is the null represen tation. Consequently U = ∅ or V = ∅ . This shows that the G -action is transitive. Con versely , assume G acts transitively on X , and supp ose ρ ∼ ρ 1  ρ 2 for some represen tations ρ i in H X i . There is then a splitting X = X 0 1 q X 0 2 , where X 0 i is measurably identiﬁed with X i and G -in v ariant. Since by h yp othesis G acts transitively on X , w e deduce that X 0 i = ∅ = X i for at least one i . Th us, ρ i is the null represen tation for at least one i ; hence ρ is indecomp osable. Let o b e an y G -orbit in H ∗ ; pick a p oin t x ∗ o , and let S ∗ o denote its stabilizer group. The orbit can b e iden tiﬁed with the homogeneous space G/S ∗ o . Let also S ⊂ S ∗ o b e an y closed subgroup of S . Then X := G/S is a measurable G -space (see Lemma 122 in the App endix). The canonical pro jection on to G/S ∗ o deﬁnes a G -equiv arian t map χ : X → H ∗ . This map is measurable: to see this, write χ = π s , where s is a measurable section of G/S as in Lemma 123 , and π : G → G/S ∗ o is the measurable pro jection. Hence, the pair ( o, S ) deﬁnes a measurable represen tation; this representations is clearly indecomp osable. Next, consider the representations giv en by t wo pairs ( o, S ) and ( o 0 , S 0 ). When are they equiv a- len t? Equiv alence means that there is an isomorphism f : G/S → G/S 0 of measurable G -equiv ariant bundles ov er H ∗ . Such isomorphism exists if and only if the orbits are the same o = o 0 and the sub- groups S, S 0 are conjugate in S ∗ o . Hence, there is class of inequiv alen t indecomp osable represen tations lab elled b y an orbit o in H ∗ and a conjugacy class of subgroups S ⊂ S ∗ o . No w, let ρ b e any indecomp osable representation on H X . Thm. 85 says X is a transitive measurable G -space. T ransitivity forces the G -equiv arian t map χ : X → H ∗ to map on to a single orbit o ' G/S ∗ o in H ∗ . Moreov er, it implies that X is isomorphic as a G -equiv ariant bundle to G/S for some closed subgroup S ⊂ S ∗ o . Hence, ρ is equiv alen t to the representation deﬁned by the orbit o and the subgroup S . These remarks yield the following: Corollary 86 Inde c omp osable r epr esentations ar e classiﬁe d, up to e quivalenc e, by a choic e of G - orbit o in the char acter gr oup H ∗ , along with a c onjugacy class of close d sub gr oups S ⊂ S ∗ o of the stabilizer of one of its p oints. Irretractable representations are c haracterized by the following theorem: Theorem 87 (Irretractable representations) L et ρ b e a me asur able r epr esentation, given by a me asur able G -e quivariant map χ : X → H ∗ , as in Thm. 56 . Then ρ is irr etr actable if and only if χ induc es a G -sp ac e isomorphism b etwe en X and a single G -orbit in H ∗ . 88 Pro of: First observe that, since a G -orbit in H ∗ is alwa ys nonempty , and the null representation is not irretractable, the theorem is obvious for the case X = ∅ . W e ma y thus assume ρ is not the n ull representation. No w suppose ρ is irretractable, and consider a single G -orbit X ∗ con tained in the image χ ( X ) ⊂ H ∗ . X ∗ is a measurable subset (see Lemma. 121 in the App endix), so it naturally b ecomes a measurable G -space, with G -action induced b y the action on H ∗ . The canonical injection X ∗ → H ∗ mak es X ∗ a measurable equiv arian t bundle o ver the c haracter group. These data give a non-n ull represen tation ρ ∗ of the 2-group on the measurable category H X ∗ . W e wan t to show that ρ ∗ is a retract of ρ . T o do so, w e ﬁrst construct an X -indexed family of measures µ x on X ∗ as follows: if χ ( x ) ∈ X ∗ , we c ho ose µ x to b e the Dirac measure δ χ ( x ) whic h c harges the p oin t χ ( x ); otherwise w e choose µ x to b e the trivial measure. This family is ﬁb erwise b y construction; the cov ariance of the ﬁeld of characters ensures that it is also equiv ariant: δ g χ ( x ) = δ χ ( x ) g = δ χ ( xg ) . T o c heck that the family is measurable, pic k a measurable subset A ∗ ⊂ X ∗ . The function x 7→ µ x ( A ∗ ) coincides with the c haracteristic function of the set A = χ − 1 ( A ∗ ), whose v alue at x is 1 if x ∈ A and 0 otherwise; this function is measurable if the set A is. Now, since w e are working with measurable representations, the map χ is me asurable: therefore A is measurable, as the pre-image of the measurable A ∗ . Th us, the family of measures µ x is m easurable. So, together with the µ - classes of one-dimensional ﬁelds of Hilb ert spaces and iden tity linear maps, it deﬁnes an in tertwiner φ : ρ ∗ → ρ . Next, we wan t to construct a X ∗ -indexed equiv arian t and ﬁb erwise family of measures ν x ∗ on X . T o do so, pick an elemen t x ∗ o ∈ X ∗ , denote by S ∗ o ⊂ G its stabilizer group. W e require some results from top ology and measure theory (see App endix A.4 ). First, S ∗ o is a closed subgroup, and the orbit X ∗ can b e measurably identiﬁed with the homogenous space G/S ∗ o ; second, there exists a measurable section for G/S ∗ o , namely a measurable map n : G/S ∗ o → G such that π n = Id, where π : G → G/S ∗ o is the canonical pro jection, and nπ ( e ) = e . Also, the action of G on X induces a measurable S ∗ o -action on the ﬁb er ov er x ∗ o ; any orbit of this ﬁb er can thus b e measurably identiﬁed with a homogeneous space S ∗ o /S , on whic h nonzero quasi-in v ariant measures are known to exist. So let ν x ∗ o b e (the extension to X of ) a S ∗ o -quasi-in v ariant measure on the ﬁb er o ver x ∗ o . Using a measurable section n : G/S ∗ o → G , each x ∗ ∈ X ∗ can then b e written unambiguously as x ∗ o n ( k ) for some coset k ∈ G/S ∗ o . Deﬁne ν x ∗ := ν n ( k ) x ∗ o where by deﬁnition ν g ( A ) = ν ( Ag − 1 ). W e obtain b y this pro cedure a measurable ﬁb erwise and equiv arian t family of measures on X . T ogether with the ( ν -classes of ) constan t one-dimensional ﬁeld(s) of Hilb ert spaces C and constan t ﬁeld of identit y linear maps, this deﬁnes an intert winer ψ : ρ → ρ ∗ . W e can immediately c heck that the comp osition ψ φ of these tw o in tertwiners deﬁned ab o ve is equiv alen t to the identit y intert winer 1 ρ ∗ , since the comp osite measure at x ∗ , Z X d ν x ∗ ( x ) µ x = ν x ∗ ( χ − 1 ( x ∗ )) δ x ∗ is equiv alen t to the delta function δ x ∗ . This sho ws that ρ ∗ is a retract of ρ . No w, b y h yp othesis ρ is irretractable; since the retract ρ ∗ is not null, it must therefore b e equiv alen t to ρ . W e know by Thm. 70 that this equiv alence gives a measurable isomorphism f : X ∗ → X , as G -equiv arian t bundles ov er H ∗ . In our case, f being a bundle map means χ ( f ( x ∗ )) = x ∗ . 89 T ogether with the inv ertibilit y of f , this relation sho ws that the image of the map χ is X ∗ , and furthermore that χ = f − 1 . W e hav e thus prov ed that χ : X → X ∗ is an inv ertible map of X onto the orbit X ∗ ⊆ H ∗ . Con versely , supp ose χ is in v ertible and maps X to a single orbit X ∗ in H ∗ and consider a non-n ull retract ρ 0 of ρ . W e denote by X 0 the underlying space and by χ 0 the ﬁeld of characters asso ciated to ρ 0 . Pic k tw o intert winers φ : ρ 0 → ρ and ψ : ρ → ρ 0 suc h that ψ φ ' 1 ρ 0 . These tw o in tertwiners pro vide an X -indexed family of measures µ x on X 0 and a X 0 -indexed family of measures ν x 0 on X whic h satisfy the prop ert y that, for each x 0 , the comp osite measure at x 0 is equiv alen t to a Dirac measure: Z X d ν x 0 ( x ) µ x ∼ δ x 0 (78) An obvious consequence of this prop ert y is that the measures ν x 0 are all non-trivial. Since ν x 0 concen trates on the ﬁb er ov er χ 0 ( x 0 ) in X , this ﬁber is therefore not empt y . This sho ws that χ 0 ( X 0 ) is included in the G -orbit χ ( X ) = X ∗ . The G -in v ariance of the subset im χ 0 sho ws furthermore that this inclusion is an equality , so χ ( X ) = χ 0 ( X 0 ). Consequently the map f = χ − 1 χ 0 is a well deﬁned measurable function from X 0 to X ; it is surjectiv e, commutes with the action of g and obviously satisﬁes χf = χ 0 . Now, b y hypothesis, the ﬁb er o ver χ 0 ( x 0 ) in X , on whic h ν x 0 concen trates, consists of the singlet { f ( x 0 ) } : we deduce that ν x 0 ∼ δ f ( x 0 ) . The property ( 78 ) th us reduces to µ f ( x 0 ) ∼ δ x 0 for all x 0 , which requires f to be injectiv e. Th us, w e hav e found an in vertible measurable map f : X 0 → X that is G -equiv ariant and preserves ﬁb ers of χ : X → H ∗ . By Thm. 70 , the representations ρ and ρ 0 are equiv alen t; hence ρ is irretractable. An y irretractable represen tation is indecomposable; up to equiv alence, it thus tak es the form ( o, S ), where o is a G -orbit in H ∗ and S is a subgroup of S ∗ o . How ever, the conv erse is not true: there are in general many indecomp osable represen tations ( o, S ) that are retractable. Indeed, ( o, S ) deﬁnes an inv ertible map χ : G/S → G/S ∗ o only when S = S ∗ o . The existence of retractable but indecomp osable represen tations has been already noted by Barrett and Mac k aaa y [ 18 ] in the con text of the represen tation theory of 2-groups on ﬁnite dimensional 2-vector spaces. W e see here that this is also true for representations on more general measurable categories. Corollary 88 Irr etr actable me asur able r epr esentations ar e classiﬁe d, up to e quivalenc e, by G -orbits in the char acter gr oup H ∗ . 4.6.2 In tertwiners Because 2-group represen tation theory inv olv es not only intert winers b etw een representations, but also 2-intert winers b et w een intert winers, there are ob vious analogs for intert winers of the concepts discussed in the previous section for representations. W e deﬁne sub-intert winers, retracts and 2- summands of intert winers in a precisely analogous wa y , obtaining notions of irreducibility , irre- tractabilit y , and indecomp osabilit y for intert winers, as for representations. Deﬁnition 89 An intertwiner φ 0 : ρ 1 → ρ 2 is a sub-intert winer of φ : ρ 1 → ρ 2 if ther e exists a monic 2-intertwiner m : φ 0 ⇒ φ . W e remind the reader that a 2-intert winer m : φ 0 ⇒ φ is monic if whenever n, n 0 : ψ ⇒ φ 0 are 2-in tertwiners suc h that m · n = m · n 0 , we hav e n = n 0 . Deﬁnition 90 An intertwiner φ 0 : ρ 1 → ρ 2 is a retract of φ : ρ 1 → ρ 2 if ther e exist 2-intertwiners 90 m : φ 0 ⇒ φ and n : φ ⇒ φ 0 such that the vertic al pr o duct n · m e quals the identity 2-intertwiner of φ 0 ρ 1 φ 0 " " φ / / φ 0 < < ρ 2 m   n   = ρ 1 φ 0 ) ) φ 0 5 5 ρ 2 1 φ 0   Deﬁnition 91 An intertwiner φ 0 : ρ 1 → ρ 2 is a summand of φ : ρ 1 → ρ 2 if φ ∼ = φ 0  φ 00 for some intertwiner φ 00 . An y summand is a retract, and any retract is a sub-intert winer. Recall from Section 4.5.1 that the n ull in tertwiner betw een measurable representations on H X and H Y is deﬁned by the trivial family of measures, µ y = 0 for all y . It is easy to see that the null in tertwiner is a summand (hence also a retract, and a sub-in tertwiner) of any intert winer. Deﬁnition 92 An intertwiner φ is irreducible if it has exactly two sub-intertwiners, up to 2- isomorphism, namely φ itself and the nul l intertwiner. Deﬁnition 93 An intertwiner φ is irretractable if it has exactly two r etr acts, up to 2-isomorphism, namely φ itself and the nul l intertwiner. Deﬁnition 94 An intertwiner φ is indecomp osable if it has exactly two summands, up to 2- isomorphism, namely φ itself and the nul l intertwiner. According to these deﬁnitions, the null represen tation is neither irreducible, nor indecom posable, nor irretractable. An irreducible intert winer is automatically irretractable, and an irretractable in tertwiner is automatically indecomp osable. A priori, neither of these implications is rev ersible. T o dig deep er into these notions, w e need some concepts from ergo dic theory: ergo dic measures, and their generalization to measurable families of measures. In what follows, we denote b y 4 the symmetric diﬀerence op eration on sets: U 4 V = ( U ∪ V ) − ( U ∩ V ) When U is a subset of a G -set X , we use the notation U g = { ug | u ∈ U } . Deﬁnition 95 A me asur e µ on X is ergo dic under a G -action if for any me asur able subset U ⊂ X such that µ ( U 4 U g ) = 0 for al l g , we have either µ ( U ) = 0 or µ ( X − U ) = 0 . In the case of quasi-inv ariant measures, there is a useful alternative criterion for ergo dicit y . Roughly sp eaking, an ergo dic quasi-in v arian t measure has as many n ull sets as possible without v anishing en tirely . More precisely , we ha ve the follo wing lemma: Lemma 96 L et µ b e a quasi invariant me asur e with r esp e ct to a G -action. Then µ is er go dic if and only if any quasi-invariant me asur e ν that is absolutely c ontinuous with r esp e ct to µ is either zer o or e quivalent to µ . Pro of: Assume ﬁrst µ is ergo dic. Let ν b e a quasi-in v ariant measure with ν  µ . Consider the Leb esgue decomp osition µ = µ ν + µ ν . As shown in Prop. 107 , the tw o measures are m utually singular, so there is a measurable set U suc h that µ ν ( A ) = µ ν ( A ∩ U ) for every measurable set A , 91 and µ ν ( U ) = 0. Hence, for all g ∈ G , µ ν ( U g − U ) = 0. Now, ν  µ implies µ ν ∼ ν , so we know µ ν is also quasi-inv ariant. This implies µ ν ( U − U g ) = µ (( U g − 1 − U ) g ) = 0 for all g . W e then hav e µ ( U 4 U g ) = µ ( U g − U ) + µ ( U − U g ) = 0 for all g ∈ G . Since µ is ergo dic, we conclude that either µ ( U ) = 0, in which case µ ν = 0 and therefore ν = 0, or µ ( X − U ) = 0, in which case µ ∼ µ ν , and hence µ ∼ ν . Con versely , suppose every quasi-inv ariant measure sub ordinate to µ is either zero or equiv alen t to µ . Let U b e a measurable set such that µ ( U 4 U g ) = 0 for all g ∈ G . Deﬁne a measure ν b y setting ν ( A ) = µ ( A ∩ U ) for eac h measurable set A . Obviously ν  µ . Since U 4 U g is µ -null, ν ( A ) = µ ( A ∩ U ) = µ ( A ∩ U g ) for all g and ev ery measurable set A . In particular, applying this to Ag , ν ( Ag ) = µ ( Ag ∩ U ) = µ (( A ∩ U ) g ) , so quasi-in v ariance of ν follo ws from that of µ . Thus, ν is a quasi-in v ariant measure suc h that ν  µ ; this, by hypothesis, yields either ν = 0, hence µ ( U ) = 0, or ν ∼ µ , hence µ ( X − U ) = 0. W e conclude that µ is ergo dic. The notion of ergodic measure has an important generalization to the case of measurable families of measures: Deﬁnition 97 L et X and Y b e me asur able G -sp ac es. A Y -indexe d e quivariant family of me asur es µ y on X is minimal if: (i) ther e exists a G -orbit Y o in Y such that µ y = 0 for al l y ∈ Y − Y o , and (ii) for al l y , µ y is er go dic under the action of the stabilizer S y ⊂ G of y . Notice that an ergodic measure is simply a minimal family whose index space is the one-point G -space. The criterion given in the previous lemma extends to the case of minimal equiv ariant families of measures: Lemma 98 L et µ y b e an e quivariant family of me asur es. The family is minimal if and only if, for any e quivariant family ν y such that ν y  µ y for al l y , ν y is either trivial or satisﬁes ν y ∼ µ y for al l y . Pro of: The ‘only if ’ part of the statemen t is a direct application of Lemma 96 ; let us prov e the ‘if ’ part. Supp ose every equiv arian t family sub ordinate to µ y is either zero or equiv alen t to µ y . W e ﬁrst sho w that µ y satisﬁes prop ert y ( i ) in Def. 97 . Assuming the family µ y is non-trivial, let Y o b e a G-orbit in Y on which µ y 6 = 0. Deﬁne an equiv arian t family ν y b y setting ν y = µ y if y ∈ Y o and 0 otherwise. This family is non-trivial and obviously satisﬁes ν y  µ y ; this by h yp othesis yields µ y ∼ ν y . Therefore µ y = 0 for all y ∈ Y − Y o . W e no w turn to prop ert y ( i ) in Def. 97 . Fix y o ∈ Y o , and let S o ⊆ G b e its stabilizer. T o show that µ y o is ergo dic under the ation of S o pic k a measurable subset U such that µ y o ( U 4 U s ) = 0 for all s ∈ S o . By equiv ariance of the family µ y , this implies µ y o g ( U g 4 U sg ) (79) 92 for all s ∈ S o and all g ∈ G . Then, for every y = y o g in Y o , deﬁne a measure ν y b y setting ν y ( A ) = µ y ( A ∩ U g ). This is w ell deﬁned, since an y g 0 suc h that y = y o g 0 is given by g 0 = sg for some s ∈ S o , and by ( 79 ) w e hav e µ y ( A ∩ U g ) = µ y ( A ∩ U sg ). The family ν y is equiv arian t; indeed, for an y g ∈ G , and y = y o g 0 ∈ Y o : ν y g ( Ag ) = µ y g (( A ∩ U g 0 ) g ) , so equiv ariance of ν y follo ws from that of µ y . Since w e also obviously hav e ν y  µ y for all y , b y h yp othesis the family ν y is either trivial, or satisﬁes ν y ∼ µ y for all y . In the former case, µ y o ( U ) = 0; in the latter, µ y o ( X − U ) = 0. Thus, µ y o is ergo dic under the action of S o . Since y o w as arbitrary , ( ii ) is prov ed, and the family µ y is minimal. T r ansitive families of measures, for which there exists a G -orbit o in Y × X such that µ y ( A ) = 0 for ev ery measurable { y } × A in the complemen t Y × X − o , are particular examples of minimal families. Indeed, the obvious pro jection Y × X → Y maps the orbit o in to an orbit Y o suc h that µ y = 0 unless y ∈ Y o ; furthermore for all y ∈ Y o , µ y is quasi-inv ariant under the action of the stabilizer S y of y and concen trates on a single orbit, so it is clearly ergo dic. It is useful to inv estigate the con verse: Is a minimal family of measures necessarily transitive? This is not the case, in general. T o understand this, w e need to dwell further on the notion of quasi-in v ariant ergo dic measure. First note that each orbit in X naturally deﬁnes a measure class of such measures: we indeed kno w that an orbit deﬁnes a measure class of quasi-inv ariant measures; no w the uniqueness of such a class yields the minimality prop ert y stated in Lemma. 96 , hence the ergo dicit y of the measures. Ho wev er, not every quasi-in v ariant and ergodic measure need b elong to one of the classes deﬁned b y the orbits. In fact, given a measure µ on X , quasi-inv ariant and ergo dic under a measurable G -action, there should b e at most one orbit with p ositive measure, and its complement in X should b e a null set. If there is an orbit with p ositiv e measure, µ b elongs to the class that the orbit deﬁnes. But it ma y also very well b e that al l G -orbits are n ull sets. Consider for example the group G = Z , acting on the unit circle X = { z ∈ C | | z | = 1 } in the complex plane as e iθ 7→ e iθ + απ , where α ∈ R − Q is some ﬁxed irrational n umber. It can b e shown that the linear measure d θ on X is ergodic, whereas the orbits, which are all coun table, are null sets. This mak es the classiﬁcation of the equiv alence classes of ergodic quasi-inv ariant measures quite diﬃcult in general. Luckily , there is a simple criterion, stated in the following lemma, that precludes the kind of behaviour illustrated in the ab o v e example. F or X a measurable G -space, we call a measurable subset N ⊂ X a measurable cross-section if it intersects each G -orbit in exactly one p oin t. Lemma 99 [ 69 , Lemma 6.14] L et X b e a me asur able G -sp ac e. If X has a me asur able cr oss- se ction, then any er go dic me asur e on X is supp orte d on a single G -orbit. Roughly sp eaking, the existence of a measurable cross-section ensures that the orbit space is “nice enough”. Th us, for example, making such assumption is equiv alent to requiring that the orbit space is countably separated as a Borel space; or, in the case of a contin uous group action, that it is a T 0 space [ 38 ]. Ha ving introduced these concepts, we now b egin our study of indecomp osable, irretractable and irreducible intert winers. Consider a pair of representations ρ 1 and ρ 2 on the measurable categories H X and H Y ; denote by χ 1 and χ 2 the corresp onding ﬁelds of characters. Let φ : ρ 1 → ρ 2 b e an in tertwiner; denote b y µ y the corresp onding equiv ariant and ﬁb erwise family µ y of measures on X . The following prop osition gives a necessary condition for the intert winer to b e indecomp osable (hence to b e irretractable or irreducible): 93 Prop osition 100 If the intertwiner φ = ( φ, Φ , µ ) is inde c omp osable, its family of me asur es µ y is minimal. Pro of: Assume φ is indecomposable, and consider an equiv arian t family of measures ν y suc h that ν y  µ y for all y . The Leb esgue decompositions: µ y = µ ν y y + µ ν y y deﬁne t wo new ﬁb erwise and equiv arian t families measures. T ogether with the µ ν -classes of ﬁelds and the µ ν -classes of ﬁelds induced by the µ -classes of φ , these sp ecify t wo in tertwiners ψ , ψ . The measures µ ν y y and µ ν y y are mutually singular for all y . Using the deﬁnition of the direct sum of intert winers, w e ﬁnd φ = ψ ⊕ ψ . No w, by hypothesis φ is indecomp osable, so that either ψ or ψ is the n ull intert winer. In the former case, the family µ ν is trivial. This means that ν y ⊥ µ y for all y ; since, furthermore, ν y  µ y , it implies that ν is trivial. In the latter case, the family µ ν is trivial. This mean that ν y ∼ µ y for all y . W e conclude with Lemma 98 that the family µ y is minimal. W e can b e more precise by fo cusing on the tr ansitive intert winers, as deﬁned in Def. 63 . Supp ose the intert winer φ : ρ 1 → ρ 2 is transitive, and sp eciﬁed by the assignments φ y ,x , Φ g y ,x of Hilb ert spaces and in vertible maps to the p oin ts of an G -orbit o in Y × X . These deﬁne ordinary linear represen tations R φ y ,x of the stabilizer S y ,x of ( y , x ) under the diagonal action of G . The following prop ositions give a criterion for φ to b e indecomp osable, irretractable, or irre- ducible: Prop osition 101 (Indecomp osable and irretractable transitive in terwiners) L et φ = ( φ, Φ , µ ) b e a tr ansitive intertwiner. Then the fol lowing ar e e quivalent: • φ is inde c omp osable • φ is irr etr actable • the stabilizer r epr esentations R φ y ,x ar e inde c omp osable. Prop osition 102 (Irreducible transitiv e in terwiners) L et φ = ( φ, Φ , µ ) b e a tr ansitive inter- twiner. Then φ is irr e ducible if and only if the stabilizer r epr esentations R φ y ,x ar e irr e ducible. Let us prov e these tw o prop ositions together: Pro of: Fix a p oint y o ∈ Y such that µ y o 6 = 0, and let S y o b e its stabilizer. The action of G on X induces an action of S y o on the ﬁb er ov er χ 2 ( y o ) in X . Since b y h yp othesis φ is transitive, µ y o concen trates on a single S y o -orbit ı o ⊆ X . Next, ﬁx x o in ı o , and let S o = S y o ,x o denote stabilizer of ( y o , x o ) under the diagonal action. Let also φ o = φ y o ,x o , Φ g o = Φ g y o ,x o b e the space and maps assigned to the p oint ( y o , x o ), and let R φ o b e the corresp onding linear representation s 7→ Φ s o of S o . Note that the representations R φ y ,x are all indecomp osable (or irreducible) if R φ o is. W e b egin with Prop. 101 . Supp ose ﬁrst that φ is indecomp osable. Consider a Hilb ert space decomp osition φ o = φ 0 o ⊕ φ 00 o that is inv ariant under R φ o ; assume that φ 0 o is non-trivial. Given a p oin t ( y , x ) = ( y o , x o ) g − 1 in the orbit, the isomorphism Φ g o : φ o → φ y ,x giv es a splitting φ y ,x = φ 0 y ,x ⊕ φ 00 y ,x where φ 0 y ,x = Φ g o ( φ 0 o ) , φ 00 y ,x = Φ g o ( φ 00 o ) 94 This decomp osition is independent of the representativ e of g S o c hosen, hence dep ends only on the p oin t ( y , x ); indeed, for every s ∈ S o , we hav e Φ g s o ( φ 0 o ) = Φ g o Φ s o ( φ 0 o ) = Φ g o ( φ 0 o ) = φ 0 y ,x b y inv ariance of φ 0 o , and lik ewise for φ 00 . The decomp osition of φ y ,x is also inv ariant under the represen tation R φ y ,x of S y ,x : Φ s y ,x ( φ 0 y ,x ) = Φ sg y ,x ( φ 0 o ) = Φ s ( g − 1 sg ) ( φ 0 o ) = Φ g o ( φ 0 o ) = φ 0 y ,x since for any s ∈ S y ,x , we hav e ( g − 1 sg ) ∈ S o , and likewise for φ 00 y ,x . These data giv e us a transitive in tertwiner φ 0 = ( φ 0 y ,x , Φ 0 g y ,x , µ y ), where Φ 0 g y ,x simply denotes the restriction of Φ g y ,x to φ 0 y ,x . By construction, φ 0 is a summand of φ , distinct from the n ull in tertwiner. Now, we hav e assumed φ is indecomp osable; so we hav e that φ 0 ' φ . W e then deduce from Prop. 71 that the representation R φ o is equiv alen t to its restriction to φ 0 o . Thus, R φ o is indecomp osable. Next, supp ose that the linear represen tations R φ y ,x are indecomp osable. W e will show that φ is irretractable; since an irretractable is automatically indecomp osable, this will complete the pro of of Prop. 101 . Let φ 0 b e a retract of φ , sp eciﬁed by the family of measures µ 0 y and the assignments of Hilbert spaces φ 0 y ,x and inv ertible maps Φ 0 g y ,x . By deﬁnition one can ﬁnd 2-intert winers m : φ ⇒ φ 0 and n : φ 0 ⇒ φ such that n · m = 1 φ 0 . This last equalit y requires that the geometric mean measures p µ y µ 0 y b e equiv alen t to µ 0 y , or equiv alen tly that µ 0 y  µ y . Hence, the S o -quasi-in v ariant measure µ 0 y o concen trates on the orbit ı o . Non-trivial S o -quasi-in v ariant measures on ı o are unique up to equiv alence, so we conclude that µ 0 y o is either trivial or equiv alen t to µ y o . In the ﬁrst case, φ 0 is trivial, so we are done. In the second case, where µ 0 y ∼ µ y for all y , the linear maps m y ,x : φ 0 y ,x → φ y ,x and n y ,x : φ y ,x → φ 0 y ,x are in tertwining op erators betw een the represen tations R φ 0 y ,x and R φ y ,x ; they satisfy n y ,x m y ,x = 1 φ 0 y,x . Thus, R φ 0 y ,x is a retract of R φ y ,x , hence a direct summand. But R φ y ,x is indecomposable: so the tw o represen tations must be equiv alent. Hence the map m y ,x is inv ertible. The 2-intert winer m : φ 0 ⇒ φ is thus in vertible, which sho ws φ 0 and φ are equiv alent. W e conclude that φ is irre- tractable. W e now pro ve Prop. 102 . Suppose ﬁrst that φ is irreducible. Consider a non-trivial subspace φ 0 o ⊂ φ o that is inv ariant under R φ o . Given a point ( y , x ) = ( y o , x o ) g − 1 in the orbit, the isomorphism Φ g o : φ o → φ y ,x giv es a subspace φ 0 y ,x := Φ g y ,x ( φ 0 o ) of φ y ,x that is in v arian t under R φ y ,x . These data giv e us a transitiv e in tertwiner φ 0 = ( φ 0 y ,x , Φ 0 g y ,x , µ y ), where Φ 0 g y ,x simply denotes the restriction of Φ g y ,x to φ 0 y ,x . The canonical injections ı y ,x : φ 0 y ,x → φ y ,x deﬁne a monic 2-intert winer ı : φ 0 → φ ; this sho ws that φ 0 is a sub-intert winer of φ . But φ is irreducible: we therefore ha ve that φ 0 ' φ . This means that R φ o is equiv alen t to its restriction to φ 0 o . Thus, R φ o is irreducible. Con versely , supp ose that the represen tations R φ y ,x are irreducible. Consider a sub-intert winer φ 0 of φ , giving a family of measures µ 0 y . First of all, note that the existence of a monic 2-intert winer m : φ 0 → φ forces µ 0 to b e transitive, with ¯ µ y supp orted on the orbit o . In particular, we ha ve that µ 0 y ∼ µ y for all y . Next, ﬁx a monic 2-intert winer m . It giv es injective linear maps m y ,x : φ 0 y ,x → φ y ,x ; these deﬁne subspaces m y ,x ( φ 0 y ,x ) in φ y ,x that are inv arian t under R φ y ,x . Since the representations are b y h yp othesis irreducible, this means that the maps m y ,x , and hence m , are in vertible. W e obtain that φ 0 ' φ , and conclude that φ is irreducible. These results allow us to classify , up to equiv alence, the indecomposable and irreducible inter- t winers betw een ﬁxed measurable representations ρ 1 , ρ 2 . W e may assume that these represen tations 95 are indecomp osable, and giv en b y the pairs ( o, S 1 ) and ( o, S 2 ). They are thus sp eciﬁed by the G - equiv arian t bundles X = G/S 1 and Y = G/S 2 o ver the same G -orbit o ' G/S ∗ o in H ∗ ; S 1 and S 2 are some closed subgroups of S ∗ o . In the following, w e denote y o = S 2 e , and ﬁx a (not necessarily measurable) cross-section of the S 2 -space S ∗ o /S 1 – namely , a subset that in tersects eac h S 2 -orbit ı o in exactly one p oin t x o := S 1 k o . Let φ : ρ 1 → ρ 2 an indecomp osable (resp. irreducible) intert winer. W e will assume that φ is transitiv e , keeping in mind the following consequence of Prop. 100 and Lemma 99 : Lemma 103 Supp ose that the S 2 -sp ac e S ∗ o /S 1 has a me asur able cr oss-se ction. Then every inde- c omp osable intertwiner φ : ( o, S 1 ) → ( o, S 2 ) is tr ansitive. The intert winer φ gives a non-trivial S 2 -quasi-in v ariant measure µ y o in the ﬁb er S ∗ o /S 1 ⊂ X o ver S ∗ o e . Moreo ver, the transitivity of φ implies that this measure is supp orted on a single S 2 -orbit ı φ o in S ∗ o /S 1 . Note that an y tw o such measures are equiv alent. φ also gives an indecomp osable (resp. irreducible) linear representation R φ o of the group S o = k − 1 o S 1 k o ∩ S 2 . So, φ giv es a pair ( ı φ o , R φ o ), where ı φ o is a S 2 -orbit in S ∗ o /S 1 and R φ o is an indecomp osable (resp. irreducible) represen tation of S o . W e easily deduce from Prop. 72 that tw o equiv alen t transitive in tertwiners giv e tw o pairs with the same orbit and equiv alent linear represen tations. Con versely , given any orbit ı o and any linear representation R o of S o on some Hilb ert space φ o , there is an in tertwiner φ = ( φ, Φ , µ ) such that ı φ o = ı o and R φ o = R o . Indeed, a measurable equiv arian t and ﬁb erwise family of measures is obtained b y c ho osing a S 2 -quasi-in v ariant measure µ o supp orted on ı o and a measurable section n : G/S 2 → G , and b y setting, for each y = y o n ( k ): µ y := µ n ( k ) o T o construct the measurable ﬁelds of spaces and linear maps, ﬁx a measurable section ¯ n : G/S o → G , denote by π : G → G/S o the canonical pro jection, and consider the function α : G → S o giv en by: α ( g ) = ( ¯ nπ )( g − 1 ) g . This function satisﬁes the prop erty that α ( g s ) = α ( g ) s for all s ∈ S o . Using this, we deﬁne a family Φ g o of isomorphisms of φ o as: Φ g o = R o ( α ( g )) and construct a measurable ﬁeld Φ g y ,x b y setting, for each ( y , x ) = ( y o , x o ) k − 1 : Φ g y ,x = Φ g o (Φ k o ) − 1 . These data spe cify a transitive in tertwiner φ ; this intert winer is indecomp osable (resp. irreducible) if R o is. These remarks yield the following: Corollary 104 Inde c omp osable (r esp. irr e ducible) tr ansitive intertwiners φ : ( o, S 1 ) → ( o, S 2 ) ar e classiﬁe d, up to e quivalenc e, by a choic e of a S 2 -orbit ı o in S ∗ o /S 1 , along with an e quivalenc e class of inde c omp osable (r esp. irr e ducible) line ar r epr esentations R o of the gr oup k − 1 o S 1 k o ∩ S 2 . W e close this section with a version of Sc hur’s lemma for irreducible intert winers: 96 Prop osition 105 (Sch ur’s Lemma for In tertwiners) L et φ, ψ : ( o, S 1 ) , → ( o, S 2 ) b e two irr e- ducible tr ansitive intertwiners. Then any 2-intertwiner m : φ ⇒ ψ is either nul l or an isomorphism. In the latter c ase, m is unique, up to a normalization factor. Pro of: W e may assume φ and ψ are given b y the pairs ( ı φ o , R φ o ) and ( ı ψ o , R ψ o ) of S 2 -orbits in S ∗ o /S 1 and irreducible linear representations. Let µ y and ν y denote the tw o families of measures. If the orbits are distinct ı φ o 6 = ı ψ o , the measures µ y and ν y ha ve disjoint supp ort, so that their geometric mean is trivial. In this case, any 2-in tertwiner m : φ ⇒ ψ is trivial. Supp ose now ı φ o = ı ψ o . In this case, we hav e that µ y ∼ ν y for all y . Let m : φ ⇒ ψ b e a 2- in tertwiner, giv en by the assignment of linear maps m y ,x : φ y ,x → ψ y ,x . Because of the intert wining rule ( 69 ), the assignment is en tirely sp eciﬁed b y the data m o := m y o ,x o . No w, m o deﬁnes a standard in tertwiner b et ween the irreducible linear representations R φ o and R ψ o . Therefore m o , and hence m , is either trivial or inv ertible; in the latter case, it is unique, up to a normalization factor. 5 Conclusion W e conclude with some possible a ven ues for future in vestigation. First, it will be in teresting to study examples of the general theory describ ed here. As explained in the Introduction, represen tations of the Poincar ´ e 2-group hav e already b een studied b y Crane and Shepp eard [ 25 ], in view of obtaining a 4-dimensional state sum mo del with p ossible relations to quan tum gravit y . Representations of the Euclidean 2-group (with G = SO(4) acting on H = R 4 in the usual wa y) are somewhat more tractable. Copying the ideas of Crane and Shepp eard, this 2-group giv es a state sum mo del [ 10 , 11 ] with interesting relations to the more familiar Ooguri mo del. There are also many other 2-groups whose representations are w orth studying. F or example, Bartlett has studied representations of ﬁnite groups G , regarded as 2-groups with trivial H [ 17 ]. He considers we ak representations of these 2-groups, where composition of 1-morphisms is preserved only up to 2-isomorphism. More precisely , he considers unitary w eak representations on ﬁnite-dimensional 2-Hib ert spaces. These c hoices lead him to a b eautiful geometrical picture of representations, in- tert winers and 2-in tertwiners — strikingly similar to our w ork here, but with U(1) gerb es playing a ma jor role. So, it will b e very interesting to generalize our work to weak representations, and sp ecialize it to unitary ones. T o deﬁne unitary representations of measurable 2-groups, we need them to act on something with more structure than a measurable category: namely , some sort of inﬁnite-dimensional 2-Hilb ert space. This notion has not y et b een deﬁned. How ever, we may hazard a guess on how the deﬁnition should go. In Section 3.3 , we argued that the measurable category H X should be a categoriﬁed analogue of L 2 ( X ), with direct integrals replacing ordinary integrals. How ev er, we nev er discussed the inner pro duct in H X . W e can deﬁne this only after choosing a measure µ on X . This measure app ears in the formula for the inner pro duct of vectors ψ , φ ∈ L 2 ( X ): h ψ , φ i = Z ψ ( x ) φ ( x ) d µ ( x ) ∈ C . Similarly , we can use it to deﬁne the inner pro duct of ﬁelds of Hilb ert spaces H , K ∈ H X : hH , K i = Z ⊕ H ( x ) ⊗ K ( x ) d µ ( x ) ∈ Hilb . 97 Here H ( x ) is the complex conjugate of the Hilb ert space H ( x ), where multiplication by i has b een redeﬁned to b e m ultiplication by − i . This is naturally isomorphic to the Hilb ert space dual H ( x ) ∗ , so we can also write hH , K i ∼ = Z ⊕ H ( x ) ∗ ⊗ K ( x ) d µ ( x ) . Recall that throughout this paper we are assuming our measures are σ -ﬁnite; this guarantees that the Hilb ert space hH , K i is sep ar able . So, we ma y giv e a preliminary deﬁnition of a ‘separable 2-Hilb ert space’ as a category of the form H X where X is a measurable space equipp ed with a measure µ . As a sign that this deﬁnition is on the right track, note that when X is a ﬁnite set equipped with a measure, H X is a ﬁnite-dimensional 2-Hilb ert space as previously deﬁned [ 3 ]. Moreov er, every ﬁnite-dimensional 2-Hilb ert space is equiv alen t to one of this form [ 17 , Sec. 2.1.2]. The main thing we lack in the inﬁnite-dimensional case, whic h w e p ossess in the ﬁnite-dimensional case, is an in trinsic deﬁnition of a 2-Hilb ert space. An intrinsic deﬁnition should not refer to the measurable space X , since this space merely serves as a ‘choice of basis’. The problem is that it seems tricky to deﬁne direct integrals of ob jects without mentioning this space X . The same problem aﬄicted our treatmen t of measurable categories. Instead of giving an in trinsic deﬁnition of measurable categories, we deﬁned a measurable category to b e a C ∗ -category that is C ∗ -equiv alen t to H X for some measurable space X . This made the construction of Meas rather roundab out. W e could try a similar approac h to deﬁning a 2-category of separable 2-Hilb ert spaces, but it would be equally roundab out. Luc kily there is another approach, essentially equiv alent to the one just presen ted, that do es not men tion measure spaces or measurable categories! In this approac h, we think of a 2-Hilb ert space as a category of representations of a commutativ e von Neumann algebra. The k ey step is to notice that when µ is a measure on a measurable space X , the algebra L ∞ ( X, µ ) acts as m ultiplication operators on L 2 ( X, µ ). Using this one can think of L ∞ ( X, µ ) as a commutativ e von Neumann algebra of operators on a separable Hilb ert space. Con versely , any comm utative von Neumann algebra of op erators on a separable Hilb ert space is isomorphic—as a C ∗ -algebra—to one of this form [ 28 , Part I, Chap. 7, Thm. 1]. The technical conditions built into our deﬁnition of ‘measurable space’ and ‘measure’ are precisely what is required to make this work (see Defs. 15 and 16 ). This viewp oin t gives a new outlo ok on ﬁelds of Hilbert spaces. Supp ose A is comm utative von Neumann algebra of op erators on a separable Hilb ert space. As a C ∗ -algebra, w e ma y iden tify A with L ∞ ( X, µ ) for some measure µ on a measurable space X . Deﬁne a separable represen tation of A to b e a representation of A on a separable Hilb ert space. It can then b e shown that ev ery separable represen tation of A is equiv alent to the representation of L ∞ ( X, µ ) as multiplication op erators on R ⊕ H ( x )d µ ( x ) for some ﬁeld of Hilb ert spaces H on X . Moreov er, this ﬁeld H is essentially unique [ 28 , Part I, Chap. 6, Thms. 2 and 3]. This suggests that w e deﬁne a separable 2-Hilb ert space to b e a category of separable rep- resen tations of some commutativ e von Neumann algebra of op erators on a separable Hilb ert space. More generally , we could drop the separability condition and deﬁne a 2-Hilb ert space to b e a category of representations of a comm utative v on Neumann algebra. While elegan t, this deﬁnition is not quite right. Any category ‘equiv alent’ to the category of represen tations of a commutativ e v on Neumann algebra—in a suitable sense of ‘equiv alent’, probably stronger than C ∗ -equiv alence—should also count as a 2-Hilb ert space. A b etter approach w ould give an intrinsic c haracterization of categories of this form. Then it would b ecome a the or em that every 2-Hilb ert space is equiv alen t to the category of representations of a commutativ e v on Neumann algebra. 98 Luc kily , there is y et another simpliﬁcation to b e made. After all, a commutativ e von Neumann algebra can b e recov ered, up to isomorphism, from its category of representations. So, we can forget the category of representations and fo cus on the von Neumann algebra itself ! The problem is then to redescrib e morphisms b et ween 2-Hilb ert spaces, and 2-morphisms b et ween these, in the language of von Neumann algebras. There is a natural guess as to how this should w ork, due to Urs Schreiber. Namely , we can deﬁne a bicategory 2Hilb for which: • ob jects are commutativ e v on Neumann algebras A, B , . . . , • a morphism H : A → B is a Hilb ert space H equipp ed with the structure of a ( B , A )-bimo dule, • a 2-morphism f : H → K is a homomorphism of ( B , A )-bimo dules. Comp osition of morphisms corresp onds to tensoring bimo dules. Note also that given an ( B , A )- bimo dule and a representation of A , we can tensor the tw o and get a representation of B . This is ho w an ( B , A )-bimodule gives a functor from the category of represen tations of A to the category of represen tations of B . Similarly , a homomorphism of ( B , A )-bimo dules giv es a natural transformation b et w een such functors. Let us brieﬂy sk etch the relation b et ween this version of 2Hilb and the 2-category Meas de- scrib ed in this pap er. First, giv en separable comm utative von Neumann algebras A and B , w e can write A ∼ = L ∞ ( X, µ ) and B ∼ = L ∞ ( Y , ν ) where X , Y are measurable spaces and µ, ν are measures. Then, given an ( B , A )-bimo dule, we can think of it as a representation of B ⊗ A ∼ = L ∞ ( Y × X , ν ⊗ µ ). By the remarks ab o v e, this representation comes from a ﬁeld of Hilb ert spaces on Y × X . Then, giv en a 2-morphism f : H → K , we can represent it as a measurable ﬁeld of b ounded op erators b et w een the corresp onding ﬁelds of Hilb ert spaces. While the details still need to b e work ed out, all this suggests that a theory of 2-Hilb ert spaces based on comm utative v on Neumann algebras should b e closely link ed to the theory of measurable categories describ ed here. Ev en b etter, the bicategory 2Hilb just describ ed sits inside a larger bicategory where w e drop the condition that the v on Neumann algebras b e comm utative. Representations of 2-groups in this larger bicategory should also b e interesting. The reason is that Sc hreib er has con vincing evidence that the w ork of Stolz and T eichner [ 68 ] provides a representation of the so-called ‘string 2-group’ [ 6 ] inside this larger bicategory . F or details, see the last section of Sc hreib er’s recent pap er on tw o approaches to quan tum ﬁeld theory [ 67 ]. This is y et another hint that inﬁnite-dimensional representations of 2-groups may someday b e useful in ph ysics. Ac kno wledgments W e thank Jeﬀrey Morton for collaboration in the early stages of this pro ject. W e also thank Jerome Kamink er, Benjamin W eiss, and the denizens of the n -Category Caf ´ e, esp ecially Bruce Bartlett and Urs Schreiber, for many useful discussions. Yves de Cornulier and T o dd T rimble came up with most of the ideas in Appendix A.3 . Our w ork was supp orted in part b y the National Science F oundation under grant DMS-0636297, and by the P erimeter Institute for Theoretical Physics. A T ools from measure theory This App endix summarizes some to ols of measure theory used in the pap er. The ﬁrst section recalls basic terminology and states the well-kno wn Leb esgue decomp osition and Radon-Nik o dym theorems. The second section deﬁnes the geometric mean of tw o measures and deriv es of some of 99 its key prop erties. The third section studies measurable ab elian groups and their duals. Finally , the fourth section presents a few standard results ab out measure theory on G -spaces. Recall that for us, a measurable space is shorthand for a standard Borel space : that is, a set X with a σ -algebra B of subsets generated b y the op en sets for some second coun table lo cally compact Hausdorﬀ top ology on X . W e gav e tw o other equiv alen t deﬁnitions of this concept in Prop. 14 . Also recall that for us, all measures are σ -ﬁnite. So, a measure on X is a function µ : B → [0 , + ∞ ] suc h that µ ( [ n A n ) = X n µ ( A n ) for an y sequence ( A n ) n ∈ N of m utually disjoint measurable sets, such that X is a countable union of S i ∈ B with µ ( S i ) < ∞ . A.1 Leb esgue decomp osition and Radon-Nik o dym deriv ativ es In a ﬁxed measurable space X , a measure t is absolutely contin uous with respect to a measure u , written t  u , if ev ery u -n ull set is also t -null. The measures are equiv alen t , written t ∼ u , if they are absolutely contin uous with resp ect to each other: in other words, they hav e the same n ull sets. The tw o measures are mutually singular , written t ⊥ u , if we can ﬁnd a measurable set A ⊆ X suc h that t ( A ) = u ( X − A ) = 0 . If A ⊆ X is a measurable set with u ( X − A ) = 0 w e s a y the measure u is supp orted on A . Theorem 106 (Leb esgue decomp osition) L et t and u b e ( σ -ﬁnite) me asur es on X . Then ther e is a unique p air of me asur es t u and t u such that t = t u + t u with t u  u and t u ⊥ u. The notation chosen here is particularly useful when we ha ve more than t wo measures around and need to distinguish b et ween Lebesgue decomp ositions with resp ect to diﬀerent measures. This result is completed by the following useful propositions. Fix tw o measures t and u on X . Prop osition 107 In the L eb esgue de c omp osition t = t u + t u , we have t u ⊥ t u . Pro of: Giv en that t u ⊥ u , there is a measurable set A such that u is supp orted on A and t u is supp orted on X − A : u ( S ) = u ( S ∩ A ) t u ( S ) = t u ( S − A ) for all measurable sets S . But then absolute con tinuit y of t u with respect to u implies t u ( X − A ) = 0, and therefore t u ( S ) = t u ( S ∩ A ). That is, t u is supp orted on A , so t u ⊥ t u . Prop osition 108 Consider the L eb esgue de c omp ositions t = t u + t u and u = u t + u t . Then t u ⊥ u t and t u ∼ u t . Pro of: Giv en that t u ⊥ u , there is a measurable set A such that u is supp orted on A and t u is supp orted on X − A . Note ﬁrst that u t is supp orted on A , as u is. This shows that t u ⊥ u t . Next, ﬁx a t u -n ull set S ; we th us hav e that t ( S ) = t u ( S ). Since t u is supp orted on X − A , it follo ws that t ( S ∩ A ) = 0. Using the fact that u t  t , we obtain u t ( S ∩ A ) = 0. But u t is supp orted on A , as u is; therefore u t ( S ) = u t ( S ∩ A ) = 0. Thus, w e hav e shown that u t  t u . W e show similarly t u  u t , and conclude that t u ∼ u t . The Lebesgue decomp osition theorem is reﬁned b y the Radon–Nik o dym theorem, whic h pro vides a classiﬁcation of absolutely contin uous measures: 100 Theorem 109 (Radon-Nikodym) L et t and u b e two σ -ﬁnite me asur es on X . Then t  u if and only if t c an b e written as u times a function d t d u , the Radon–Nik o dym deriv ative : that is, t ( A ) = Z A d u d t d u A.2 Geometric mean measure Supp ose X is a measurable space on which are deﬁned tw o measures, u and t . If eac h measure is absolutely contin uous with resp ect to the other, then we ha ve the equalit y r d t d u d u = r d u d t d t so we can deﬁne the ‘geometric mean’ √ d u d t of the tw o measures to b e giv en by either side of this equalit y . In the more general case, where u and t are not necessarily m utually absolutely con tinuous, w e may still deﬁne √ d u d t , as we shall see. Using the notation of the ﬁrst section we hav e the following key fact. Recall once more that all our measures are assumed σ -ﬁnite. Prop osition 110 If u and t ar e me asur es on the same me asur able sp ac e X then r d t u d u d u = r d u t d t d t Pro of: Our notation for the Leb esgue decomp osition means u = u t + u t u t  t u t ⊥ t and likewise, t = t u + t u t u  u t u ⊥ u. Prop. 107 shows that u t and u t are mutually singular. So there is a measurable set A with t and u t are supp orted on A and, and u t supp orted on X − A . Similarly , there is a measurable set B with u and t u supp orted on B , and t u supp orted on X − B . These sets divide X in to four subsets: A ∩ B , A − B , B − A , and X − ( B ∪ A ). The uniqueness of the Leb esgue decomp osition implies the decomp osition of the res triction of a measure is given by the restriction of the decomp osition. On A ∩ B , u and t restrict to u t and t u , whic h are m utually absolutely contin uous. Hence, on this subset, we hav e r d t u d u d u = r d u t d t d t On the other three subsets of X , we hav e, resp ectively u = 0, t = 0, and u = t = 0. In each case, b oth sides of the previous equation are zero. Giv en this prop osition, w e deﬁne the geometric mean of the measures u and t to b e: √ d t d u := r d t u d u d u = r d u t d t d t Outside of this app endix, to reduce notational clutter, w e generally drop the superscripts in Radon– Nik o dym deriv ativ es and simply write, for example: d t d u := d t u d u . 101 Prop osition 111 L et t , u b e me asur es on X . Then a set is √ tu -nul l if and only if it is the union of a t -nul l set and a u -nul l set. Equivalently, expr esse d in terms of almost-everywher e e quivalenc e, the r elation ‘ √ tu -a.e.’ is the tr ansitive closur e of the union of the r elations ‘ t -a.e.’ and ‘ u -a.e.’. Pro of: First, √ tu  t and √ tu  u ; indeed √ tu is equiv alen t to b oth t u and u t . So clearly the union of a t -null set and a u -null set is also √ tu -n ull. Con versely , supp ose D ⊆ X has √ tu ( D ) = 0. Then u ( D ) = ¯ u t ( D ), and t ( D ) = ¯ t u ( D ). But ¯ u t ⊥ ¯ t u , so w e can pick a set P ⊆ X on whic h ¯ u t is supp orted and ¯ t u v anishes. Then t ( D ∩ P ) = 0 and u ( D − P ) = 0, so D is the union of a t -null set and a u -null set. Expressing this in terms of equiv alence relations, supp ose f 1 ( x ) = f 2 ( x ) √ tu - a.e. in the v ariable x ; w e will construct g ( x ) suc h that g ( x ) = f 1 ( x ) t - a.e. and g ( x ) = f 2 ( x ) u - a.e. . Let D b e the set on whic h f 1 and f 2 diﬀer, and let P b e the set deﬁned in the previous paragraph. Set g ( x ) := f 1 ( x ) = f 2 ( x ) on X − D , g ( x ) := f 1 ( x ) on D − P , and g ( x ) := f 2 ( x ) on D ∩ P . This deﬁnes g on all of x . No w f 1 and g diﬀer only on D ∩ P , whic h has t -measure 0; f 2 and g diﬀer only on D − P , whic h has u -measure 0. No w supp ose we hav e three measures t , u , and v on the same space. How are the geometric means √ d t d u and √ d t d v related? An answer to this question is given by the following lemma, whic h is useful for rewriting an integral with resp ect to one of these geometric means as an integral with resp ect the other. Lemma 112 L et t , u , and v b e me asur es on X . Then we have an e quality of me asur es √ d t d u r d v u d u = √ d t d v r d v t d t r d u v d v r d t u d u Pro of: Let us ﬁrst deﬁne a measure µ by the left side of the desired equalit y: d µ = √ d t d u r d v u d u W e then hav e, using the deﬁnition of geometric mean measure, d µ = d t r d u t d t r d t u d u = (d t v + d t v ) r d u t d t r d t u d u where the latter expression gives the Lebesgue decomp osition of µ with resp ect to v . How ever, as w e sho w momentarily , the singular part of this decomp osition is identically zero. Assuming this result for the moment, w e then hav e d µ = d t v r d u t d t r d t u d u = d v d t v d v r d u t d t r d t u d u = √ d t d v r d v t d t r d u v d v r d t u d u 102 as we wished to show. T o complete the pro of, we thus need only see that the t v part of µ v anishes: d t v r d u t d t r d v u d u = 0 That is, we m ust show that µ v ( X ) = Z X d t v r d u t d t r d v u d u = 0 . Let Y ⊆ X b e a measurable set suc h that t v is supported on Y , while v and t v are supported on its complemen t: v = v | X − Y t v = t v | X − Y t v = t v | Y Similarly , let A ⊆ X b e suc h that t = t | A u t = ut | A u t = u t | X − A Note that d u t d t v anishes t – a.e. , and hence t v – a.e. on X − A . Thus the measure d t v r d u t d t is zero on X − A . Since we also ha ve t v v anishing on X − Y , we ha ve µ v ( X ) = Z Y ∩ A d t v r d u t d t r d v u d u . No w by construction of Y , w e hav e v ( Y ∩ A ) = 0, and hence v u ( Y ∩ A ) = 0. So d v u d u v anishes u – a.e. , and hence u t – a.e. , on Y ∩ A . If C ⊆ Y ∩ A is the set of p oin ts where the latter Radon–Nik o dym deriv ativ e do es not v anish, then u t ( C ) = 0 implies that r d u t d t v anishes t – a.e. , hence t v – a.e. on C. Thus µ v ( X ) = Z C d t v r d u t d t r d v u d u = 0 , so µ is absolutely contin uous with respect to v . Prop osition 113 L et t, u b e me asur es on X , and c onsider the L eb esgue de c omp ositions t = t u + t u and u = u t + u t . Then: d u t d t d t u d u = 1 √ tu − a.e. Pro of: Applying Lemma 112 with v = u we get √ d t d u = √ d t d u r d u t d t r d t u d u Th us the function d u t d t d t u d u diﬀers from 1 at most on a set of √ tu -measure zero. 103 A.3 Measurable groups Giv en a measurable group H , it is natural to ask whether H ∗ is again a measurable group. The main goal of this section is to present necessary and suﬃcien t conditions for this to be so. These conditions are due to Yv es de Corn ulier and T o dd T rimble. W e also sho w that when H and H ∗ are measurable, a con tinuous action of a measurable group G on H gives a contin uous action of G on H ∗ . Recall that for us, a measurable group is a locally compact Hausdorﬀ second coun table top o- logical group. Any measurable group becomes a measurable space with its σ -algebra of Borel subsets. The multiplication and in verse maps for the group are then measurable. How ev er, not every mea- surable space that is a group with measurable multiplication and inv erse maps can b e promoted to a measurable group in our sense! There ma y b e no second coun table lo cally compact Hausdorﬀ top ology making these maps contin uous. Luc kily , all the counterexamples are fairly exotic [ 19 , Sec. 1.6]. Lemma 114 A me asur able homomorphism b etwe en me asur able gr oups is c ontinuous. Pro of: V arious pro ofs can b e found in the literature. F or example, Kleppner show ed that a measurable homomorphism b et ween locally compact groups is automatically contin uous [ 45 ]. Giv en a measurable group H , we let H ∗ b e the set of measurable — or equiv alently , by Lemma 114 , contin uous — homomorphisms from H to C × . W e make H ∗ in to a top ological space with the compact-op en topology . H ∗ then b ecomes a top ological group under p oin twise multiplication. The ﬁrst step in analyzing H ∗ is noting that every con tinuous homomorphism χ : H → C × is trivial on the commutator subgroup [ H , H ] and thus also on its closure [ H , H ]. This lets us reduce the problem from H to Ab( H ) = H / [ H, H ] , whic h b ecomes a top ological group with the quotient top ology . Let π : H → Ab( H ) be the quotien t map. Then w e hav e: Lemma 115 Supp ose H is a me asur able gr oup. Then Ab( H ) is a me asur able gr oup. Ab( H ) ∗ is a me asur able gr oup if and only if H ∗ is, and in this c ase the map π ∗ : Ab( H ) ∗ → H ∗ χ 7→ χπ is an isomorphism of me asur able gr oups. Pro of: Supp ose H is a measurable group: that is, a second countable lo cally compact Hausdorﬀ group. By Lemma 122 , the quotient Ab( H ) is a second countable lo cally compact Hausdorﬀ space b ecause the subgroup [ H , H ] is closed. So, Ab( H ) is a measurable group. The map π ∗ is a bijection b ecause every con tinuous homomorphism φ : H → C × equals the iden- tit y on [ H , H ] and thus can b e written as χπ for a unique contin uous homomorphism φ : Ab( H ) → C × . W e can also see that π ∗ is contin uous. Suppose a net χ α ∈ Ab( H ) ∗ con verges uniformly to χ ∈ Ab( H ) ∗ on compact subsets of Ab( H ). Then if K ⊆ H is compact, χ α π conv erges uniformly to χπ on K b ecause χ a con verges uniformly to χ on the compact set π ( K ). It follows that π ∗ : Ab( H ) ∗ → H ∗ is a contin uous bijection b etw een second countable lo cally compact Hausdorﬀ spaces. This induces a measurable bijection b etw een measurable spaces. Such a map alwa ys has a measurable inv erse [ 57 , Chap. I, Cor. 3.3]. (This reference describ es measurable spaces in terms of separable metric spaces, but we hav e seen in Lemma 14 that this characterization 104 is equiv alent to the one we are using here.) So, π ∗ is an isomorphism of measurable spaces. Since it is a group homomorphism, it is also an isomorphism of measurable groups. Thanks to the ab o ve result, w e henceforth assume H is an ab elian measurable group. Since C × ∼ = U(1) × R as top ological groups, w e hav e H ∗ ∼ = hom( H , U(1)) × hom( H, R ) as topological groups, where hom denotes the space of con tinuous homomorphisms equipped with its compact-op en top ology and made in to a top ological group using p oin twise multiplication. The top ological group ˆ H = hom( H , U(1)) is the sub ject of Pon trjagin dualit y so this part of H ∗ is well-understoo d [ 1 , 54 , 58 ]. In particular: Lemma 116 If H is an ab elian me asur able gr oup, so is its Pontrjagin dual ˆ H . Pro of: It is well-kno wn that whenever H is an ab elian lo cally compact Hausdorﬀ group, so is ˆ H [ 54 , Thm. 10]. So, let us assume in addition that H is second coun table, and sho w the same for ˆ H . F or this, ﬁrst note by Lemma 14 that H is metrizable. A lo cally compact second-countable space is clearly σ -compact, so H is also σ -compact. Second, note that a lo cally compact Hausdorﬀ ab elian group H is metrizable if and only ˆ H is σ -compact [ 54 , Thm. 29]. It follows that ˆ H is also σ -compact and metrizable. Since a compact metric space is second coun table (for eac h n it admits a ﬁnite co vering by balls of radius 1 /n ), so is a σ -compact metric space. It follo ws that ˆ H is second countable. The issue th us boils do wn to: if H is an abelian measurable group, is hom( H , R ) also measurable? Sadly , the answer is “no”. Supp ose H is the free ab elian group on countably man y generators. Then hom( H , R ) is a countable product of copies of R , with its pro duct top ology . This space is not lo cally compact. Luc kily , there is a sense in whic h this coun terexample is the only problem: Lemma 117 Supp ose that H is an ab elian me asur able gr oup. Then hom( H , R ) is me asur able if and only if the fr e e ab elian gr oup on c ountably many gener ators is not a discr ete sub gr oup of H . Pro of: First supp ose H is an ab elian lo cally compact Hausdorﬀ group. Then H has a compact subgroup K suc h that H/K is a Lie group, p erhaps with inﬁnitely many connected comp onen ts [ 42 , Cor. 7.54]. Since an y connected ab elian Lie group is the pro duct of R n and a torus, we can enlarge K while keeping it compact to ensure that the identit y comp onen t of H /K is R n . An y contin uous homomorphism from a compact group to R must hav e compact range, and th us b e trivial. It follows that K lies in the kernel of an y χ ∈ hom( H, R ), so hom( H , R ) ∼ = hom( H /K, R ) . So, without loss of generality we can replace H by H /K . In other words, we may assume that H is an ab elian Lie group with R n as its identit y comp onen t. The only subtlety is that H may ha ve inﬁnitely many comp onents. 105 Since R n is a divisible ab elian group, the inclusion j : R n → H comes with a homomorphism p : H → R n with pj = 1, so w e actually hav e H ∼ = R n × A as abstract groups, where A is the range of p . Since A ∩ R n is trivial, A is actually a discrete subgroup of H . So, as a top ological group H m ust b e the pro duct of R n and a discrete ab elian group A . It follows that hom( H , R ) ∼ = R n × hom( A, R ) , so without loss of generalit y w e may replace H by the discrete abelian group A , and ask if hom( A, R ) is measurable. Since homomorphisms χ : A → R v anish on the torsion of A , w e ma y assume A is torsion-free. There are tw o alternatives now: 1. A has ﬁnite rank: i.e., it is a subgroup of the discrete group Q k for some ﬁnite k . If w e c ho ose the smallest such k , then A contains a subgroup isomorphic to Z k suc h that the natural restriction map hom( A, R ) → hom( Z k , R ) is an isomorphism (actually of top ological groups). Since hom( Z k , R ) is lo cally compact, Hausdorﬀ, and second coun table, so is hom( A, R ). So, in this case our original top ological group hom( H , R ) is measurable. 2. A has inﬁnite rank. This happens precisely when our original group H contains the free ab elian group on a coun table inﬁnite set of generators as a discrete subgroup. In this case we can sho w that hom( A, R ) and th us our original top ological group hom( H , R ) is not lo cally compact. T o see this, let U b e any neighborho o d of 0 in hom( A, R ). By the deﬁnition of the compact- op en top ology , there is a compact (and thus ﬁnite) subset K ⊆ A and a num ber r > 0 suc h that U contains the set V consisting of χ ∈ hom( A, R ) with | χ ( a ) | ≤ r for all a ∈ K . It suﬃces to show that V is not relativ ely compact. T o do this, w e shall ﬁnd a sequence χ n ∈ V with no cluster p oin t. Since A has inﬁnite rank, w e can ﬁnd a ∈ A such that the subgroup generated b y a has trivial in tersection with the ﬁnite set K . F or each n ∈ N , there is a unique homomorphism φ n from the subgroup generated by a and K to R with φ n ( a ) = n and φ n ( K ) = 0. Since R is a divisible abelian group, we can extend φ n to a homomorphism χ n : A → R . Since χ n v anishes on K , it lies in V . But since χ n ( a ) = n , there can b e no cluster p oint in the sequence χ n . Com bining all these lemmas, we easily conclude: Theorem 118 Supp ose H is a me asur able gr oup. Then H ∗ is a me asur able gr oup if and only if the fr e e ab elian gr oup on c ountably many gener ators is not a discr ete sub gr oup of Ab( H ) . This is true, for example, if H has ﬁnitely many c onne cte d c omp onents. W e also hav e: Lemma 119 L et G and H b e me asur able gr oups with a left action B of G as automorphisms of H such that the map B : G × H → H is c ontinuous. Then the right action of G on H ∗ given by χ g [ h ] = χ [ g B h ] is also c ontinuous. 106 Pro of: Recall that H ∗ has the induced top ology coming from the fact that it is a subset of the space of con tinuous maps C ( H , C × ) with its compact-op en top ology . So, it suﬃces to sho w that the follo wing map is is contin uous: C ( H, C × ) × G → C ( H , C × ) ( f , g ) 7→ f g where f g [ h ] = f [ g B h ] . This map is the comp osite of tw o maps: C ( H, C × ) × G 1 × α − → C ( H , C × ) × C ( H, H ) ◦ − → C ( H , C × ) ( f , g ) 7→ ( f , α ( g )) 7→ f ◦ α ( g ) = f g . where α ( g ) h = g B h. The ﬁrst map in this comp osite is con tinuous because α is: in fact, any con tinuous map B : X × Y → X determines a contin uous map α : Y → C ( X, X ) b y the ab o v e formula, as long as X and Y are locally compact Hausdorﬀ spaces. The second map C ( H, C × ) × C ( H, H ) ◦ → C ( H , C × ) is also contin uous, since comp osition C ( Y , Z ) × C ( X, Y ) ◦ → C ( X , Z ) is contin uous in the compact-op en top ology whenever X, Y and Z are lo cally compact Hausdorﬀ spaces. A.4 Measurable G -spaces Supp ose G is a measurable group. A (righ t) action of G on a measurable space X is a measurable if the map ( g , x ) 7→ xg of G × X into X is measurable. A measurable space X on whic h G acts measurably is called a measurable G -space . In fact, we can alwa ys equip a measurable G -space with a top ology for which the action of G is con tinuous: Lemma 120 [ 19 , Thm. 5.2.1] Supp ose G is a me asur able gr oup and X is a me asur able sp ac e with σ -algebr a B . Then ther e is a way to e quip X with a top olo gy such that: • X is a Polish sp ac e—i.e., home omorphic to sep ar able c omplete metric sp ac e, • B c onsists pr e cisely of the Bor el sets for this top olo gy, and • the action of G on X is c ontinuous. 107 Moreo ver: Lemma 121 [ 69 , Cor. 5.8] L et G b e a me asur able gr oup and let X b e a me asur able G -sp ac e. Then for every x ∈ X , the orbit xG = { xg : g ∈ G } is a me asur able subset of X ; mor e over the stabilizer S x = { g ∈ G | xg = x } is a close d sub gr oup of G . This result is imp ortan t for the following reason. Given a point x o ∈ X , the measurable map g 7→ x o g from G in to X allo ws us to measurably identify the orbit x o G with the homogeneous space G/S x o of righ t cosets S x o g , on which G acts in the obvious wa y . No w, suc h spaces enjoy some nice prop erties, some of which are listed b elo w. Fix a measurable group G and a closed subgroup S of G . Lemma 122 [ 49 , Thm. 7.2] The homo gene ous sp ac e X = G/S , e quipp e d with the quotient top ol- o gy, is a Polish sp ac e. Sinc e the action of G on X is c ontinuous, it fol lows that X b e c omes a me asur able G -sp ac e when endowe d with its σ -algebr a of Bor el sets. Let π : G → G/S denote the canonical pro jection. A measurable section for G/S is a measur- able map s : G/S → G suc h that π s is the identit y on G/S and s ( π (1)) = 1, where 1 is the identit y in G . Lemma 123 [ 48 , Lemma 1.1] Ther e exist me asur able se ctions for G/S . Next w e presen t a classic result concerning quasi-in v ariant measures on homogeneous spaces. Let X b e a measurable G -space, and µ a measure on X . F or each g ∈ G , deﬁne a new measure µ g b y setting µ g ( A ) = µ ( Ag − 1 ). W e sa y the measure is in v arian t if µ g = µ for each g ∈ G ; we say it is quasi-in v arian t if µ g ∼ µ for eac h g ∈ G . 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