Categorifying measure theory: a roadmap

CA TEGORIFYING MEASURE THEOR Y: A RO ADMAP G. RODRIG UES Abstract. A program for c a tegorifying me asur e the ory is outlined. Contents 1. Int r o duction 2 Conten ts 6 Ac knowledgements 7 2. Measurable bundles of Hilbe rt spaces 7 3. T ow ar ds ca teg oriﬁed measur e theory 11 3.1. Hilb as a categor iﬁed ring 12 3.2. Measurability of bundles 15 3.3. Linearity o f the direct in tegral 20 3.4. Cont inuit y of the dire ct integral 22 3.5. Univ ers al pro p er ty of the direct integral 24 4. F rom Hilbert to Banach spa ces 28 4.1. The failure of the Radon-Nik o dym pro p e rty 33 5. F ore g rounding: measure alge br as and integrals 37 5.1. The Bana ch algebra L ∞ (Ω) 40 5.2. The Bo chner integral 43 5.3. The Stone space of a Boo lean alg ebra and weak integrals. 46 6. Banach 2-s paces 51 6.1. Presheaf ca tegories 53 6.2. F ree Banach 2-spaces over sets 59 6.3. Categoriﬁed measur e theory: the discrete case 62 7. Categoriﬁed meas ures and in tegra ls 65 7.1. Banach sheaves 65 7.2. Cosheav es and inverters 72 7.3. The cos he a ﬁﬁcation functor 79 7.4. The sp ectra l measure of a coshea f 81 References 87 This w ork w as supported b y the Pr o gr ama Op er acional Ciˆ encia e Inova¸ c˜ ao 2010 , ﬁnanced by the F und a¸ c˜ ao p ar a a Ciˆ encia e a T e cnolo gia (FCT / Portug al) and coﬁnanced by the Europ ean Commun i ty fund FEDER, in pa r t through the researc h pro ject Quant um T op ology POCI / MA T / 60352 / 2004. 1 2 G. RODRIGUES 1. Introduction Begin, epheb e, by p erce iv ing the idea Of this in ven tion, this inv ented world, The inco nceiv able idea o f the sun. Y ou m ust become an ignora nt man aga in And see the sun again with an igno rant eye And see it clearly in the idea of it. Never supp o s e an in ven ting mind as sour ce Of this idea nor for that mind compo se A voluminous master folded in his ﬁre. How clean the sun w he n seen in its idea, W ashed in the remotest cleanliness of a heav en That ha s exp elled us and our images. . . — W allace Stevens, fro m Notes T o war d a Supr eme Fiction . The word “ categor iﬁcation” was ﬁr st c o ined in [CF94]. The con text of the paper is the construction of four-dimensional top olo gic a l quantum ﬁeld the ories (TQFT’s for short) via state sums. TQFT’s were ﬁrst rigor ously deﬁned by M. Atiy ah in [A ti88], mo delled on G. Segal’s deﬁnition o f conformal ﬁeld theory ([Seg04]), with the inten tion of formalizing E . Witten’s w or k [Wit88] on 4-ma nifo ld inv ariants com- ing from the gauge-theore tic Donaldson theo ry . It was the critica l obser v ation of the authors o f [CF94] that (at least in low dimensions) there is a deep connection betw een o n one side, how the ( n + 1)-Pac hner mov es relate to the n -Pac hner mov es and on the other, how the alge braic structures in n - categor ies that pr ovide TQ FT’s get “catego riﬁed” in ( n + 1)-categor ies (the so-c alled “categoric a l ladder”). Besides [CF94], we refer the rea der to the introductio n of [Mac00] for a very lucid account of the stor y . Here, it suﬃces to say that in t wo dimensions, the constr uction of ma n- ifold inv a riants takes as input a semisimple algebr a ([FHK94]). The com binatoric s of the Pachner mov es rela ting equiv alent tria ngulations of the same s ur face exactly match the algebr aic laws of asso cia tive algebra s: the 2 -2 Pachner mov e corresp o nds to ass o ciativity of the multiplication and the 1 -3 mov e (the “triangula tion reﬁne- men t” mov e) c o rresp o nds to s emi-simplicity , a nd in essence it is a cutoﬀ, ﬁniteness condition. The construction of 3 - manifold in v ariants takes as alg ebraic input a cer- tain kind of mo noidal ca tegory ([BW96]) and thes e a rise naturally a s catego ries of representations of qua ntum groups ([T ur94]). The situation in four dimensions is more m ys terious (in mo r e senses than one), but following the categor iﬁc a tion cue of [CF94], [Ma c99] constructed 4-manifo ld in v aria nt s via state sums tha t take as al- gebraic input cer tain kinds o f monoidal 2- categor ies. W e currently lack non-trivial examples of suc h 2-ca tegories 1 , but once aga in following the categoriﬁc a tion hin t, it is expected that they arise a s monoida l 2-ca tegories of re presentations. The question w e are interested in answ ering is r epr ese nt ations in what ? Q ua ntu m groups ar e represented in ﬁnite-dimensiona l linea r space s so w e w ant some sort of linear 2-spaces. Since linear spaces are free, o ne plausible candidate for the categoriﬁe d analog ues of linear space s a re categ ories equiv alent to V ect n where 1 Non-triviality of the 2-category here can b e tak en to m ean that when f ed as algebraic input int o the state sum mac hinery , it is able to detect mor e than j ust the homotop y ty p e of the manifold. CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 3 V ect is the category of ﬁnite-dimensio nal K -linear spaces with K a ﬁeld 2 . This is precisely the deﬁnition of linear 2-space adv anced in [KV94]. These categor ies ca n be characterized intrinsically by ab elia nness plus semi-simplicity (see [Y et ]). The crucial fact is that just as the linear spaces K n hav e a canonica l basis, the categ ories V ect n come equipp ed with a canonical basis o f ob jects e i deﬁned by: e i ( j ) = def ( K if i = j , 0 otherwise. There is now a canonica l decomp osition with e a ch V ∈ V ect n isomorphic to a direct sum of e i tensored with linear spac e s V i ∈ V ect : V ∼ = M i V i ⊗ e i (1.1) Another impor tant fact is that the categor iﬁcation T : V ect n − → V ect n of the notion of linear map can b e in trinsica lly deﬁned, but turns out to b e equiv alen t to a square matrix of linear spa ces  T i j  . The action of T o n an ob ject V = ( V i ) ∈ V ect n is the v ector whos e jth compo ne nt is T ( V )( j ) ∼ = M i T i j ⊗ V i Comp osition is g iven by the us ual matrix comp ositio n formula by replac ing sums with direct sums ⊕ and pro ducts b y tensor products ⊗ . Note that with these fo r - m ula s , c omp osition is only a sso ciative up to unique, c anonic a l line ar isomorph ism . This means that w e end up with a bic ate gory of 2-s paces, morphisms T and 2 - morphisms b etw een them. The latter can b e identiﬁed with matrices of linear maps τ i j : T i j − → S i j with comp o s ition b eing point wise compositio n of matrices. Up to now, these ar e all featur e s one co uld rea s onably exp ect from a c ategori- ﬁcation of linear algebra. But w e so on r ealize that these catego riﬁed linear spaces exhibit some unexp ected b ehavior. The gist is that the notion of basis allowing the decomp osition (1.1) is subtler than app ears to b e at a ﬁr st glance. F or example, it implies that every self-equiv alence V ect n − → V ect n amounts to a p ermutation of the canonica l ba sis ( e i ) of V ect n . As ﬁrst obser ved in [BM06 b], since Lie groups G tend to ha ve few p ermutation r epresentations, this en tails that if the group o b ject of a 2-g roup G is s uch a L ie g roup, then there a re few repr esentations of G in V ect n . This indicates that to obtain non-trivia l representation 2 -categor ies, one m ust lo ok around for monoidal categories with larger groupal g roup oids of self-equiv a- lences. If we think ab out the constructio n o f V ect n one immediate poss ibilit y is to replace n with inﬁnite sets X . S ince the categor y V ect X is equiv alent to the category of bund les of line ar sp ac es over X , the next lo g ical step is to imp ose some structure on the ba s e set X (a top olog y , a measurable structure, etc.) and then consider the category o f corr e sp onding (contin uous, mea s urable, etc.) bundles. At this p oint, it may ev en co me to the re a der’s mind the analogous diﬃculties that one faces when r epresenting locally compact groups: ﬁnite-dimens io nal spaces are not enough a nd w e hav e to enlarge the r e presenting c ategory to inc lude inﬁnite- dimensional ones. And just as measure theory is the natura l pla ce to ﬁnd suc h inﬁnite-dimensional s pa ces larg e eno ug h to yield non-tr iv ial represe ntations, we 2 In concrete constructions, K is usually the complex num b ers but at the level of generality we are working, it makes no diﬀerence whatso ev er what base ﬁeld one takes. 4 G. RODRIGUES exp ect some so rt of ca tegoriﬁed meas ur e theory to b e a natural place to ﬁnd inﬁnite- dimensional analogues of V ect n that (may) yield non-trivial representations o f 2- groups. Note the emphasis here: wha t we a re really after are inﬁnite-dimensional categorie s, and mo re gener ally , c ate goriﬁe d functional analysis . Measure theory is pla ying the role of the middle man, providing us with a whole class of na tur al examples. This is the p oint of depar ture for [Y et05 ] and its concomita nt suggestion of lo oking at categor ies whose o b jects are measurable ﬁelds o f Hilbert spaces and with mor phisms the measurable ﬁelds of b ounded linear ma ps. These are class ical analytical ob jects of study and their theory is explained in several tex tbo o ks of which w e c a n cite [Mac76], [Dix81] and [T ak 02]. An imp or tant ingredient is that the canonical decomp ositio n (1.1 ) is replac ed by a dir e ct inte gr al de c o mp osition in the for m, ξ ∼ = Z ⊕ X H x d µ (1.2) where H x is a measurable ﬁeld on X and µ a scala r meas ure. A pro blem with these categor ies of measurable ﬁelds is that while in the pre- vious discrete situation the right notion of categoriﬁed linear ma p is more or less immediate and c an ev en b e intrinsically characterized, the case is not so clear-c ut for what is the right notion of ca teg oriﬁed (b ounded) linear ma p b etw een them. Nevertheless, a solution was found in [Y et05], even if needing s omewhat obscure, techn ica l hypotheses on the bas e measure spaces. The integral notation for the ob ject on the righ t hand s ide of (1.2) is a cute but suggestive no ta tion for a s pace of squa re-integrable sec tions of a bundle. In this pap er we in tend to take the no tation (1.2) seriously as a c ate goriﬁe d inte gr al , or in o ther words, to a rgue that the constructions of [Y et0 5] (and such followups as [BBFW08]) can b e interpreted as c ate gorify ing me asur e the ory . T o motiv ate and explain such an interpretation is a long v oyage with a couple of false starts, the inevitable detours and a g o o d deal o f backtrac king. In the words of W. Stevens’ int ense po em, w e m ust b eco me an ignorant man again and per ceive anew the in- conceiv able idea of a bright sun. I dir ect the r eader to the end of this section for a brief description of the co nten ts of the pap er and end this intro duction with a motley collectio n of remarks. First, the ambitions of the pap e r are far more mo dest than what the title may indicate. Although certain re ﬂe c tions of wider signiﬁca nce on the sco p e and con- ceptual meaning of catego r iﬁcation will b e made in due course, I will not attempt to be systematic or construct a ny sor t of coherent framework. Other categor iﬁcations of measur e theory exhibiting other prop er ties may very well exist, and although at times I may hin t at poss ible generaliza tions or further av enues o f r esearch, the choices and comments ar e alwa ys guided and framed by the desire to in terpret the sp eciﬁc set of constructions in [Y et05] as a c ategoriﬁca tion, not the categoriﬁca- tion, of mea s ure theor y . Also, since the main c o ncern is with c o nceptual issues, no applications will b e given. The ragtag reﬂectio ns o n o ther, inciden tal matters, ar e relegated to the footnotes, where their small prin t is not able to dis tr act innocent readers, ca use a fuss o r otherwise do an y ha rm. Throughout, I hav e concen tra ted on general ideas, no t on the spec iﬁc details of which ther e ar e to o many to b e cra mmed in a pap er with an ything r esembling CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 5 a rea sonable size. The basic results will b e exp ounded in the sto ut handb o o k deﬁnition-theorem-co rollar y format, but the pro ofs are only sketc hed, if at all. Pre- po sterous as it ma y s o und, there is a real po s sibility that these so-called theorems are not theo rems at a ll and I have b otched things hor ribly somewher e along the wa y . An y lingering doubts in the reader ’s mind will only be alleviated in future pap ers ([Ro da] and [Ro db]), where I intend to pro vide all the missing details. I have also endeav o red to make the pap er accessible to the widest audience po ssible, by surveying, even if sketch ily , the necess a ry background mater ial, so that the reader not acquainted with it may at least be a ble to pick up the bas ic ideas. There are many ma thematical ﬁelds that will b e to uched up o n a nd for what cannot b e reviewed within a re a sonable spa c e , I hav e tried to provide reference s to the relev ant literature. Naturally enoug h, some k nowledge of the tw o central pillars, functional analys is and measure theory on one side and catego ry theory on the o ther, is required. Despite ﬁrst app ear ances, the amount of functional analysis and measure theo r y needed is ac tua lly very mo dest (it is all that this author knows anywa y). F o r functional a nalysis, o ur basic reference is [Con9 0] and for measure theory it is [Hal7 4]. As far a s categor y theor y is concerned, the situation is diﬀerent and s ome so phisticated ar tillery will be deploy ed. F or any undeﬁned terms, the reader is referred to [Lan71] or the tw o v olumes [Bor94a] and [B o r94b]. F or mono idal categor ies, cons ult [Lan71, chapter VI I], while 2 -catego ries and their weak er siblings, bicategor ie s , are considered in [Bor94a, chapter 7]. Mor e sp eciﬁc accounts can b e found in [K S74] or in the mor e recent [Lac07] and refer ences therein. It is traditional in mathematical pape r s to put the conten ts in p ersp ective, re late them to earlier w or k and then demonstrate in a positive and asser tive wa y that mathematics has just taken a quantum lea p forward. Surveying the pap er, the reader will ha ve oppo rtunity to observe that all the ma jo r ideas in it (a nd mos t of the minor ones) have b een bor r ow ed (adapted, stolen, whatev er) from previous work, sca ttered throug ho ut numerous sources . Once the ma jor conceptual hurdles hav e b een overcome and the cor rect fra mework ident iﬁed, it is basically a ma tter of cranking the categor ial machinery and let it op erate its magic. Although at times I hav e b een forc e d to baptize some ob ject with a more descriptive name (e.g. Banach 2 -sp ac e instead of the uninspiring and opaque c o c omplete Ban c -enriche d c ate gory ), there are no essentially new concepts. The tra in o f idea s leading to the present pap er was put in motion by a n oﬀ-hand commen t o f L. Crane made during a minicour se on unitary repre s entations of lo cally compact gr oups (Lisbo n, 1 998-1 999), to the eﬀect that mea s urable ﬁelds of Hilber t spaces are a “ sort of shea ves.” They were developed intermitten tly during a long p erio d, in isolatio n a nd at a time of great per sonal diﬃcult y . At some p o int, I have lo st tr a ck of what was originally my own lab or a nd what was b or row ed 3 . I hav e tried to pro vide as ma ny references as po ssible (which is basically everything c onnected with this work that I hav e laid m y eyes upon, and then some more), which accounts for the inordina te length o f the bibliogra phy , but have swerved awa y from settling matters of prec edency . If the r eader feels that I a m being to o cav alie r ab out an is sue of such a paramount impo rtance, I am all to o happy to acknowledge that Someo ne Else is to b e blamed for this sk ein of thor ns. 3 Dr. Johnson, the mortal god of my imaginings, f amously retorted to an author: “Y our man- uscript is b oth go o d and or iginal; but the parts that are go o d are not original, and the parts that are ori ginal ar e not go o d.” 6 G. RODRIGUES A ﬁnal note o n style is in order. This pap er being just an initial mapping stage, I have allow ed myself mor e freedo m in writing it. Some reader s may ﬁnd the o cca- sional quirkiness distasteful, others may dis like the displa y of liter ary p edantry . I will not apolo gize for it, but if it is any consolation I remark that initia l versions featured such precious tidbits as a go ofy limerick; Biblical passages with a ccom- panying theo lo gical commentary; several jokes in p o or taste and an even larger nu mber of failed puns. Whatever remained either slipp ed b y the Censor’s vigila nt watc h or is s o inextricably tied to the matter that it could no t be deleted without injuring the in tegrity of the whole. Such as the following q uotation: THE Neces sity of this Dig r ession, will easily excuse the Leng th; and I hav e chosen for it as pr o p er a Pla c e as I could rea dily ﬁnd. If the judicious Reader can a ssign a ﬁtter, I do here emp ow er him to remov e it into any other Corner he pleases. And so I return with great Alacrit y to pursue a mor e imp ortant Concern. — Jonathan Swift, from the section A Digr ession in Pr aise of Di- gr essions of A T ale of a T u b . Con tents. In section 2, the fundamental co ncept of measurable ﬁeld of Hilb er t spaces is discussed. Sections 3 and 4 form the hea r t and soul of the pape r . In them, I try out a pr eliminary des cription intended to yield a model of ca tegoriﬁed measure theor y . The discussion is somewhat long -winded, ev en repe titious , but I feel it ne c e ssary if the certitude and pre cision of the model is to be established. In section 3 , measur able ﬁelds of Hilb ert s paces a nd the foundational pap er [Y et0 5] are revisited in order to extra ct the most important principles of ca tegoriﬁed measure theory , a nd then in section 4 the mos t impor tant obstr uctions are faced, namely the need to include Banach s pa ces and the failure of the Radon-Nikodym theor em in inﬁnite dimensions. Ha ving raised the dust and t hen swept it out, we pro ceed in the next thre e sections w ith whatever unguen ts o f revelation w e ma y p o s sess and ﬂesh out the basic principles exp ounded in s ections 3 a nd 4. In section 5 we enter in review mo de and anchor the required measure theory on measur e algebra s instead of mea sure spaces. This will not only en tail some technical simpliﬁcatio ns 4 , but it will open imp o rtant new vistas on the ca teg oriﬁcatio n of mea sure theor y . Subsec- tion 5.1 constr ucts the Ba nach algebr a L ∞ (Ω), the universal home for all measures on Ω and s ubs ection 5.2 constructs the Bochner integral for an y vector measure ν o n a measure algebr a and discusses some of the basic theor ems of measure the- ory . Section 6 starts by discussing very brieﬂy Banach 2-spa ces, the categoriﬁed analogue of Banach s paces. The main bulk of the section is subsection 6.1 that is devoted to r e v iewing basic concepts o f enriched categor y theor y like Ka n ex tensions and coends. This is then applied in subsection 6.2 to a treatment of the Banach 2-spaces of Banach bundles a nd in s ubsection 6.3 to a complete treatmen t of cate- goriﬁed measure theory in the discr ete σ -ﬁnite case , tha t is, the case o f the atomic complete Bo olean alg ebras 2 X for a co unt a ble s et X . In the ﬁnal section 7 we give some of the details of the fundamen tal constructions o f ca teg oriﬁed measur e theory . The ﬁrst subsection 7.1 starts by de ﬁning the three Gro thendieck to po logies on a Bo olean algebra and ends with the fundament a l theorem 7.8 that ident iﬁes s heav es for the ﬁnite to po logy with the internal Banach spaces of the topos Shv (S(Ω)). 4 The reader wi ll ﬁnd no mention of descriptive set theory and suc h concepts as analytic set or Luzhin space, that were needed in [Y et 05 ] to deﬁne the correct notion of categ ori ﬁed linear map. CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 7 The next three subsections are devoted to co sheav es. Subsection 7.2 shows that Sh v (Ω) is the B anach 2-space birepresenting coshea ves, subsection 7.3 constr ucts the coshea ﬁﬁcation functor and subsection 7.4 co nstructs the sp ectral meas ur e of a cosheaf. In this ﬁnal subsec tio n, the Rado n-Nikodym pr o p erty c ha racterizing dir ect int eg rals is ﬁna lly dis closed. Ac kno wledg ements. Several pa rticipants of the math newsgr o up sci.math hav e kindly a nswered my questions and help ed me sort out some of the technical details. A special thanks to G. A. Edgar (measurabilit y in Banac h spaces and the Radon- Nikodym pro pe rty), A. N. Niel (mea surability) and M. Olshok (density of functors). Many many thanks are ow ed to R. Pick en for his genero us encouragement, his help in tracking do wn some imp orta nt pap er s and for his comments on initial versions of this paper that help ed impr oving it enormously . 2. Measurable bundles of Hil ber t sp a ces The pap er [Y et05] is founded up on the clas s ical analytical to ol of me asur able ﬁelds of Hilb ert sp ac es , so we be g in at the b eginning a nd des crib e these ob jects, even if in a sketc hy wa y . F or more informa tion, we refer the r eader to [Dix81, chapter I I] or [Mac76, chapter 2]. Let ( X, Ω , µ ) be a me asur e sp ac e (or simply 5 X ), wher e ( X, Ω) is a measurable space (a set X tog ether with a σ -a lgebra of subsets of X ) and µ is a (positive, σ - additive) measure on Ω. T o disp el any p ossible co nfusio ns, the term σ -alg ebra as we employ it always ca r ries with it the implicit ass umption that the who le space X is measurable. W e will a lso assume that µ is a c omplete pr ob abi lity . The completeness condition can always b e arra nged for b y adding to Ω the σ -ideal of subsets o f µ -null sets. Being a probability measure is not as restrictive as it sounds, b eca us e in the class of σ -ﬁnite s paces this can also b e arra ng ed for by the following device: let E b e a p artition of u nity , that is, a countable s e t of pa ir wise disjoint measura ble sets of non-zero ﬁnite measur e such that X = S E n ∈E E n . Deﬁne the seque nc e of functions ( f n ) by , f n = def χ ( E n ) µ ( E n )2 n where χ ( E n ) is the characteristic function of E n , and put g n = def P n m =1 f m . Then g n is a n integrable po sitive function with k , k 1 -norm eq ual to n X m =1 Z X f m d µ = n X m =1 1 2 m An application of Leb esg ue’s dominated co nv ergence theor em ([Hal7 4 , theorem D, page 1 10]) tells us that the sequence ( g n ) conv erges in L 1 ( X ) to an almost everywhere 6 strictly positive function g with k g k 1 = 1. T aking the measure, ν ( E ) = def Z E g d µ we obtain a probabilit y space suc h that the iden tity induces an isomorphism with X in the category of measurable spaces a nd meas urable maps. 5 As i s customary , we denote structured sets b y their carrier s . Ironically , in this case this is actually a bad choice as wil l b e seen in section 5. 6 F r om no w on, we will use a.e. as an abbreviation for almost everywher e . 8 G. RODRIGUES With these pr eliminaries out of the wa y , we can deﬁne the notion of me asur able ﬁeld of H ilb ert sp ac es . Analogous to the situation in measure theory , measurability of ﬁelds is the intermediate concept allowing us to deﬁne their dir e ct inte gr al . View the set X as a discre te category . Then a functor X − → Hil b is the same thing as a family ( H x ) of Hilber t spa ces parameterized b y x ∈ X . These functor s can also b e descr ib ed as bund les . W e use the word bund le instead of the more classical termino logy of ﬁeld to emphasize their geometric kinship. Deﬁnition 2. 1 . A bund le ξ ov er a set X is a surjective map, π : P − → X such that the inv er s e image 7 π ∗ ( x ) is a Hilb ert space for every x ∈ X . The set P is the total sp ac e of the bundle and the Hilb ert spaces ξ x = π ∗ ( x ) a re the ﬁb ers ov er x ∈ X . The map π is the bu n d le pr oj e ction m ap . A map τ : ξ − → ζ b etwe en X -bund les ξ and ζ is a map b etw een the total spaces τ : P ξ − → P ζ such that diagram 2.1 is commu tative and for e very x ∈ X , the restriction τ | ξ x : ξ x − → ζ x to the ﬁber ξ x is a bounded linear map. P ξ π @ @ @ @ @ @ @ τ / / P ζ π ~ ~ ~ ~ ~ ~ ~ ~ ~ X Figure 2. 1 . Fibe r -preser ving co ndition for bundle maps. There is a n eq uiv alence betw een the catego ry Bun ( X ) of bundles ov er X as deﬁned in 2.1 and the categ o ry Hilb X of functors X − → Hilb giv en b y sending each bundle ξ to the functor x 7− → ξ x . F rom no w on, w e will routinely iden tify these tw o ca tegories via this equiv alence. Deﬁnition 2. 2. A se ction o f an X -bundle ξ is a map s : X − → P s uch that π s = 1 X . If ξ is the constant bundle x 7− → K then the set of sections o f ξ is iso mo rphic to the set of functions X − → K and we can deﬁne the Hilber t space L 2 ( ξ ) of squar e- inte gr able se ctions a s the s pace L 2 ( X ) o f square -integrable functions X − → K . The dir e ct int e gr al is the extension of this constr uctio n to a functor on bundles. But just as in or dinary integration theory we do not (and b y the axiom of c hoice, cannot) exp ect to integrate an a r bitrary function, we should not exp ect to b e able to assig n a “Hilb ert spa ce of square-integrable sections” to every bundle. The crux of the matter is that since the ﬁb ers ξ x v ary from p oint to p oint, there is no notion o f measurability av ailable and thus no w ay to fo r m the line ar subsp ac e of me asur able se ctions . The naive solution of ta king the set o f se c tio ns s : X − → P such that the function x 7− → k s ( x ) k is measurable do es not w or k, b ecause this set is not clo sed for addition. The following exa mple of [Mac 7 6, chapter 2, page 90 ] shows this. 7 W e denote the in verse i mage of a set F ⊆ Y along a map f : X − → Y b y f ∗ ( F ). This is at o dds with the more usual f − 1 ( F ) but it has the con venien t adv ant age of ha ving one and the same notation f or b oth the inv erse image and the ev aluation at F of the inv erse image function f ∗ : 2 Y − → 2 X . CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 9 Example 2.3. Let E b e a non-measurable subs e t of X and let ξ b e the constant bundle x 7− → K . Deﬁne the section s of ξ by , s ( x ) = def ( 1 if x ∈ E , − 1 otherwise. and le t t be a second se c tio n such that x 7− → k t ( x ) k is mea surable. Then t + st = (1 + s ) t and from k s ( x ) t ( x ) k = k t ( x ) k we conclude tha t k s ( x ) t ( x ) k is measurable. But now no te that, k ( t + st )( x ) k = ( 2 k t ( x ) k if x ∈ E , 0 otherwise. so that k ( t + st )( x ) k is not mea surable. By s uitably mo difying the ab ove example, the reader ca n come up with t wo natural transfo rmations τ , φ such that b o th functions k τ x k and k φ x k are meas urable and yet the norm of their composition is no t. It is clea r that whatev er meas ur able condition w e imp o se on natur a l tr a nsformations τ , the measurability of k τ x k is certainly a ne c essary one. G. Mack ey’s exa mple s hows that it is not suﬃcient . The failure exp osed by it is clear: there is no mea surable structure on the total s pace o f the bundle and hence no way to form the linear subspace of mea surable sections. What w e need is a c riterion that tells us that the ﬁb ers ξ x v ary “in a mea surable wa y” from point to p oint. The classical solution (see for exa mple [Ma c76, chapter 2]) is g iven by the notion of p ervasi ve se quenc e . The idea behind the deﬁnition is tha t instead of ﬁddling with mea surable structures directly , we des c rib e the corres p o nding mo dules of measura ble sections 8 . Deﬁnition 2.4. Let X b e a measur e space . A me asur able bund le over X is a pair ( ξ , ( s n )) where ξ is a bundle ov er X and ( s n ) is a sequence of sections X − → P that is p erva sive : (1) F or a.e. x ∈ X , the set { s n ( x ) } is dense in the ﬁber ξ x . (2) F or every n, m ∈ N the function x 7− → h s n ( x ) , s m ( x ) i is measurable. Given a measur able bundle ( ξ , ( s n )), a section s : X − → P is me asur able if for every n , the function x 7− → h s ( x ) , s n ( x ) i is measura ble. Now, if s and t are t wo mea surable sections then the function x 7− → h s ( x ) , t ( x ) i is measur able 9 . In particular, the nor m function x 7− → k s ( x ) k of any measurable sec tion is measur able. If X is a measure spa ce, taking the integral of the inner pro duct, h s, t i = def Z X h s ( x ) , t ( x ) i d µ we obtain a semi-inner pro duct o n the linear space of measur a ble sectio ns of s quare- int eg rable norm. Quotienting by the linear subspa ce of s ections a.e. eq ua l to 0 , we obtain the inner pro duct spa ce L 2 ( ξ ) which, by co pying the clas sical pr o of, is easily 8 The desc r i ption of top ological ob jects via modules o ver appropriate function algebras is at the heart of mo dern topology and geometry . The Serre-Swan theorem establishing an equiv alence betw een the category of vector bundles o ver a top ological space X and the category of pro jective modules o ver the algebra C ( X ) of con tinuous f unctions is certainly a ma jor example, but the whole ﬁeld of non-commuta tive geometry can b e seen as a v ast elab or ation and extension of these ideas. 9 This is most easily shown wi th the aid of theorem 3.5 b elow. 10 G. RODRIGUES shown to be complete. Since the p er v asive sequence is a c o untable dense set, the space L 2 ( ξ ) is sep ar able . The space L 2 ( ξ ) is also a Banach mo dule ov er the Banach algebra L ∞ ( X ) where the action is scalar mu ltiplicatio n ( f , s ) 7− → ( f s : x 7− → f ( x ) s ( x )) The asso ciatio n ξ 7− → L 2 ( ξ ) deﬁnes the ob ject part of a functor on measurable bundles. T o deﬁne it on morphisms w e need to imp os e one more condition o n a bundle mor phism. Deﬁnition 2.5. A morphism τ : ξ − → ζ is me asur able if it preser ves the measur a- bilit y of sectio ns: for every measura ble sectio n s o f ξ , the sectio n τ ( s ) is mea surable. T o illustrate the fundamental character of the mea surability condition, let τ be a measurable bundle morphism ξ − → ζ and ( s n ) the per v asive sequence of ξ . W e start by cutting o ﬀ ea ch s n aw ay from the unit ball. Deﬁne the sets E n = def { x ∈ X : k s n ( x ) k ≤ 1 } By measurabilit y of s n , ea ch E n is mea surable. Deﬁne, t n = def χ ( E n ) s n The sections t n are all measurable and clearly , the sequenc e ( t n ( x )) is dense in the unit ball of the ﬁber ξ x . Therefore : k τ x k = sup {k τ x ( t n ( x )) k : n ∈ N } (2.1) By mea surability of τ , each τ ( t n ) is meas ur able and in particular, the norm function x 7− → k τ x ( t n ( x )) k is meas urable. Since the s upremum o n the r ight-hand side of (2 .1) is tak en o ver a c ountable set, the next theorem follo ws. Theorem 2.6. If τ : ξ − → ζ is a me asur able bu nd le morphism, the fun ction x 7− → k τ x k is me asur able. Denote b y τ ∗ the linear map L 2 ( ξ ) − → L 2 ( ζ ) given b y s 7− → τ ( s ). On the hypothesis that x 7− → k τ x k is e ssentially bo unded, we hav e the equalit y k τ ∗ k = k τ x k ∞ (2.2) F or the pro of, we refer the reader to [Dix81, chapter I I, sectio n 2 , prop o sition 2]. These c o nstructions yield the morphism part o f the L 2 -functor. Its do main is the ca tegory 10 L ( X ) of measurable bundles and a.e. equal equiv alence classes of essentially b ounded, measurable mor phisms. 10 The reader s hould note that w e hav e dropped the subscript 2 fr om the category L ( X ) of measurable bundles. F or no w, this has the adv an tage of av oiding any p ossibl e confusion with the Hilb ert spaces L 2 ( X ). Later on, we will b e delib erate and b end our notation so as to maximi ze the chances of confusion with the ordinary decategoriﬁed situation. The absence of the subscript will then b e explained a wa y by the fact that in the world of the categories we will consider suc h concept s as b ounded, summable, etc. are either v acuous or sim ply make no sense. CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 11 3. Tow ards ca tegorified measure theor y In se c tion 2 we in tr o duced the categor y of measura ble bundles L ( X ) and the functor ass igning to ea ch measur able bundle ξ its Hilb ert space of square-integrable sections L 2 ( ξ ). This space is also ca lled the dir e ct inte gr al of ξ a nd is denoted classically by Z ⊕ X ξ d µ (3.1) If we drop the ⊕ from (3.1), we end up with ordinary integral notation, and it is our in tention to take this seriously . This entails in terpreting the co nstructions of section 2 as a c ate goriﬁe d inte gr al . If we s urmise the evidence from [Y et05 ], we can gather that the bundles are o ver a measurable s pa ce and that the L 2 -space of sections is denoted b y an in tegra l sign. The pap er [BBFW08] presents in section 3.3 a table of c ategoriﬁed analo g ues for the bas ic ingre die nts of linear a lgebra. The last tw o rows of the table give measurable bundles and the direct int eg ral as the catego r iﬁed analogues of resp ectively , measurable functions and the ordinary int eg ral, but the obvious analo gy with mea sure theo ry is not pursued a ny further. T o quote a lmost v erba tim from the in tro duction to [DB98], at its hear t, cate- goriﬁcatio n is based on a n analo gy b etw een sets and c a tegories as pre s ented in table 3.1. Sets X Categorie s X Elements Ob jects x ∈ X x ∈ X Equations Isomorphisms x = y x ∼ = y functions F unctors f : X − → Y F : X − → Y Set Set ( X , Y ) of Category Cat ( X , Y ) of functions X − → Y functors X − → Y T able 3.1. Analo gy b etw een sets a nd ca tegories Building on table 3.1 , w e adduce table 3.2 that pr esents the catego riﬁed ana- logues of some basic mea sure-theor etical co ncepts. Ordinary integrals Direct integrals Measurable functions Measurable bundles f : X − → K ξ : X − → Hilb Int eg ral Direct in teg ral R X f d µ ∈ K R X ξ d µ ∈ Hilb Int eg ral map Int eg ral functor R X d µ : L 2 ( X ) − → K R X d µ : L ( X ) − → Hilb T able 3.2. Ca tegoriﬁed ana logues of o rdinary integrals The entries of table 3.2 are imp osed on us by the constructions of se c tion 2, but they leave a lot of questions unanswered. F o r one, while the codoma in K is 12 G. RODRIGUES categoriﬁe d to Hil b , the doma in (a measure space) r e ma ins untouched. Similarly , lo oking at the seco nd r ow of 3.2 we see that while the integrand of a dir ect integral is a categoriﬁed function, a bundle, the measure is still the same uncategor iﬁed scalar meas ure µ . Some of these questions will b e a nswered o nce w e star t clea ring up the fog, but for now just ta ke table 3.2 as a starting p oint (with emphasis on the start ing ). In the rest of the section, the tw o main issues to b e tac kled are the deﬁnition of me asur ability and the notion of c ate goriﬁe d inte gr al . W e will go through the constructions of section 2 and ask ho w and in what sense they can be interpreted as the categ oriﬁcation of the familiar no tions of meas ure theory . In the spirit of an inductiv e in vestigation, the ﬁndings will be crystalized in ten principles that will then serve as a guide to the construction of a pr op er, fully-ﬂedged theor y of categoriﬁe d measure s and integrals. The quest w ill b e long, but we ho p e that in the end it will be clear that a co mprehensive, conceptual picture of categoriﬁed measure theory has emerged. Instead of a mishmash of lo o s e analogies , more o r less felicitous gues ses, juxtap ositions and ﬂights of fancy , we will hav e at o ur dispos a l a conceptual framework in which it is p oss ible to s ystematically answer the many questions that arise . I am also aw are that at e very s tep of the a rgument ther e a re some g e nu ine pro blems to solve and ga ps to ﬁll, so I am co mp elled to a sk the benign reader to overlook , not in the sense of ignoring but of seeing past, whatever str ikes him as inadequate or simply wrong. 3.1. Hi lb as a categoriﬁed ring. Per the third row of table 3.2 and the construc- tions of section 2, the in tegr al is a functor Z X d µ : L ( X ) − → Hilb Let us lea ve a s ide for the momen t the c a tegoria l structure of L ( X ) and in what sense it ca n b e considered a catego riﬁed Hilb e rt s pace, and instead concentrate on the catego r iﬁcation of the co domain K . Ideally , an integral is not just a functor on some categor y of bundles, but a functor satisfying some sort of line arity and c ontinuity pr op erties , the s ame as ordina ry integrals. In order to fo rmulate these linearity prop erties, the ﬁrst task is to iden tify wha t are the analo gues of the basic op erations needed to do integration, sum a nd m ultiplicatio n, that is , we need the notion of c ate goriﬁe d ring . W e adv ance with table 3.3 for the s p ec iﬁc case o f the category Hil b of Hilb ert spaces. Commutativ e ring Categoriﬁed co mm utative ring Field K Category H i lb Sum + Direct s um ⊕ Multiplication × T ensor pro duct ⊗ Additiv e zero 0 Z ero o b ject 0 Multiplicative unit 1 T ensor unit K T able 3.3. Ca tegoriﬁed ana logues of the ring op er a tions In the cour s e of this work, w e will have oppo r tunit y to commen t on this and other similar tables; for the moment , we just refer the r e a der to the ﬁr st section of [Bae97] (from whic h the table was b or row ed) for a pro p er motiv ation and instead CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 13 mov e on to recall the co nstruction o f dir e ct su ms in H ilb . If ( H x ) is a family of Hilber t spa c es, take the linear direct sum L x H x with the p oint wise-sum inner pro duct h ( h x ) , ( k x ) i = def X x ∈ X h h x , k x i (3.2) If the indexing set is inﬁnite, the spa ce L x H x is not co mplete. The direct sum P x ∈ X H x is the completion of L x H x under the inner pro duct (3.2). Denote by i x the inclusion H x − → P x ∈ X H x assigning to h ∈ H x the element i x ( h ) ∈ P x ∈ X H x given by i x ( h ) y = def ( h if x = y , 0 otherwise. The universal proper ty of the space P x ∈ X H x is de s crib ed and prov ed in the next theorem in the ca se where X is ﬁnite. Theorem 3.1. L et X b e a ﬁn ite set and ( H x ) an X -p ar a met erize d family of Hilb ert sp ac es or a bund le over X . Then for every c one ( T x ) of b ounde d line ar maps 11 H x − → H ther e is a unique b ounde d line ar map T : P x ∈ X H x − → H such that for every x ∈ X the triangle 3.1 is c ommutative. P x ∈ X H x T / / _ _ _ H H x i x O O T x : : v v v v v v v v v v Figure 3. 1 . Universal prop erty of P x ∈ X H x . The map ( T x ) 7− → T is a c ontr active line a r bije ctio n, X x ∈ X Hilb ( H x , H ) ≃ Hilb X x ∈ X H x , H ! (3.3) wher e the Banach sp ac e on t he left-hand side is endowe d with the Hilb ertian 2 -norm. Pr o of. The crucial fact is that g iven a cone ( T x ) there is a unique linear map T : P x ∈ X H x − → H closing the triangle 3.1 deﬁned by: b = ( b x ) 7− → X x ∈ X T x b x The b o undedness of T is now a simple applica tion of the CBS inequality ([Con90, chapter I, section 1]): k T ( b ) k =      X x ∈ X T x b x      ≤ X x ∈ X k T x k k b x k ≤ k ( T x ) k 2 k ( b x ) k 2 11 A c one fr om a functor F to an obje ct a is simply a natural transformation F − → ∆ ( a ) with ∆( a ) the constan t functor with v alue a . In the case at hand, since the domain is a discrete category , a cone is just a family of bounded li near maps all with the same codomain. Cones to a functor are deﬁned simil arly; f or reasons of euphon y we r efrain fr om usi ng suc h terms as cocone. 14 G. RODRIGUES This implies that T is bounded with k T k ≤ k ( T x ) k 2 . It is easy to see that the map is a bijection: just note that if T is a b ounded linear map P x ∈ X H x − → H then we hav e a cone ( T i x ) and, since the i x are is o metries, X x ∈ X k T i x k 2 ≤ X x ∈ X k T k 2 = n k T k 2 with n the cardinality of X .  The rea der should note t wo things in theorem 3 .1. First, that the repr esentabilit y isomorphism (3.3) is not an isometr y and there is no natur al nor m on the space of cones P x ∈ X Hilb ( H x , H ) that makes it so . In this context, the choice of the Hilber tian norm for the space of cones ends up be ing a tad arbitra ry 12 . Seco nd, a nd more impo rtantly for what follows, is the inevitable app earance of Bana ch spaces via spac es of b o unded linear maps . Recalling the deﬁnition of copro ducts ([Lan71, chapter I I I, section 3]), w e hav e the following immediate c orollar y . Corollary 3.2. The c ate gory Hilb has al l ﬁn ite c op r o ducts. The restr iction to ﬁnite X is necessary . The pro of of theorem 3.1 br eaks down if X is inﬁnite and later in section 4 we will g ive the simple argument that pr ov es that Hilb do es not have inﬁnite (co)pro ducts. This is a v ery serious drawback of H ilb as will b e seen in the contin uation. On the other hand, since Hi lb is a dditive it follows that it has a ll pro ducts and they are in fact, bipro ducts (see [L a n71, chapter VII I, section 2] for the deﬁnition and basic proper ties of a dditive categor ie s). If copro ducts g ive categor iﬁed s ums, tensor pro ducts g ive the categoriﬁed m ulti- plication. Reca ll tha t the tensor pro duct of Hilb ert s paces is the r epresenting ob ject for the bifunctor of b ounded bilinea r maps. This univ ersa l prop er t y is not as slick as that of theor em 3 .1, beca use although Hilb is a symmetric monoidal catego ry , it is n ot closed. T his is the s e cond ser io us drawback o f Hi lb . W e note how ever that the tensor bifunctor, b eing linear in eac h v ariable, automatically preserves bipro d- ucts up to linea r homeomor phism which are the isomo rphisms in Hilb by the op en mapping theorem ([Con90, chapter II I, section 1 2]). The next theorem improv es on this a little bit. Theorem 3.3 . L et X b e a ﬁnite set, H x and K Hilb ert sp ac es. The c anonic al map X x ∈ X H x ⊗ K − → X x ∈ X H x ! ⊗ K (3.4) is an isometric isomor phism . One imp ortant detail that table 3.3 manages to hide is that the alg ebraic laws of rings g et pr omoted in the categor iﬁed setting to speciﬁed is o morphisms that satisfy equations o f their own, the so-called c oher enc e laws . These coher ence laws are the more m yster ious a nd fascinating part of an y catego riﬁcation en terpris e. There is a v ast literature on the sub ject, from whic h we can cite [La p72] a nd [KL80] that int r o duced and formalized the notion of symmetric ring c ate gory . F or the most part, it is gobbledyg o ok co mpletely irrelev ant to our purp oses. In a wonderful ﬂash 12 W e could ha ve endo wed L x ∈ X H x with any p -norm, with p ∈ [1 , ∞ ]. Since X i s ﬁnite, these norms are all equiv alent and almost all the basic theorems inv olving the Hi lb ertian norm would still b e true by replacing the Hilb ertian norm by any other p -norm. CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 15 of ins ight, [Y e t ] observed that a ll the ope rations in table 3.3 are deﬁned b y un iversal pr op erties and w e can a void entirely the discussion of their c oherence laws. The notion of universal pro p erty is, in its v a rious guises, the ce nt r a l c o ncept of category theory and D. Y etter’s obser v ation s trikes us as so imp orta nt that w e elect it as our ﬁr st g uiding principle 13 . Principle 1. Un iversal pr op ertie s ar e one honking gr e at ide a – let’s do mor e of those! Principle 1 is the ov erar ching orga nizational principle of this pape r and its conse- quences will b e follo wed thro ugh doggedly . F or e x ample, we ment ione d above tha t we can a void a discussion o f coherence la ws b ecause all the en tries in table 3.3 ar e deﬁned b y univ ers al prop erties. This was so mething o f a rhetorical exag geration made in order to driv e a p oint ho me. The actual truth is that tensor pro ducts are pr oblematic, pre c isely b ecause Hilb is not closed. If w e res tricted ours elves to ﬁnite dimensional Hilb ert spaces, then we w ould hav e a symmetric monoidal closed category , meaning that there is a natural iso metric iso morphism Hilb ( H ⊗ K , L ) ∼ = Hilb ( H , Hilb ( K, L )) (3.5) Closedness is an extremely use ful prop erty . It not o nly allows a decent theor y of enriched categor ies but it turns tenso r pro ducts into tensors ([Kel05, c hapter 3 , section 7]). Now, it just happens that tensors are a sp ecial exa mple of weighte d c olimits and thus it is obvious what it mea ns for a functor to preserve them – and a gain no need to discuss an y coherence la ws. As things stand w e will need to hav e to re course to the notion o f mo dule functors ([Y e t, sectio n 1 ]). The upshot of these contortions is simple: Hilb is the “wrong ca tegory” 14 . There a re tw o cog ent reasons why this remark is not erected in to a principle; one, that we ar e not y et quite ﬁnished with badmouthing Hilb ; the o ther, that at this p oint it is not clea r what catego ries would even count as “r ight”. 3.2. M easurability of bundles. Let X b e a measurable s pa ce. Recall that a function f : X − → K is me asur able if the inv erse imag e f ∗ ( E ) of every (Borel) measurable set E ⊆ K is measur able. Mea surability is a pr op erty : a function either is or is not measurable. Mea surability for bundles a s in deﬁnition 2.4 is an extr a structur e on the bundle, the choic e of a per v asive s equence. This, sur e enough, is a common theme in ca tegoriﬁcatio n. Poten tially , there are many wa ys to make a category in to, sa y , a symmetric ring ca tegory and each choice is a n e xtra structure piled on the ca tegorial structure. But as dis c us sed in the pr e ceding subsection, the categ oriﬁed algebr aic op er a tions we are interested in are given by universal prop erties (even colimits), and a g iven categor y either ha s them o r not. W e will argue that deﬁnition 2.4 of measurable bundle is similarly misguided. First, there is the lur k ing suspicio n that the choice of p er v asive seque nc e is fairly arbitrar y , almost irrelev ant. T his is b est seen by starting at the end: if ( h n ) is a countable dense seq ue nce o f L 2 ( ξ ), then w e obtain a p erv asive seq uence for ξ that for all purp o ses is equiv alen t to the original p erv asive sequence, tha t is, the 13 Users of the Python programming language wil l readily r ecognize the source f or the formu- lation of pri nciple 1. 14 One of the more striking teac hings of modern algebraic geomet r y a la A. Grothendiec k is that it is far more pr eferable to hav e a well-behav ed category of (possibl y) badly b ehav ed ob jects, than well-b eha ved ob jects but constituting a badly b ehav ed category . 16 G. RODRIGUES resp ective s paces o f squa re-integrable sec tions ar e canonically isomor phic 15 . What seems to b e happ ening here is that we do hav e to make a choice, but then all choices a re eq ua lly go o d (o r equa lly bad, dep ending on one’s p ers onal philo sophy). Whenever ther e is a larg e num ber of equally go o d c hoices to ma ke, it is a n hint that there is an action of some symmetry g roup. The p erv asive sequence solution is to factor out this symmetry group by mak ing a choice of an or igin, but mo dern geometry shows us that there are other wa ys to ta ckle this problem. Second, although the choice of a perv asive s equence allows us to pic k a linea r space of sections with the required measura bility prop er ties, in the case of char ac- teristic bund les there alr e ady is a notion of measurability av ailable. Let E ⊆ X b e a measurable s ubset and H a separable Hilb er t spa ce. The characteristic bundle χ ( E ) H is deﬁned as: χ ( E ) H = def ( H if x ∈ E , 0 otherwise. (3.6) The measur a ble structure o f χ ( E ) H is given by pic king a coun table dens e se- quence ( h n ) of H and taking the seq uence of co nstant sectio ns s n ( x ) = def ( h n if x ∈ E , 0 otherwise. Note that the par ticular choice of the dense sequence is irr elev ant and a ny other such would lead to an equiv alent measurable structure. O n the other hand, since sections s of χ ( E ) H can b e identiﬁed with functions E − → H we say that s is Bor el me asur able if the asso cia ted function is measur a ble when H is given the Borel measurable str ucture. Recall that if X is a top ologica l space, the Bor el structure of X is the σ -algebr a generated by the o pen sets. W e now hav e: Theorem 3.4. A se ction s of χ ( E ) H is Bor e l me asur able iﬀ for every n , the map x 7− → h s ( x ) , s n ( x ) i is me asur able. A direct pro of of the ab ov e theo rem is not diﬃcult, but since it is a sp ecial case o f the Pettis measur ability theorem to which we will hav e to return to la ter in subsection 4.1, we lea ve it to the interested reader. Note also that a bundle morphism τ : χ ( E ) H − → χ ( F ) K can b e identiﬁed with a map τ : E ∩ F − → Hi lb ( H , K ) (3.7) The co domain is a Banach space a nd ha s a Bor el structur e . The question of the eq uiv alence of the measurability of τ with the Borel measurability of (3.7) is more delicate 16 , but since the Bore l measura bility of the latter already implies the measurability of x 7− → k τ x k w e ar e o n the right track. But the story do e s not end here. Le t ξ be a mea surable bundle. Then fro m a per v asive sequence ( t m ) we can construct a p ointwise orthonormal se quenc e ( s n ). Since this construction is imp or ta nt , b eing the foundation of later results, we recall it in the nex t theorem. F or the pro of, we refer the r eader to [Dix81 , chapter II, section 4]. 15 If this last p oint is not clear, consult theorem 3.7 b elow and the comments surr ounding i t. 16 The diﬃculty l ies partly i n the fact that the space Hilb ( H, K ) i s not separable when H and K are inﬁnite dimensional and th us, Borel measurabilit y is not adequate. T o make matters mor e complicated, a deep result of A. Szank owski ([Sza81]) s ho ws that this space does not ha ve the appro ximation prop erty . CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 17 Theorem 3.5. L et ξ b e a bund le of Hilb ert sp ac es and ( t m ) a p ervasive se quenc e. Then ther e exists a se quenc e ( s n ) of se ctions of ξ such that: (1) F or every n, m , the function x 7− → h s n ( x ) , s m ( x ) i is me asur able. (2) F or a.e. x ∈ X , the set { s n ( x ) } is an orthonormal b ase of ξ x if dim( ξ x ) = ω . If dim ( ξ x ) = m then the set { s n ( x ) : n ≤ m } is an orthonormal b ase of ξ x and s n ( x ) = 0 for n > m . (3) A se ction s of ξ is me asur able iﬀ the function x 7− → h s ( x ) , s n ( x ) i is me a- sur able for every n ∈ N . The conv er se pass age from p oint wise ortho no rmal s equences to p erv asive ones is even easier. Given a sequence ( s n ) sa tisfying (1) and (2) of theorem 3 .5 , then taking rational linear combinations of s n and reindexing, we obtain a p erv asive se q uence of ξ and the res pec tive L 2 -spaces of sections are isometrica lly isomo rphic. In other words, a s far as bundle measurability is concer ned the tw o co ncepts ar e equiv alent. Before putting theorem 3.5 to w or k, we m ust mak e a s mall detour and discuss one piece of the categorial structure of L ( X ). Wh a t w e need is the simple fact that dir ect sums in Hilb lift to L ( X ). If ( ξ , ( s n )) and ( ζ , ( t m )) ar e tw o measurable bundles, their p ointwise dir e ct su m is the bundle x 7− → ξ x ⊕ ζ x with per v asive sequence (after suitable reindexing ) given by all sums of the form s n + t m It is not diﬃcult to s ee that ξ ⊕ ζ is a copro duct in L ( X ). Since the zer o bundle x 7− → 0 is a zero in L ( X ), we have the fo llowing theor em. Theorem 3. 6. The c ate gory L ( X ) is additive. The construc tio n of p oint wise direct sums can b e extended to a countable 17 sequence ( ξ n , ( s m,n )) of measurable bundles. On the ﬁb ers , the direct sum is given by X n ξ n : x 7− → X n ( ξ n ) x The mea surable structure is the s equence (after reindexing ) given by sums of the form, X n ∈ I s n,m n where I ⊆ N is a ﬁnite subset and s n,m n is a section of ξ n in the res pe c tive per v asive sequence. F o r muc h the sa me reasons as in Hi lb (see section 4 b elow), the co un table po int wise direct sum P n ξ n is not a co pro duct in L ( X ). Back to measur ability iss ue s . If s is a measurable section of a measurable bundle ξ , then the supp o rt of s is the set supp( s ) = def { x ∈ X : s ( x ) 6 = 0 } (3.8) Since s is measurable, supp( s ) is measurable. Thu s, the section s generates the characteristic line bundle χ (supp( s )) K co ncentrated o n supp( s ) and from p o int wise orthonor mality it follows that: 17 But no larger. Indexing sets of l ar ger cardinality lead to non-separable ﬁb ers. 18 G. RODRIGUES Theorem 3.7. L et ξ b e a me asur able bun d le and ( s n ) a p ointwise orthonormal se quenc e in ξ . Put E n = def supp( s n ) . Then ther e is an isometric bund le isomorphism ξ ∼ = X n χ ( E n ) K (3.9) Theorem 3.7 des crib es every measurable bundle as a gluing of character is tic line bundles by a coun table co llection of meas ur able bundle isomo rphisms φ n,m : χ ( E n ∩ F m ) − → χ ( E n ∩ F m ), or by a me asur able non- z er o function E n ∩ F m − → K . Normalizing, that is, dividing b y | φ n,m | , w e obtain a function with v alues in the unit spher e of K : the unit c ircle S 1 in the complex sca la r case, the zer oth sphere S 0 = {− 1 , 1 } in the real sca lar case 18 . In a nut shell, we can do aw ay with pe rv asive sequences by working with countable direct sums of characteristic bundles a nd co unt a ble direct sums of mea s urable mor- phisms. In more categor ial terms, we exp ect that s omething very clo se to principle 2 is true. It can be seen as the categoriﬁed analogue of the fact that the space of measurable functions is, in the appropriate top olog y , the sequentially closed linea r span of the characteristic functions. Principle 2 . Th e c ate gory of me asur able bund les is the closur e of the sub c ate gory of char a cteristic bund les and me asur able morphisms under c ountable c olimits. The pr o cess o f completing categ ories b y classes o f (co )limits is covered in [Kel0 5] (starting in c hapter 5, section 7) and then in a string of pap ers of which we can men tion [AK88], [KS05] and [KL00]. In ge ne r al, such completions inv o lve transﬁnite constructions, on which the deﬁnitive pap er is the fair ly indigestible [K el80]. L a ter, we will s e e that it is b oth technically simpler and conceptually so under to simply add al l (small) c o limits. The main po int of principle 2 is that under lying it, there is at work a deep analog y with geometry where the space s (sc hemes, bundles, manifolds, etc.) are built b y gluing lo cal mo dels (aﬃne schemes, triv ia l or constant bundles , o pe n subsets of euclidean space , etc.) along a top olo gy by using an a ppropriate clas s of g luing maps. In o ur case, the lo cal mo dels are the characteristic bundles and the top ology is repla c ed b y the Bo ole an algebra of measurable sets. In this view, a bundle is measurable b ecause it can “b e r eached” by a c o untable colimit of c hara cteristic bundles a nd the problem has shifted from the deﬁnition o f measurability to the existenc e of inﬁnitary co limits. An impor tant diﬀer e nc e is tha t while in geometry , the gluing is in general made along an o p en cov er with arbitrary c ardinality , with measurable bundles we have to restrict to c ount able co vers since a nything of higher cardinality is b ound to destr oy mea surability . The next ex ample lifted from [Y et05] shows this. Example 3.8. L e t E ⊆ X be a non-measurable set and let ( E i ) be a, nece ssarily uncountable, par tition of E into measurable sets. F or example, if singleton sets a r e 18 This cocycle description is familiar to an yone conv ersant with ve ctor bundles. What ma y not b e so famili ar is that these gluing functions φ n,m are precisely the extr eme p oints of the unit ball of certain Banac h space s. At this point this may sound l i ke a usel ess piece of inf ormation, but the reader should store i t in the bac k of his mind as extreme p oint phenomena will r esurface later. CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 19 measurable just tak e {{ x } : x ∈ E } . Then: χ ( E ) K ∼ = X i χ ( E i ) K But the iden tity map on χ ( E ) K do es not hav e mea s urable norm. Even though exa mple 3.8 shows that there ar e diﬀerences, the ana logy with algebraic geometr y and sc hemes is to o g o o d to pas s up and it is worth elab ora ting upo n it. The ﬁrs t thing to remember is that to deﬁne sheav es w e do not need a top ologica l s pace X , we only need the lo c ale o f o pen sets o pe n( X ). The σ -Bo ole a n algebras of measure theory ar e not lo cales beca use they are not order complete, only σ -complete (o r c ountably co mplete), so w e hav e to res o rt to the next b est thing: Gr othendie ck top olo gies 19 on categ ories. In simple ter ms, a Gr othendieck top ology is a ga dget that allows us to deﬁne sheaves over the ca tegory . In our case, a Bo o lean alg ebra is viewed as a catego ry in the usual w ay: there is a (unique) arrow E − → F iﬀ E ⊆ F . The ex act deﬁnition of Gro thendieck topology can b e found in many textbo oks (e.g. [LM92, c hapter I I I]); for now, we con tent ourselves with conveying the basic ideas. This leads us to o ur third principle. Principle 3. The c ate gory of me asur able bund les is a c ate gory of “lo c al ly c onstant” she aves of Hilb ert sp a c es on a suitable site. The Grothendieck top olog y of principle 3 we hav e in mind is the so- c alled c ount- able joi n t op olo g y (or σ -top olo gy ). A rigorous deﬁnition will b e g iven later, but for the moment it suﬃces to think o f it as restricting the arbitra ry cardinality cov er s by open sets in the topolo gical cas e to countable cov er s by measurable sets. The “lo cally cons ta nt ” buzzw or d is supposed to remind the reader that a general mea- surable bundle is o btained from c hara cteristic bundles in a manner reminiscent o f how schemes a re o btained from aﬃne schemes. Another connection with sheaves on top olo gical spaces can b e established by recalling that Stone duality (for example, see [Joh82, chapter II]), given by asso ci- ating to a Bo o lean algebr a Ω its Stone space S(Ω), y ields an equiv alence b etw een the category of Bo olea n a lgebras and the dual o f the categor y of compact Haus- dorﬀ, totally disconnec ted spaces. This sugges ts that it may be p ossible to tr a nsfer sheav es o ver the site Ω to sheav es as we kno w and love them over the top o lo gical space S(Ω). This is indeed p os s ible but it is mor e delicate 20 , so we leave it aside for now. There is yet a third, more analy tical, descr iption of measurable bundles a s certain mo dules over the Banach alge br a L ∞ ( X ). This description is equa lly impor tant but since we hav e no ne e d of it right no w, w e pass on immediately to our se cond concern in this section. 19 P . Johnstone’s terminology of co vera ge is closer to con ve ying the underlying i dea: the form al- ization of the basic properties of cov ers in top ological spaces p ermitting the deﬁnition of sheav es. 20 The ﬁrst problem is that the unit isomorphism η : Ω − → clop en(S(Ω)) of the Stone eq uiv- alence is only ﬁnitely order con tinuous s o that the transfer of a sheaf ov er Ω to S(Ω) gives a presheaf over the clop ens and satisf ying the patch i ng condition f or ﬁnite c overs only . The second problem i s with the deﬁnition of a Banac h space-v alued sheaf on top ological spaces. The t wo ar e int im ately r elated – see subsection 7.1 for the details. 20 G. RODRIGUES 3.3. Line arit y of the direct integral. As is w ell- k nown, even b y most calculus student s, the integral is linear in the in tegrands a nd w e ha ve the equalit y , Z X ( k f + l g ) d µ = k Z X f d µ + l Z X g d µ (3.10) for every pair of integrable functions f , g and ev ery pa ir of scalars k , l ∈ K . By subsection 3.1 we alr eady know how to in terpret the r ight-hand side of (3.10) in the categoriﬁed setting. The bit of categ o rial structure of L ( X ) that we need to int er pret the left-hand side is that o f a mo dule over a symmetric ring c ate gory as deﬁned for example, in [Y et]. The next table contains the rele v an t infor mation. Mo dule M Catego riﬁed mo dule L ( X ) Zero 0 ∈ M Zero bundle x 7− → 0 Sum v + w ∈ M Poin twise direct sum ξ ⊕ ζ Scalar actio n kv ∈ M Poin twise tensor product H ⊗ ξ T able 3.4. Ca tegoriﬁed ana logues of the module op erations Poin twise direct sums hav e alrea dy b een deﬁned in subsection 3.2 a nd they int er act well with the direct integral functor. Theorem 3. 9 . L et ( ξ n ) b e a se quenc e of me asur able bun d les. Then ther e is an isometric isomorp hism X n Z X ξ n d µ ∼ = Z X X n ξ n ! d µ (3.11) Pr o of. T o a sequence of sections s n of ξ n living in R X ( P n ξ n ) d µ w e asso cia te the section, x 7− → X n s n ( x ) where we identify a section of ξ n with a section of P n ξ n via the obvious isometric embedding. Tha t this map is measura ble and bijective is mo re or less clear and that it is an iso metr y amounts to the norm equality , X n k s n k 2 2 =      X n s n      2 2 which is true b e cause the s n are o rthogona l in R X ( P n ξ n ) d µ .  Theorem 3.9 no t only encompasses the ﬁnite linearity of the in tegr al but it is natural to view it as the expr ession of the fact that the dir e ct inte gr al is σ - additive . Next, we turn to the point wise tenso r pro duct. If ( ξ , ( s n )) and ( ζ , ( t m )) are tw o measurable bundles, their p oint wise tensor pro duct is the bundle x 7− → ξ x ⊗ ζ x with per v asive s e quence (a s alwa ys, after suitable reindexing) given by r ational linear combinations of elemen ts of the form, s n ⊗ t m where s n and t m are s e c tions in the p erv asive sequences of the resp ective bundles. The bifunctor ( ξ , ζ ) 7− → ξ ⊗ ζ endo ws L ( X ) with a symmetric mono idal structur e. CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 21 Unfortunately , the formulation o f the universal prop erty is even more diﬃcult bec ause of the fact that Hilb is not closed is now c omp ounded by diﬃcult bundle measurability issues. Nevertheless, if we make use of the notion of mo d ule functor (see [Y et ]) the categoriﬁed sca lar inv a riance of the in tegr al is easy to for m ula te. If H is the scala r ﬁeld K , then the characteristic line bundle χ ( E ) K introduced in subsection 3 .2 will b e deno ted simply by χ ( E ). Identifying an Hilber t spa c e H with the c ha r acteristic bundle χ ( X ) H and making use of the p oint wise tensor pro duct, we hav e the isometric isomo r phism χ ( E ) H ∼ = χ ( E ) ⊗ H (3.12) As p er the last row of 3 .4, the a ction of Hilb on L ( X ) is given by ( ξ , H ) 7− → ξ ⊗ H The next theore m now shows that the direct integral comm utes with the Hilb action – the c ate goriﬁe d version of the fact that the inte gr al is e quivariant for the action of the sc alar ﬁeld . Theorem 3.1 0. L et ξ b e a me a su r able bund le. Then ther e is an isometric isomor- phism Z X ( ξ ⊗ H ) d µ ∼ =  Z X ξ d µ  ⊗ H (3.13) Pr o of. See [Dix81, chapter II, section 8].  Since the direct integral is the extensio n of the L 2 -functor on measure spa ces to a functor on the category of measurable bundles, the co mputation o f the direct int eg ral of c hara cteristic line bundles is trivial. This isomor phism is s o imp ortant that we recor d it in the next theorem. Theorem 3. 11. If ξ is the char acteristic line bund le χ ( E ) of a me asur able subset E ⊆ X , then ther e is an isometric isomorphism: Z X χ ( E ) d µ ∼ = L 2 ( E , µ ) (3.14) In o rdinary measure theo ry we hav e the equality R X χ ( E ) d µ = µ ( E ). If we take our notations ser iously , then isomorphism (3.14 ) is telling us that E 7− → L 2 ( E , µ ) is playing the r ole of c ate goriﬁe d me asur e . The ﬁrst thing to notice ab out this functor is tha t it is c ovariant with domain the σ -algebra Ω of mea surable sets of X – it is a pr e c oshe af 21 . And we ca n ev en is olate the crucial σ -additivity pr op erty : Theorem 3.12. L et ( E n ) b e a c ountable p artition of E . D enote by ι E n ,E the inclusion map L 2 ( E n , µ ) − → L 2 ( E , µ ) . Then the induc e d map X n ι E n ,E : X n L 2 ( E n , µ ) − → L 2 ( E , µ ) (3.15) is an isometric iso morphism. Pr o of. In essence, this is a sp ecial case of theorem 3.9.  21 Not b eing a native Engli sh sp eak er, I do not know w hi c h, if any , of c opr eshe af or pr e c oshe af is more correct. The l atter j ust sounds sligh tly less horrible to my in ternal ear. 22 G. RODRIGUES W e will not prove it here, but it can b e shown that theor em 3.12 implies that the functor E 7− → L 2 ( E , µ ) satisﬁes the dual p atch ing c ondition for ﬁnite c overs and it fails the patching condition for c ountable c overs simply because the countable direct sums in theor em 3.12 a re not copro ducts. W e hav e a rrived at the fourth principle of categoriﬁed measur e theory . Principle 4 . A c ate goriﬁe d me asur e is a c oshe af of Hilb ert sp ac es over the σ - algebr a of me asur able sets and any me asure µ on X yields a c ate goriﬁe d me asur e via E 7− → L 2 ( E , µ ) . 3.4. Co ntin uit y of the di rect int eg ral. Int eg rals are (or ought to be) limits of sums. Recall that a function f : X − → K is simple if there is a ﬁnite family of pairwise disjoint measura ble sets E n ⊆ X and sca lars k n ∈ K such tha t, f = a.e. X n k n χ ( E n ) The connection b etw een the int eg ral map R X d µ and the measure µ is the equality Z X f d µ = X n k n µ ( E n ) (3.16) The left-hand s ide of (3 .1 6) is then extended to the whole space of in tegra ble functions by a limiting pro cedure. In subse ctions 3.2 a nd 3.3 the calculations with measurable bundles and direct integrals s how that once the direct integral of char- acteristic line bundles in known, the direct integral of ev er y measurable bundle is known. Pursuing this analog y seriously , we are inevitably led to o ur ﬁfth pr inciple. Principle 5. In ﬁnite dir e ct sum s and mor e gener al ly inﬁn itary c olimi t s , give the c ate goriﬁe d notion of c onver genc e. The need for inﬁnitary colimits has arisen from a consideration of σ -additivity , but it has alr eady app eare d explicitly b efore in principle 2 when w e ven ture d that the category of measurable bundles is the clo sure of the category of characteris tic bundles under coun table colimits. Principle 5 is form ulated in a necessarily v ague wa y , not the lea s t b eca use as we hav e b een insistently re pe a ting, the categ ory Hilb do es not have inﬁnite (co)pro ducts. Let us for the moment forget s uch a n unfor tu- nate circumsta nc e , and pro ceed a s if all the categories in sight were co complete. Denote by S ( X ) the c ate gory of char acteristic line bund les . The ob jects are the characteristic line bundles χ ( E ), which we can ide ntify with the mea surable subsets E ⊆ X , a nd a morphism E − → F is an esse ntially bo unded mea surable morphis m χ ( E ) − → χ ( F ). The co nstructions of sectio n 2 res tr icted to this category yield a functor µ : S ( X ) − → Hi lb (3.17) given on ob jects by E 7− → µ ( E ) = L 2 ( E , µ ). On the o ther hand, consider the fully-faithful inclusion of S ( X ) int o the catego r y of measurable bundles L ( X ). The integral functor is then a lift of µ : E 7− → µ ( E ) as in dia gram 3.2 . Diagram 3.2 is not commutativ e, but only commutativ e up to a unique iso - morphism. This isomor phism, the inner 2 -cell ﬁlling the triang le , is the na tural isomorphism of theorem 3.11. Thinking in categoria l terms, this suggests a sixth principle. CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 23 L ( X ) R X d µ / / _ _ _ ⇑ Hilb S ( X ) µ ; ; w w w w w w w w w O O Figure 3. 2 . Universal pro pe r ty of the direct integral functor. Principle 6 . The int e gr al functor is t he left Kan ex tension of µ along the inclusion of the sub c ate gory of char acteristic line bund les. W e exp ect that the in tegr al functor R X d µ is an extension of the µ - functor . That it is a Kan extension is the pavlovian reﬂe x resp o ns e of any one tr ained in catego ry theory (seriously , it is just another instance of principle 1 at work). That it is a left Kan extension (as opp osed to a right one – no pun in tended) is related to the fact that it is the co limits that are imp ortant, no t limits. If Hilb were co complete, b y the coend form ula for le ft Kan extensions 22 , and dropping the X from the morphism space s o f L ( X ), we would have Z X ξ d µ ∼ = Z E ∈ Ω L ( χ ( E ) , ξ ) ⊗ µ ( E ) ∼ = Z E ∈ Ω ξ ( E ) ⊗ µ ( E ) (3.18) F ormula (3.18 ) shows that just as in the ordina ry integral case, the direct integral is uniquely determined by its v alues on the c hara cteristic line bundles. No te that the right-hand side of the ﬁrst iso morphism do es not really make sense, beca use Hilb is not closed and th us L ( χ ( E ) , ξ ) is no t a Hilbert space. The last term do es make sense (forgetting momentarily that Hilb is not co complete) and the second isomorphism is due to the following theorem, whos e easy pro of is left to the reader . Theorem 3.13. L et ξ b e a me asur able bund le and E ⊆ X a me asur able set. Then ther e is an isometric isomorphism, L ( χ ( E ) , ξ ) ∼ = ξ ( E ) ( 3.1 9) wher e ξ ( E ) denotes the r estriction of ξ to E . Theorem 3.13 lo o ks remar k ably like Y oneda lemma fo r the inclusion S ( X ) − → L ( X ). Coincidence? There are no coincidences in mathematics. An yway , the preceding discussio n sugges ts another imp or tant principle. Principle 7 . Th e inte gr al functor has str ong co contin uity prop erties . Combining the principles exp ounded up to now, we see that just as the or dinary int eg ral is a t wo-stage pro cess so is the categoriﬁed integral. Given a coshea f (a categoriﬁe d meas ure), we can deﬁne the int eg ral of c har acteristic bundles. The int eg ral of a general measur able bundle is then obtained by a limiting pro cedure, in this cas e , a left Kan extension. The prop erties of the integral, including its 22 Left Kan extensions and coends wi ll b e revisited later in subsection 6.1. F or no w, w e oﬀer the reader the reference [Lan71, chapte r X, section 4]. As usual, we use in tegral notation for coends. They should not b e confused with the dir ect i n tegrals R X ξ d µ . 24 G. RODRIGUES co contin uity , should follow from this tw o-step constructio n. F o rmulated in these terms, w e can ex tend this categor iﬁed integral to a larg e num ber of other ca tegories since what we need is basica lly , the representabilit y of certain functors (to have a t our disp osal the integral of c har acteristic bundles) a nd the exis tence of certa in colimits (to b e able to construct the left Kan extension). What we hav e done was actually the re verse of the historical pro cess, since this categ o riﬁed in tegr a l construction has a pp e ared b efor e in other contexts such as top oi 23 . As will b e seen below, there is a link betw een categor iﬁed mea sure theory a nd topos theory . Stay tuned. Given a measurable subset E ⊆ X we can consider the categor y of measurable bundles L ( E ) on E . W e exp ect it is a 2-Hilb ert space, whatever that may b e. V arying E ∈ Ω, w e should o btain the categoriﬁed version of a categ oriﬁed mea- sure, that is, a precosheaf of 2-Hilb ert spaces that is a c ostack 24 for the countable Grothendieck top olog y o n Ω. The pattern of higher-orde r categoriﬁca tio ns should now b e clearer. W e record this as our last pr inciple for this subsection. Principle 8. The pr e c o she af E 7− → L ( E ) on Ω is a c ostack of 2 -Hilb ert sp a c es for the Gr othendie c k c ountable joi n t op olo g y. 3.5. Universal prop ert y o f the direct in tegral. In subsection 3.3 we have in- tro duced the categ o riﬁed analogues o f measur e s : c o sheav es. Given a coshea f µ a nd a measur able bundle ξ , we exp ect to obtain another cosheaf b y taking the indeﬁnite 23 In [Cur90, chap ter 4, section 3], the great German mediev alist and literary cri tic E. R. Cur- tius, inform s us that ancient rhetoric had ﬁve divisions. Of these, the most important was “In- v ention” (Latin inventi o ) ab out which: It is divided according to the ﬁv e parts which mak e up the judicial oration: 1. int r oduction ( exor dium or pr o o emium ); 2. “narrative” ( narr a ti o ), that is, exposition of the facts i n the matter; 3. evidence ( ar gumentatio or pr ob atio ); 4. refutation of opposing opinions ( r efutatio ); 5. close ( p er or a tio or epilo gus ). About the third division, “evidence” , E. R. Curtius writes: As f or the “evidence”, antique theory made this divi sion the ﬁeld of supersubtle distinctions in to whic h we may not en ter. Essentially , every oration (including panegyrics) mu st make some proposition or thing plausible. It must adduce in its fav or arguments whi c h address themselve s to the hearer’s m ind or heart. Now, there is a whole seri es of such arguments, which can b e used on the most dive rs e o ccasions. They ar e intellectual themes, suitable for developmen t and modi ﬁcation at the orator’s pleasure. In Gr eek they are called κoιν oι τ oπ oι ; in Latin, lo ci c ommunes ; in earlier German, Gemein¨ orter . Lessing and Kant still use the word. Ab out 1770, Gemeinplatz w as for med after the English “com- monplace.” W e cannot use the word, since i t has lost i ts original application. W e shall therefore retain the Greek top os . T o elucidate its meaning—a topos of the most general sor t is “emphasis on inability to do justice to the sub ject”; a top os of panegyric: “praise of f orebears and their deeds.” In Antiquit y col- lections of suc h top oi were made. The science of top oi—called “topics”—was set forth in separate treatises. With due deference to those that prefer the plural “top oses”, there is some inexpressible p o esy in using one and the same word f or a mathematical univ erse and a technica l term of ancien t rhetoric designating a con ve ntional – in the v arious senses of the word including structur al or informing – li terary idea. 24 Stac ks are categoriﬁed v ersions of sheav es. V ery roughly , they are pr eshea ves with v alues in a 2-category s atisfying the patc hing condition up to coheren t isomorphis m. CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 25 inte gr al : E 7− → Z E ξ d µ = def Z X χ ( E ) ⊗ ξ d µ (3.20) This yields an action of the category of measurable bundles on the category of cosheav es. It also express es the integral R X ξ d µ as the evaluation of a measure at the whole spa ce. The Radon-Niko dym theo rem holds if this action yields an equiv alence betw een the categ ories of measurable bundles and cosheav es . In this case, the measura ble bundles app ear as r elative densities , or suitable quotients , betw een cosheaf catego riﬁed measures. As a rgued in subsection 3.4, (3.2 0) is obtained as a left Kan extension of the functor that a coshe a f µ deﬁnes on S ( X ) and her e in lies the rub. F or L 2 -cosheaves coming from scala r measur es it is ob vio us that this functor is just the functor co n- structed in section 2 restricted to the full sub categor y of characteristic line bundles , but what a bo ut for a general co s heaf ? The co nclusions drawn in subsections 3 .2, 3.3 and 3.4 w ere based on formal analo gies b etw een the ordinary in tegr al of mea- surable functions a nd the dir ect integral of measura ble bundles. In this subsection we will try to dig deep er; pragmatically this means a nswering the question if ther e is some univ ersal pr o p erty a ttached to R X χ ( E ) d µ . In order to make any kind of progres s let us take a step back and direct our attent io n to the simplest possible c a se of direct integrals: take the measure space X to b e a ﬁnite set and for the measure µ take the counting measure. Then the dir e c t integral R X ξ d µ of a bundle ξ over X always e xists and is simply the direct sum P x ∈ X ξ x as deﬁned in subse c tion 3.2. Pr inciple 1 to the for e: the most impo rtant fact related to direct sums is that they are a copro duct and deﬁne d by a universal pr op erty . Recall that if ( a x ) is a ﬁnite family of ob jects in a categ ory (additive o r not), their c opr o d uct is an ob ject P x ∈ X a x together with a cone ( i x ) of maps a x − → P x ∈ X a x such tha t for every cone ( f x ) there is a unique a rrow f making the diagram 3.3 commutative for ev ery x ∈ X . P x ∈ X a x f / / _ _ _ b a x i x O O f x ; ; w w w w w w w w w w Figure 3. 3 . Universal pro pe r ty of the copro duct. If ( T x ) is a ﬁnite cone of bounded linear maps H x − → H then the induced map T is the ma p given by ( h x ) 7− → P x ∈ X T x ( h x ). In particular , if H x = H a nd the maps T x are the iden tity , then the induced map is the sum map : X x ∈ X H − → H : ( h x ) 7− → X x ∈ X h x (3.21) In o ther words, the direc t sum is not just a categ oriﬁed analo gue o f sums, but is dir e ctly c onne cte d with or dinary sums via the co unit map (3.21 ). The direct sum ob ject P x ∈ X H x is the “ ob ject of sums P x ∈ X h x of elemen ts of H x ”. The sums P x ∈ X h x are s ent to a concrete sum via the counit (3.21) as so o n a s we have a cone ( T x ) of bounded linear maps. 26 G. RODRIGUES Another p o int of view on the universal prop erty o f a copro duct is given by observing that a co ne ( T x ) of maps H x − → H with H x = K is by the isometric isomorphism Hi lb ( K , H ) ∼ = H , the same thing as a s e c tion of the bundle with tota l space ` x ∈ X H o r a function X − → H . Rewriting the universal prop e rty of the copro duct in these terms , we see that every function f : X − → H induces a unique bo unded linea r map P x ∈ X K − → H a s in diagr am 3.4. P x ∈ X K / / _ _ _ H X ∆ O O f ; ; v v v v v v v v v v Figure 3. 4 . Discrete Rado n-Nikodym. The ma p ∆ is the diagonal map giv en b y x 7− → δ x ∈ P x ∈ X K with, δ x ( y ) = ( 1 if x = y , 0 otherwise. and the unique map in the factoriza tion is the sum map applied to f : X x ∈ X K − → H : ( k x ) 7− → X x ∈ X f ( k x ) But this is nothing els e than a ﬁnite, Hilb ert sp ac e, ve ctor versio n of t he R adon- Niko dym the or em . If this is no t clea r, just put H = K . T he n the universal pr op erty states that every b ounded linear functional o n P x ∈ X K ∼ = ℓ 2 ( X ) a rises from a func- tion X − → K by taking the indeﬁnite in tegr al. Since X is discr e te, the in tegra l amounts to a simple ﬁnite sum. This is the ca se for the co nstant line bundle x 7− → K . Let us generalize a little bit and take the constan t bundle x 7− → K with K a n a rbitrar y Hilb ert space. The cones h x ∈ H are replaced by cones T x ∈ Hi lb ( H x , H ) with H x constant and eq ual to K . But since Hi lb is a symmetric monoidal categor y and the tensor pro duct commutes with direct sums, w e hav e the chain o f isomorphis ms , Hilb X x ∈ X K, H ! ∼ = Hilb X x ∈ X K ⊗ K , H ! ∼ = Hilb K ⊗ X x ∈ X K ! , H ! ∼ = Hilb X x ∈ X K , Hil b ( K, H ) ! This means that the Radon- Nikodym prop erty for the v ector case follo ws, that is, we have the isomorphism, Hilb X x ∈ X K, H ! ∼ = Z X H d µ CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 27 if the characteristic line bundles hav e the Ra don-Nikodym prop erty with r esp e ct to Banach sp ac es , that is, w e ha ve the isomorphism Hilb X x ∈ X K , Hil b ( K, H ) ! ∼ = Z X H d µ Once again, Banach space s are se en to be essential to the prop er for m ula tion of the universal pro pe r ties. Ultimately , this is due to the fact that Hilb is not closed 25 . Now, let us test these ﬁndings on the measur able situation, starting with a characteristic line bundle χ ( E ) with E ⊆ X a meas ur able subset. Its direct in tegr al was computed in theorem 3.11. The univ er sal prop erty of diagra m 3.4 now tak es the for m of diag ram 3.5. R X χ ( E ) d µ / / _ _ _ H X ∆ O O : : t t t t t t t t t t t Figure 3. 5 . Meas urable Radon-Niko dym. But in order to make sense of dia gram 3.5 we ne e d to have the integral version of the counit sum ma p, or we need int e gr als of ve ctor-value d funct ions X − → H . Assuming that all this works, then rep eating the argument ab ov e, in order to de- rive the Radon-Nikody m prop er t y for more genera l characteristic bundles than line bundles, not only w e need the measurable Ra don-Nikodym prop erty with resp ect to Banach spaces, we need the distributivit y of ⊗ with resp ect to direc t integrals – but this is just theorem 3.1 0. Thus, diag ram 3.5 is telling us that the c ate gori- ﬁe d inte gr al of char acteristic bund les is char acterize d by a universal pr op erty . This suggests that just a s the existence of direct sums is not an extra structure but a prop erty of a category , the same happ ens with direct in tegra ls. W e ele ct this as another of our guiding pr inciples. Principle 9. The dir e ct inte gr al of a char acteristic bund le is an obje ct char acterize d by a universal pr o p erty, the Ra don-Nikodym prop er t y . Principle 9 is a concrete em b o diment of pr inciple 1 . W e managed to isolate the basic ob ject and characterize it by a universal pr op erty . Instea d of b eing guided by more or less v ague a nalogies , we now have a systematic means of in vestigation. It allows us to br ing in the full force of categ o ry theory a nd pr ompts us to ﬁnd s uitable replacements for certain measure-theor e tic and functional-analytic concepts in order to b e able to also c ate gorify the pr o ofs . Unfortunately , the reader should curb his natural enth usias m b ecaus e as will b e seen b elow in subsection 4.1, principle 9 cannot stand as currently formulated. 25 In connection wi th these matters, it is worth remarking that although in [BBFW08, section 5] the authors suggest h H, K i = H ∗ ⊗ K for the categoriﬁed inner pro duct, the f act is that the space on the right-hand si de is the space of Hil b er t-Sc hmidt operators which, in the inﬁnite-dimensional case, is but a tiny subspace of Hilb ( H, K ). 28 G. RODRIGUES 4. From Hilber t to Banach sp a ces In section 3, time and aga in w e hav e hinted a t the fact that an extension of measurable bundles to Banach spaces is b ound to b e necess ary . The ﬁrst reason is tha t Hi lb , while symmetric monoidal is not clo sed, so that enric hed category theory with Hilb as a base is vir tually imp o ssible. F o r example, if ξ , ζ ar e tw o X -bundles, it is natural to co nsider the bundle ζ ξ of ma ps ξ − → ζ given by: x 7− → Hilb ( ξ x , ζ x ) But this is a bundle of Banach s pa ces. In other words, no t only the categ ories of Hilber t space bundles ar e b o und to la ck such a basic construction as exp o nent ia ls, but the formulation o f the critical Radon-Nikodym universal pr op erty is right-do wn impo ssible. The constr uction o f ﬁnite co pro ducts in Hilb has alr eady be en given in theorem 3.1. Since Hilb is additive it has all products, and both kernels and cokernels are constructed in the usua l ma nner . In other words, Hilb is ﬁnitely complete and ﬁnitely co complete. T o emphas iz e the fact that in inﬁnite dimensions substantially new phenomena arise, we r ecall the follo wing well-known r esult. Theorem 4. 1. The c ate gory Hilb is not ab eli an. Pr o of. This follo ws from the fact that a monic in Hilb is an injective map and a n epi is a b ounded linea r map with dense image. It is easy to ﬁnd epi- monics in Hilb that are not isomor phisms, th us Hilb cannot b e ab elian.  But the r e al ly damaging fact is that Hilb do es not hav e inﬁnite (co)pro ducts. Let ( H x ) b e an inﬁnite family of Hilber t spaces and let us assume that the pro duct Q x ∈ X H x existed and let ( p x ) b e the co ne of pro jections. If ( T x ) is a co ne of bo unded linear maps, then there is unique b ounded linear ma p T closing triangle 4.1 for e very x ∈ X . Q y ∈ X H y p x   K T x / / T ; ; v v v v v H x Figure 4. 1 . Universal prop erty of ( p x ). Commutativit y of triangle 4.1 implies the nor m inequality k T x k = k p x T k ≤ k p x k k T k But c ho o s ing a cone with a subsequence ( T x n ) such that k T x n k → ∞ , this is clearly imp ossible. A similar reaso ning shows tha t inﬁnite copr o ducts cannot exist. The problem in b oth non-ex istence statements lies in the fac t that we are trying to parameterize al l families of oper ators. In the pro duct case, this is not p oss ible bec ause there is no meaningful inner pr o duct in the inﬁnite pr o duct linear space Q x ∈ X H x . In the copro duct ca se, this is not p o s sible b ecause to deﬁne the map, Y x ∈ X Hilb ( H x , H ) − → Hilb X x ∈ X H x , H ! CA TEGORIFYING ME ASURE THEOR Y: A R OADMAP 29 we need to sum families of element s, which is impossible in any reasonable w ay if the family is ar bitrary . In fact, the tw o problems a r e re la ted beca use the sum map gives the representabilit y isomorphism, Hilb X x ∈ X H x , H ! ∼ = Y x ∈ X Hilb ( H x , H ) but as we hav e noticed alr eady the linear space on the right-hand side is not even normed. Given the cr ucial r ole o f co unt a ble colimits and in particular , countable copro d- ucts, this would s eem to deal a death blow to a prop er theory of categorial integrals. The solution is to extend the codomain of the measurable bundles from Hilbe r t to Banach s paces. At ﬁrst sight, the rea der may rightly wonder why this is a solution at all, since the previous example that show ed that inﬁnite copro ducts of Hilb ert spaces do not exist a lso works for the ca tegory Ban of Banach spaces and bo unded linear maps. This is indeed true, but there is a wa y out. Let ( B x ) b e an X -family of Bana ch spaces. W e will employ the same functor and bundle terminolog y introduced in section 2 in co nnection with bundles of Hilbert spaces. T he pro duct of the family ( B x ) exists in the catego ry V ect of linear spa c e s and its elements are X -parameter iz ed families ( b x ) of elements b x ∈ B x . Deﬁne the linear subspace, X x ∈ X B x = def ( ( b x ) : X x ∈ X k b x k < ∞ ) and equip it with the norm k ( b x ) k 1 = def X x ∈ X k b x k W e also deﬁne the linear subspace, Y x ∈ X B x = def { ( b x ) : sup {k b x k : x ∈ X } < ∞} equipp e d with the uniform no rm k ( b x ) k ∞ = def sup {k b x k : x ∈ X } A simple example illustrates these deﬁnitions. Example 4.2. Consider the constant functor x 7− → B with B a Banach space. Then P x ∈ X B is the Banach space ℓ 1 ( X, B ) o f absolutely summable families in B indexed by X and Q x ∈ X B is the Banach space B ( X, B ) o f bo unded functions X − → B with the suprem um nor m. The fac t that b oth P x ∈ X B x and Q x ∈ X B x are B anach spa c es is prov ed in exactly the same way as in the aforementioned cla s sical ca ses. F or ea ch x ∈ X there is an isometric em b edding i x : B x − → P x ∈ X B x that to b x ∈ B x asso ciates the abs olutely summable family δ b x ( y ) = def ( b x if y = x , 0 otherwise. 30 G. RODRIGUES Similarly , for each x ∈ X there is a contractive linear map p x : Q x ∈ X B x − → B x given by ( b x ) 7− → b x . The next theorem states the univ er sal pro p e rty of these tw o spaces and their asso cia ted co nes. Its pr o of is left as an ex e rcise to the reader . Theorem 4.3. F or any fami ly ( T x ) of b ounde d line ar maps B x − → A such that sup x ∈ X {k T x k} < ∞ , t her e is a unique b oun de d line ar map T : P x ∈ X B x − → A such that for every x ∈ X , dia gr am 4.2 is c ommutative. P x ∈ X B x T / / _ _ _ A B x i x O O T x ; ; v v v v v v v v v v Figure 4.2. Universal prop erty of injections i x . The asso ciation ( T x ) 7− → T is an isometric isomorphism Y x ∈ X Ban ( B x , A ) ∼ = Ban X x ∈ X B x , A ! (4.1) F or any family ( T x ) of b oun de d line ar maps A − → B x in Q x ∈ X Ban ( A, B x ) , ther e is a unique b oun de d line ar map T : A − → Q x ∈ X B x such t hat for every x ∈ X , diagr am 4.3 is c ommutative. Q x ∈ X B x p x   A T ; ; v v v v v T x / / B x Figure 4. 3 . Universal prop erty of the pro jections p x . The asso ciation ( T x ) 7− → T is an isometric isomorphism Y x ∈ X Ban ( A, B x ) ∼ = Ban A, Y x ∈ X B x ! (4.2) Theorem 4.3 exhibits s o me no table improv ements o ver theorem 3.1. The indexing set X is arbitrary and the re presentabilit y isomor phisms are isometries. And a s men tione d previous ly , a lthough inﬁnite (co)pro ducts do not e x ist in Ban , it is clear that theorem 4.3 is express ing a universal pro p e rty of a (co)pro duct ob ject. It expresses a uniq ue nes s factorization prop erty for b ounde d c ones , precisely the elements of the Banach space Q x ∈ X Ban ( A, B x ). Denote by Ban c the categor y of Banach spaces and linear contractions, that is, those b o unded linear maps T with op erator norm k T k ≤ 1. I f we restrict a tten tion to elements in the unit b al l of Q x ∈ X Ban ( A, B x ), that is, c ont r active c ones , then theore m 4.3 implies: Theorem 4.4. The c ate gory Ban c has al l smal l pr o ducts and al l smal l c opr o duct s. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 31 Equalizers and coe q ualizers ar e constructed in the usual manner so tha t Ban c is bo th co mplete and co co mplete. F or the b eneﬁt of the reader , we reca ll that if A and B are Ba nach spa ces, the pr oje ctive tensor pr o duct 26 A ˆ ⊗ B is the completion of the linear tensor product A ⊗ B under the pr oje ctive norm , k u k ˆ = inf ( X n k a n k k b n k : u = X n a n ⊗ b n ) (4.3) where the inﬁm um is tak en o ver all the represe nt a tio ns o f u a s a sum P n a n ⊗ b n of elementary tensors . The pro jective tensor pro duct mak es of Ban a clo s ed s y mmetric monoidal cat- egory . Since this structure descends to Ban c (all the coher ence isomorphisms are isometries), it follows that w e hav e a go o d theory of c ate gories enriche d in Ban c . The tw o single big gest problems detected in section 3 , the lack of clos edness a nd inﬁnitary co limits ha ve just b een solved. Categorie s o f Banach s paces have r e ceived a go o d deal of attention in the 1970’s . F or our purp o s es, the monogra ph [CLM7 9] con tains most of what we need. While not couc hed in the la nguage of enriched category theo ry that is, in practice, what the autho r s ar e doing. Our basic refere nce for enr iched ca tegory theor y is [Kel05], but to simplify the discussio n it is co nv enient to introduce some terminolo gy sp eciﬁc to the Banach space cas e. Deﬁnition 4.5. A Ban c -enriched c a tegory will be called a Banach c ate gory 27 . A Ban c -enriched (or str ong , or even c ontr active ) functor b e t ween Banach catego ries is an o rdinary functor F : A − → B such that the map f 7− → F ( f ) is a line a r co ntraction. If F, G : A − → B ar e tw o con tr a ctive functors, a Ban c -enriched na tural tra nsfor- mation F − → G is an o rdinary natural tra nsformation τ : F − → G that is uniformly b oun de d , that is: sup {k τ a k : a ∈ A} < ∞ It sho uld b e noted tha t the Banach s pace o f na tur al transfor mations F − → G is an ob ject of Ban c only if the do ma in category A is small. F or the moment , we forego such siz e technicalities but even tually w e will hav e to come to g rips with them. T aking unit balls, we o btain the 2-catego ry BanCat o f Banach catego ries, contractiv e functors and contractiv e natura l transfor mations. The remar ks preceding deﬁnition 4.5 sug gest the tenth and the la st of our list o f fundamen tal principles. Principle 1 0 . Cate goriﬁe d me asur e the ory lives in the 2 -c ate gory BanCat . 26 Although the pr o jective tensor pro duct is in a sense the canonical one, there are other tensor products and useful ones at that. F or example, later on we will make use of the inje ctive tensor pr o duct . Just as there is no unique wa y to norm a dir ect sum (think of the whole fami ly of p -norm s) there ar e also other reasonable norms one can put on the tensor pr oduct. The deﬁnition of what constitutes a reasonable norm, their conne ctions wi th spac es of op erators, etc. is an imp ortant area in Bana ch space the ory . Goo d references are [R ya02] and, more demanding but also more exhaustiv e in its survey , [DF93]. 27 Deference to tr adition and settled custom has l ed me to this c hoice of terminology , but I ha ve to confess m y envy of a physicist’s f reedom for scav enging Finne gans Wake for accurate and descriptiv e names. 32 G. RODRIGUES Later, we w ill s harp en princ iple 1 0 by taking into account principle 5 and thr ow- ing co co mpleteness co nditions into the mix. What is to b e emphasized here is that the imp ortant 2-category is the 2 -categor y of Ba na ch categ ories and that all no- tions are relative to (or enriched in) Banach spa c es. This has some noteworth y consequences. F or example, since the isomo rphisms in Ban c are the linear bijec- tive isometries we will be interested mainly in isometric Banach sp ac e the ory ; the distinction betw een copro ducts and pro ducts b eco mes importa nt beca use ev en in the ﬁnite case they a r e not isometrically iso morphic, etc. While the passa ge to Ban c -enriched categorie s gives us the neces sary categor ial completeness conditions, it also lands us in the very harsh Bana ch space la nd. The long litany of wild and int r icate counterexamples is not o nly a testament to the ingenuit y o f Ba nach space theo rists but also the deﬁnitiv e pro of of the unsuitabil- it y o f Banach spaces for many purp oses 28 . L e t one example suﬃce for many . In a Hilb ert space , the exis tence of the orthogona lit y co ncept implies that for every closed linea r subspace M there is a canonical nor m 1 pro jection with r ange M and a canonical topolo gical co mplement M ⊥ . The relatively simple and tame linear geometry of Hilbert spaces allows so me dee p theorems on b ounded linear op er ators (e.g. the sp ectr al theorem) which in its turn allows a reas o nably s a tisfying represe n- tation theor y for many ob jects (e.g . lo cally compact groups). This idyllic situation is blown to smithereens with Banach space s b ecause their linear geo metry can b e extremely complicated. While in a Hilbert spa ce ev ery closed subspace is co mple- men ted, in a genera l Ba nach spa ce the b e st gener al result is tha t ﬁnite-dimensional and ﬁnite-codimens io nal subspaces are complemented. In [L T77] it is proved that every inﬁnite-dimensional B a nach space not isomorphic to a Hilbe rt space has one non-trivial (that is, neither ﬁnite-dimensional nor ﬁnite-co dimens io nal) uncomple- men ted subspa ce. In [GM93], the authors constructed the ﬁrst known ex ample of a Banach space whose every non-trivia l close d subspace is uncomplemen ted. By now there are known several e x amples of such s pa ces (commonly called her e ditarily inde c omp osable ), inc luding examples o f C ( X )-spaces for cunningly crafted co mpa ct Hausdorﬀ X (see [Kos04] a nd [Ple04]). The pa ssage from Hilb ert to Banach spaces ha s other imp ortant cons equences to categoriﬁe d mea sure theo ry . Ultimately , it is the very simple isometr ic cla ssiﬁcation of separ able Hilb ert spaces tha t allows a decomp osition such as that of theor em 3.7. In fact, one can even say that the countabilit y restrictio ns in such pr inciples as principle 2 is something o f a red herring, imp osed o n us by a co ns piracy b etw e e n bad measurability deﬁnitions and the simple class iﬁcation of separable Hilb ert spa ces. With Banach s paces the situation is completely diﬀere nt . Let A and B b e tw o Banach spaces o f linear dimension n . The spaces A and B a re isomorphic in Ban but they are not neces sarily isometric isomorphic, that is, isomor phic in B an c . Deﬁne d ( A, B ) to b e the quantit y log inf  k T k   T − 1   : T is a linear is omorphism A − → B  (4.4) 28 The underlying motiv ation for undertaking the categoriﬁcation of measure theory is to buil d TQFT’s and quan tum gr avit y m o dels. In this resp ect, one should notice that the naiv e deﬁnition of Banac h-space v alued TQFT’s is bound to encoun ter diﬃculties. The reason is that every ob j ect M of the cob ordism catego r y Cob ord is r eﬂexiv e whil e there are many non-r eﬂexiv e Banac h s paces. Ev en w orse, the pro jective tensor product of tw o r eﬂexiv e spaces need not b e r eﬂexiv e. F or a striking example, we refer the reader to the computation of the tensor di agonal of ℓ p ˆ ⊗ ℓ p , p ≥ 1 in [ Rya 02 , section 2.5]. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 33 Deﬁnition (4.4 ) gives a metric to the se t M ( n ) of equiv alence cla sses of n - dimensional B anach spaces for the equiv alence relation o f iso metric iso morphism. The metric space M ( n ) is called the Bana ch-Mazur c omp actum and is a compact metric space. In [ABR04] it is proved that it is ho meomorphic to the Alex a ndroﬀ compactiﬁcation of a Hilb ert cube manifold – a humongous space. The mo ral is that a measurable bundle ξ , even if it has for v alues only ﬁnite dimensional Bana ch spaces, can be a very co mplicated b east. Our c hoice to extend Hilb to Ban is a matter o f co nv enience. The categor y Ban lies neare s t at ha nd a nd allows us to remain within the conﬁnes o f cla ssical functional analysis. Mo re generally , any sy mmetric monoidal clos ed ca tegory A that is b o th complete and co complete and in which Hilb em b eds in a suﬃciently nice wa y can serve a s a candidate for base categor y . The categor y A should also have the categoria l structure of Hil b that ma kes it so impo rtant for quantum mechanics (basically , a contrav ar iant endofunctor on A that distills the prop er ties of adjoint op erator s in Hil b – precise deﬁnitions ca n b e found in [Bae97] a nd [Sel07]) and the em b edding should prese rve it 29 . Indep endently of the em b edding Hi l b − → A we choose and ass uming it exists in the ﬁr st place, we b elieve that the categ oriﬁed measure theor y br ieﬂy descr ib ed in sec tion 7 is suﬃciently ro bust and gener al that it extends to this new se tting in a straightforw ard way . 4.1. The failure of the Radon-N ik o dym prop ert y . As remarked in the b e - ginning of this section, while pass a ge to the ca tegory of Banach spaces gives us the necessar y categorial completeness co nditions, we pay a heavy price in hav- ing to deal with ob jects far mo re co mplicated than Hilbert spaces. This fact will surface on many o cc asions, the ﬁr st and most imp orta nt being the failure of the Radon-Nikodym pro p erty , which, as argued in section 3, is fundamental for a g o o d categoriﬁe d measure theory . The plain matter of fa ct is that the Radon-Nikodym prop erty fails in inﬁnite dimensions and fails badly . Let us re turn to the principles formu la ted in section 3, esp ecially principle 9. The universal prop erty implied by it was depicted in diagram 3.5 , which we depict again next. R X χ ( E ) d µ R X f d µ / / _ _ _ H X ∆ O O f : : t t t t t t t t t t t Figure 4. 4 . Meas urable Radon-Niko dym. In ter ms of repr esentabilit y isomor phisms, it says that the map f 7− → R X f d µ establishes a natural isometric is omorphism, L ∞ ( X, B ) ∼ = Ban ( L 1 ( X ) , B ) 29 How ever, note that a general observ able in quan tum mechan i cs i s a self -adjoint, not ne c es- sarily b ounde d oper ator. 34 G. RODRIGUES betw een a space of meas urable functions X − → B and the space of bo unded linea r op erator s L 1 ( X ) − → B . Using in tegra l notation, w e ca n write, Y X B d µ ∼ = Ban  Z X χ ( E ) d µ, B  (4.5) so that L ∞ ( X, B ) is a me asur able p ower while L 1 ( X ) is a me asura ble c op ower . The Radon-Niko dy m prop erty then states that the functor B 7− → L ∞ ( X, B ) is r epr esentable . The space L ∞ ( X, B ) is the Banach space of a.e. equiv a lence classes of “mea- surable” functions f : X − → B with the essential supremum norm. The adjectiv e measurable is betw een quotes b ecause measurability here is not what one would naively exp ect: measura bility of f with B endow ed with the Bor el measurable struc- ture. This latter notion of measurability is almost goo d eno ug h but the reasons why it is no t g o o d enoug h are subtle 30 . In subsection 3 .4 w e re c alled the notion of simple function : a function tha t up to a n ull-meas ure set is a linear combin a tion of characteristic functions. W e will need a generalization. Deﬁnition 4 .6. An elementary fun ction is a function f : X − → B suc h that there is a countable partition ( E n ) of X into measurable sets and a seq uence ( b n ) in B such that: f = a.e. X n χ ( E n ) b n W e say that a function f : X − → B is str ongly m e asura ble if it is the a.e. uniform limit of a sequence o f elemen tar y functions. W e now deﬁne L ∞ ( X, B ) as the space of a.e. equiv alence classes o f b ounded strong ly mea surable functions with the es sential supremum nor m. The next theorem, which w e hav e alrea dy alluded to b efore, is fundamental. The pr o of (as well as the pr o of of the Bochner theorem 4.8 b elow) can be found in [Rya02 , c hapter 2 , s e ction 3]. Chapter 2 of [DU77] is another reference. Theorem 4. 7 (Pettis) . F or a function f : X − → B the following ar e e quivalent: (1) f is st r ongly me asur able. (2) f is Bor el me asur able and ther e is a c one gligible 31 set E ⊆ X such that f ∗ ( E ) is sep ar able. (3) f is weakly mea surable , t hat is, for every b ounde d line ar funct ional b ∗ , the function b ∗ f is me asur able, and ther e is a c one gligible set E ⊆ X such that f ∗ ( E ) is sep ar able. With the strong measurability notion, the (stro ng or Bo chner) integral of func- tions is dev elo pe d just like the Leb esg ue integral. If f is an elementary function, we deﬁne its in tegr al to be, Z X f d µ = def X n µ ( E n ) b n (4.6) 30 Subtle b ecause ﬁnding an example of a Borel measurable, not strongly measurable function is directly connected with questions undecidable i n ZF C. W e r efer the reader to [Edg77] and [Edg79] for the details. 31 A measurable subset E ⊆ X is c one g ligible if its complement has null m easure. W e r emind the r eader of our standing assumption that the m easure µ i s complete. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 35 and say that f is inte gr able if the series on the r ight-hand side of (4.6) con verges. The integral is then extended in the obvious w ay to stro ngly measura ble functions. The next criterion for in tegrability is very useful. Theorem 4. 8 (Bo chner) . A str ongly m e asura ble fun ction f : X − → B is inte gr able iﬀ the function x 7− → k f ( x ) k is inte gr able. Basically , all the prop erties of scalar in tegra ls lift unscathed to the v ector case. One that we will need below, and that follows from the Pettis and Bochner theo- rems, is the following: Theorem 4.9 . The line ar sp ac e of simple functions X − → B is dense in the sp ac e L 1 ( X, B ) of inte gr able functions. The integral of a measurable vector function g yields a natural isometric embed- ding, the R adon-Niko dym map , L ∞ ( X, B ) − → B an ( L 1 ( X ) , B ) given by g 7− →  f 7− → Z X g f d µ  Bounded linear maps T that are in the image of the Radon-Niko dym map ar e called r epr esentable and the function g ∈ L ∞ ( X, B ) representing T is called its R adon-Niko dym derivative . A Banach spa c e B such tha t every b o unded linear map L 1 ( X ) − → B is repr esentable for every pro bability space X is sa id to hav e the R adon-Niko dym pr op erty , or RNP for short. The scalar Radon-Nikodym theorem (for example, s ee [Hal74, sectio n 3 1]) implies that the map is indeed surjective for B = K . Since B 7− → Ban ( L 1 ( X ) , B ) is a representable functor, it is con tinuous in the Banach space v ar iable a nd from the fact that B 7− → L ∞ ( X, B ) is ﬁnitely c ontinu ous it follows that every ﬁnite limit of Bana ch spaces with RNP has RNP . Note also that since what is in question is the su rje ctivity of the Radon-Nikodym map, it follows that the Radon-Nikodym prop erty is a n inv aria nt of the linea r homeomor phism type. Many other classes o f spaces are known to hav e the Ra don-Nikodym prop er ty , such a s all reﬂexive spaces, all separable dual spaces, countable coproducts of RNP spaces, etc. but alas , it is not universally tr ue. The problem is r eadily appa rent if we note that the unit X − → L 1 ( X ) is the inv erse image o f the identit y under the would-b e-isomor phism Radon-Nikodym map. The o bvious candidate for such a universal map is x 7− → δ x with δ x given by: δ x ( y ) = def ( 1 if y = x , 0 otherwise. But if the measure space is ato mless, δ x = a.e. 0 so that this cannot work. This is no pro o f that L 1 ( X ) do es not have the Radon- Nikodym prop erty 32 , but it sure is a stro ng indication that in inﬁnite dimensions the Rado n-Nikodym theorem fa ils. 32 This line of argumen t can b e pursued and justiﬁed somewhat. The presence of atoms in the measure space guaran tees the existence of extreme p oint s in the unit ball of L 1 ( X ) while if X is atomless i t can b e shown that the unit ball has no extreme p oi n ts w hatso ever. This s uggests that there is an intimate relation b et ween a geometric prop erty , the existence of extreme p oints, 36 G. RODRIGUES Theorem 4.10. Denote by ι the inclusion L ∞ ( X ) − → L 1 ( X ) . If the b ounde d line ar map T : L 1 ( X ) − → B is r epr esentable then the c omp osite T ι is c omp act. Pr o of. Supp ose T is representable, that is, there is g ∈ L ∞ ( X, B ) suc h that: T ( f ) = Z X g f d µ By Bo chner’s theorem, the function g is Bo ch ner integrable. Densit y of simple functions implies the existence of a s e q uence ( g n ) such tha t g n → g in L 1 ( X, B ). The linear maps T n ( f ) = def Z X g n f d µ on L ∞ ( X ) have ﬁnite rank. F rom,     Z X ( g − g n ) f d µ     ≤ k f k ∞ Z X k g − g n k d µ we can conclude that the sequence of o p erators T n conv erges to T ι , which implies that T ι is compact.  Theorem 4 .1 0 can be improv ed. A b ounded linear map T : L 1 ( X ) − → B is representable iﬀ the co mp o site map T ι is nucle ar . Reca ll that a linear map A − → B is nuclear if it is in the image of the natural ma p A ∗ ˆ ⊗ B − → Ban ( A, B ). The quotient of A ∗ ˆ ⊗ B b y the kernel of this map is the space N ( A, B ) o f nuclear op erator s. This r esult, originally due to A. Grothendieck, will no t b e needed since the theorem as stated is mo r e than enoug h for our purpo ses. If X is a n inﬁnite atomless probabilit y space suc h as the unit interv al with the Leb esgue measure, consider the identit y L 1 ( X ) − → L 1 ( X ). By theor em 4.10, if the identit y were representable, the unit ball of L ∞ ( X ) would b e relatively compact in L 1 ( X ), which is clear ly and o bviously false 33 . and an analytic one, the existenc e of Radon-Nikodym deri v atives. The single most imp ortant unsolv ed problem in the area of ve ctor measures on Banach spaces is pr ecisely the conjectured equiv alence betw een the Krein-Mil man pr op erty (ev ery closed b ounded con vex set is the closed con vex hull of its extreme points) and the Radon-Nikodym prop erty . That RNP implies KMP is kno wn and is due to J. Li ndenstrauss ([Lin66]). A s for the con verse, in [A K 06, section 5.4, page 118] the authors r emark that “It is probably fai r to say that the sub ject has r eceiv ed relativ ely little atten tion since the 1980’s and some really new ideas seem to be necessary to make further progress.” W e refer the r eader to [DU 77, c hapter VI I] for more information on this sub ject. 33 On second thought , this ma y not b e entirely obv i ous. T o show that the unit ball of L ∞ ( X ) is not relativ ely compact i t suﬃces to construct a unif ormly b ounded sequence ( f n ) in L ∞ ( X ) with no L 1 -con vergen t subsequence. The fact that X is atomless allo ws us to emplo y exhaustion (see [Hal74, section 41, exercises 2 and 3] or [F re01, c hapter 1, section 5]) to, starting with the singleton partition { X } of X , recursively build a partition E n +1 of X wi th 2 n sets b y dividi ng eac h set of E n in t wo sets of equal measure. With this setup, we will now si mulat e R ademacher functions on X . Deﬁne the sequence ( r n ) b y , r n = def X E m,n ∈E n ǫ m χ ( E m,n ) with ǫ m ∈ {− 1 , 1 } (the unit sphere of R ) deﬁned by ǫ m = def ( 1 if m is eve n, − 1 othe rw i se. It is easy to see that k r n − r m k 1 = µ ( X ) for n 6 = m . Leaping ahead of ourselves, we remark that this construction extends almost w ord f or word to σ -complete atomless measure algebras. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 37 Pondering over theorem 4.10 , the reader may suspec t that the failure of RNP is due to our deﬁnition of the s pa ce L ∞ ( X, B ), a space too s mall to represent all bo unded linear maps , a nd if only we were ingenious enough a suitable genera lized deﬁnition o f a.e. bounded measura ble functions could b e found to ma ke the Ra don- Nikodym map surjective. Unfor tunately , even a cursory perusa l o f [DS58, chapter VI] will show that to make the Radon-Niko dym map surjective, we hav e to change the space L ∞ ( X, B ) and then in all cases one ends up having one or more of the following defects: (1) The Radon-Nikodym map is not surjective for so me Banach s pa ces B . (2) The representing function g takes v alues in a spac e other than B . (3) The representing function g is not a function at all but a measure, who se Radon-Nikodym deriv ative, if it existed, would b e the representing function. In s ection 5 we will provide a Radon-Nikodym theo rem in the form (3), which will be imp or tant in o ur quest to ca teg orify measure theo ry . In subsection 5 .3 we will give an inkling of how Radon-Niko dym theore ms of the form (2) can b e obtained when discussing the fundamen tal Stone space construction of a Bo olean algebra Ω. The failure of the Radon-Nikodym prop er t y in inﬁnite dimensions is a blo w in our dr eams of building a pro pe r categoriﬁed meas ure theor y . But no t all is lost. A po ssible course o f action is to sa lv ag e what can b e sa lv ag ed: w e alr eady know that RNP ho lds uniformly for all ﬁnite dimensional spaces so this could be exploited to ﬁnd weaker forms of the RNP universal proper ty . This co urse of action co uld be pursued, but not only it w o uld lead us far in to the so-called lo c al Banach sp ac e the ory (see [Pie99] a nd r eferences therein), but at this p oint it is not even clear that it brings any signiﬁcant dividends, so w e leave it aside fo r some b etter o pp o rtunity . The more obvious course of action howev er, is to simply forget (half of ) principle 9 and pro ce e d with what we have and this is precisely wha t we will do. As section 7 unfolds, the imp o rtance of the Radon- Nikodym prop erty will re cede into near ly microscopic prop or tions, until ﬁnally our p erse verance is rewarded and in the end, almost out of the blue, the univ er sal prop erty for R X χ ( E ) d µ will fall on our laps and the ﬁna l truth revealed. But this universal prop erty depe nds on some of the deep e r prop erties of the categ o ry of cosheaves a nd to g et ther e, a sig niﬁcant amount of gro und will ha ve to b e cov ered. 5. Foregr ounding: measure algebras and integrals If the re a der lo oks closely on the w or k done up to now, he will notice that the underlying s e t X of a measur e space was har dly ever needed a nd was dra gged along like an appe nda ge whose origina l function has b een long forgotten. In this s ection we will excise it, throw it in the trash bin, and conce nt r a te on the rea l co re of measure theory : a Bo ole a n algebr a Ω and a measure µ deﬁned o n it, o r a me asure algebr a as will b e deﬁned b elow. A t this juncture, so me readers will ra ise their hands in pro test and r emark that the underly ing set X do es fulﬁll a function: it allo ws us to deﬁne the L p -spaces of functions. The answer to this ob jection is that the L p -spaces c an be constructed for a general Bo olean algebra and even throw a considera ble lig ht on the whole categoriﬁc a tion pro c e ss. The reaso n for this is ac tua lly quite simple: the abstra ct construction of the L p -spaces bring s to the foreground the universal pr op erties of 38 G. RODRIGUES these sp ac es (principle 1 aga in!) and these dep end not on the underly ing set X but on the mea sure a lgebra (Ω , µ ). In gener al, we b e lieve that a g o o d deal o f conceptual clariﬁcation is achiev ed and, most imp ortantly for us , they are an e ssential s tepping stone for ca tegorifying measure theo ry . Since most readers will proba bly no t ha ve seen these univ ers al prop erties 34 we will dev ote this section to them 35 . F or measure algebr a s, the obvious r eference is the encyclop edic [F re02]. The reader ca n also ﬁnd there appro priate reference s for the theory of Bo olean a lgebras, including Stone duality . F or the latter, [Jo h8 2] is eminently suitable. F or vector measures, there is nothing b etter than to turn to [DU77]. As it happe ns frequently , many of the results we ﬁnd ours elves quo ting do not come in the exact versions we require. Many a re but tr iﬂe mo diﬁcations of known res ults, while others r equire o ne or tw o additional ideas, usually o f a categor y-theoretic nature. Since their n umber is Legio n, w e will herd them all together in a future surv ey paper ([Rodc]). W e start by int r o ducing a couple of deﬁnitio ns. Deﬁnition 5.1 . A me asur e algebr a is a pair (Ω , µ ) where Ω is a Bo olean algebra and µ is a b ounded, positive, ﬁnitely additiv e map Ω − → R . Comparing with the corres po nding deﬁnition in [F r e02, chapter 2, section 1], there ar e a few diﬀerences. First, we w or k only with b ounded (or total ly ﬁnite ) measures. This is necessar y if we are to exploit the strong link b etw een the measure- theoretic and the functiona l-analytic sides. Seco nd, w e do not require the meas ur e µ to be non-de gener ate , tha t is: µ ( E ) = 0 = ⇒ E = 0 This is a mino r technicalit y since we can alwa ys quotient Ω b y the ideal of null- measure sets (see b elow). As will b e seen in the contin uation, vir tually every functor that we deﬁne on the category of meas ure alg ebras will factor through the subca t- egory o f non-degener ate meas ure a lgebras. The measure alg e br a deﬁnition 5.1 just happ ens to be slight ly more conv enient for our categor ially-minded pur po ses. The more substa ntial diﬀerence is how ever, that we do not r e quire either σ - completeness of Ω o r σ - additivity of µ . There a re many reasons for this laxness, starting with the fact that the more imp ortant measur es that we will meet are sp ectral measures and these are never σ -additive except in triv ia l cases. Restrict- ing o urselves to ﬁnitely additive measure s also has a didactic purp ose. All the deep theo rems of measure theor y need s ome sort of σ -co mpleteness hypothesis (Lebe s gue’s dominated theorem, Radon- Nikodym theorem, etc.). Cont r ap ositively , all the pro ofs sk etched b elow for ﬁnitely additive mea sures ar e not only easy but they are e asy to c ate gorify . This is not to say that σ - completeness hypothesis ar e not needed o r important; they ar e a nd v ery m uch so. But by restricting ourselves 34 Understandably , textbo oks on measure theory do not men tion these universal properties as they call for conce pts suc h as v ector measures, Banac h spaces, semiv ariation, etc., which hav e no place i n i n tro ductory texts. 35 As said elsewhere, the main motiv ation for this work arises from physical considerations related with TQFT’s and quan tum grav ity models. One of the mai n ob jective s of quan tum grav ity is to derive space-time f rom the physical theory and one ob vious place to start f rom is the structure of the (state indep enden t) propositions that can b e made about the system. Classi cally , these are Bo olean algebras (e.g. see [Jau68, chapt er 5]). See recen t work connecting top os theory with quan tum mechanics [DI08] and [ HLS07]. The connection of this pap er with these w orks will become clearer in subsection 7.1. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 39 to ﬁnitely additive meas ur es, we can more e a sily underline the categ o rial underpin- nings esse ntial for categorifying mea sure theor y . It is easy to see that µ is monotone , tha t is, if E ⊆ F then µ ( E ) ≤ µ ( F ). Because of this, µ is b o unded iﬀ µ (1) < ∞ . W e rep eat that unless sp eciﬁed o therwise, we will only cons ider b ounded measures. W e denote by N ( µ ) the ideal of µ -null ele ments. In symbols: N ( µ ) = def { E ∈ Ω : µ ( E ) = 0 } (5.1) W e can get rid o f these µ -null sets by taking the quotient π : Ω − → Ω / N ( µ ). The universal prop erty of this q uo tient with resp ect to measures is easy to state. A ﬁnitely additiv e ν is µ -supp orte d (or has µ -supp ort ) 36 if for every E ∈ Ω with µ ( E ) = 0 then ν ( E ) = 0. W e now hav e that a meas ur e ν on Ω is µ -suppo rted iﬀ there is a (unique) ﬁnitely additiv e b ν on the quotient Ω / N ( µ ) s uch that diagram 5.1 is comm utative. Ω / N ( µ ) b ν / / _ _ _ V Ω π O O ν : : v v v v v v v v v v Figure 5. 1 . Universal pro p erty of quotien t π : Ω − → Ω / N ( µ ). This universal prop erty s ays that for most purp oses we can forget ab out the ideal of µ - null s ets and simply assume that µ is non-de gener ate , that is N ( µ ) = 0 . A morphism (Ω , µ ) − → (Σ , ν ) of measure alg ebras is a Bo o le a n a lgebra morphism φ : Ω − → Σ such that ν φ has µ -supp or t. Note that formally , a measure algebra (Ω , µ ) c a n b e identiﬁed with the measure since the Bo olean a lgebra can alwa ys b e recov er e d as the domain of µ . Thus, the categ ory of measur e algebra s and measure algebra ma ps will b e denoted by Meas . Without any sor t of o rder c ompleteness prop erties (and e.g. Radon-Nikody m’s theorem) the morphisms a r e to o weak for some purp o ses a nd a t times it will b e nece ssary to r equire mo re o f φ . The mor phism φ is Lip schitz if there is a constant C suc h that: ν φ ( E ) ≤ C µ ( E ) , ∀ E ∈ Ω (5.2) If φ is a Lipschit z morphism w e denote b y k φ k L , its Lipschitz norm , the qua nt ity inf C φ where C φ is the set of consta n ts C in the conditions of (5.2). Lipschitz morphisms are needed fo r example, to es tablish the functorialit y of the L 1 -functor on measur e algebras . 36 Classically , for σ -additive me asur e s on σ -co mplete Bo ole an algebr as , having µ -supp ort is equiv alent to a “uniform con tinuit y” prop erty and is ca l l ed absolute µ -c ontinuity . Reusing the classical termi nology w ould b e a mi stak e how ever, because µ -supp ort is a very weak requiremen t when the σ -completeness conditions are dropped. The paper [ W en94] uses the term me asur e r e ﬂe cting , but the dir ection is reversed si nce the author is working with measure spaces. 40 G. RODRIGUES 5.1. The Banac h algebra L ∞ (Ω) . One of the more striking theor e ms on linear op erator s is the sp ectra l theorem, which in one form ([Con90, c ha pter IX, section 2]) represents every b ounded normal op erato r T o n a separ able Hilbert H as an int eg ral over a canonical spectral measur e on the B orel sets of the sp ectrum of T . The extension of this cycle of ideas to Banach space s w as done b y W. G. Bade in the 1950’s . The origina l pap ers a re [Ba d55] and [Bad59], but s ee also [DS71] or the more rece nt monogr aph [Ric99], whose o utlo ok is clo ser to our needs. As we no ticed in section 4 ther e ar e kno wn by now a handful of Banach spaces with very few pro jections so it is matter of some imp or tance to know conditions that guarantee the existence of eno ugh pro jections. W e oﬀer to the re a der [OV89] as a n ent r y p oint int o this v a st sub ject. In this sketc hy subsection we are interested only in some very simple a sp ects of these o b jects which will b e enoug h to construct the sp ectral mea sure asso c ia ted to a cosheaf. The theory of L ∞ (Ω) dep ends o nly on the fact that Ω is a Boo le an algebra a nd necessitates no order completeness prop erties or the choice of a mea sure on Ω. The ﬁrst step is the construction of the space of “s imple functions on Ω”. This is simply the quo tient of the free linear space F(Ω) by the subspace generated b y , ι ( E ∪ F ) − ι ( E ) − ι ( F ) for a ll pairs o f disjoint E , F ∈ Ω. Denote the quotient space, the spac e of simple elements , by S (Ω), and by π the quotient linea r map F(Ω) − → S (Ω). The comp osite Ω ι − → F(Ω) π − → S (Ω), the char acteristic map , is denoted by χ . By construction it is ﬁnitely additive and the universal such among ﬁnitely additive maps with v a lues in linear spaces – see diag ram 5.2 . S (Ω) / / _ _ _ V Ω χ O O ν = = { { { { { { { { { Figure 5. 2 . Universal pro pe r ty of χ : Ω − → S (Ω). T o describ e the universal prop erty inside the categor y of Banach spaces , we start b y intro ducing a norm on S (Ω). First, w e need a lemma to the eﬀect that every simple element ha s a repr esentation as a linea r combination of characteristic elements. Lemma 5.2. F or every non-zer o f ∈ S (Ω) ther e is a p artition ( E n ) and non-zer o sc alars k n ∈ K , su ch that: f = X n k n χ ( E n ) (5.3) Pr o of. F or eac h f ∈ F(Ω) there are elements E n ∈ Ω and scalar s k n ∈ K suc h that f = P n k n χ ( E n ). The lemma is no w proved by induction on the num b e r of terms and using some basic Bo o lean alg ebra manipulations.  W e now introduce a norm on S (Ω) by picking a repres e ntation of f as in lemma 5.2 and putting: k f k ∞ = def max {| k n |} (5.4) CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 41 It is not diﬃcult to see that (5.4 ) do es no t dep end on the chosen r epresentation. The completion of S (Ω) under the nor m (5.4) is denoted by L ∞ (Ω). F or the additive maps side, denote by parts ( E ) the set o f ﬁnite partitions o f E ∈ Ω. If E = 1 then this set is als o denoted b y 37 parts (Ω). This set is order ed by r eﬁnement : a partition E r eﬁnes F if for ev er y F ∈ F there is E ∈ E s uch that E ⊆ F . Under this pa r tial order, parts ( E ) is ﬁlter e d . Deﬁnition 5.3. Let B b e a Bana ch spac e and ν : Ω − → B an additiv e map. The semivariatio n o f ν o n E is the (p ossibly inﬁnite) quantit y: k ν k ∞ ( E ) = s up ( X F ∈E |h b ∗ , ν ( F ) i| : b ∗ ∈ ball( B ∗ ) , E ∈ parts ( E ) ) (5.5) The ma p ν ha s b ounde d semivariation if for every E ∈ Ω, k ν k ∞ ( E ) < ∞ . It is ha rd to gra sp w ha t exactly is the semiv ariation measuring and to b e totally honest, I do not have any illuminating tho ughts to oﬀer on this matter. The s emiv ar iation deﬁnes a map E 7− → k ν k ∞ ( E ) o n Ω that is a p os itive, mono- tone a nd ﬁnitely subadditive, th us ν has bounded semiv aria tion iﬀ k ν k ∞ (1) < ∞ . Only slig ht ly mor e diﬃcult than this triviality is the fact that ν has b ounded semiv ar iation iﬀ it has bounded r ange. W e deno te b y A (Ω , V ) the linear space o f additive maps Ω − → V and by BA (Ω , B ) the normed space of b ounded ﬁnitely additive maps Ω − → B with the semiv ariation nor m. Theorem 5.4. F or every b ounde d additive map ν : Ω − → B with B a Banach sp ac e, t her e is a unique b ounde d line ar map R d ν : L ∞ (Ω) − → B such that the triangle 5.3 is c ommutative. L ∞ (Ω) R d ν / / _ _ _ B Ω χ O O ν ; ; x x x x x x x x x x Figure 5. 3 . Universal pro p erty o f L ∞ (Ω). The map ν 7− → R d ν is a natur al line ar isometric isomorphism BA (Ω , B ) ∼ = Ban ( L ∞ (Ω) , B ) (5.6) Pr o of. The pro of is mostly a matter of checking details. Firs t, if T is a bounded linear map then T χ is an additive map with k T χ k ∞ ≤ k T k k χ k ∞ . The trickiest thing may now b e to prov e that the s e miv ar ia tion of χ is ≤ 1 – if the r eader is having diﬃculties we r efer him to [DU77, page 3 , example 7]. On the conv erse direction, the univ ers al proper ty 5.2 gives the lift o f ν to S (Ω) and a simple computation shows it to b e bounded for the nor m (5.4), thus it lifts uniquely to L ∞ (Ω).  37 Or the other wa y around, the unit or top elemen t 1 of a Bo olean algebra Ω will also b e denoted by Ω. 42 G. RODRIGUES Theorem 5.4 can seem quite inno cent, but when coupled with other high-p owered to ols such as Stone duality , it can yield some sig niﬁcant results. F o r example, w e refer the reader to [Gar7 3] for a very elegant pro of of the Riesz representation theorem. An exp o sition of this pro o f can als o b e found in [Car 05, chapter 16]. W e emphasize the conceptual role o f theor em 5.4. The integral map is introduce d as a b ypr o duct of the universal prop erty of L ∞ (Ω), exactly the t yp e of result we need to guide us in ca tegorifying measure theory . Op erator alge braists will readily recognize it. Note also that some of the formal pro p erties o f the in tegr al are implicit in 5.4. F or exa mple, the natur a lity of the isomorphis m (5.6 ) in b o th v ariables amounts to the equalities, T  Z f d ν  = Z f d T ν Z φ ∗ ( f )d ν = Z f d φ ∗ ν (5.7) for every b ounded linear map T and every B o olean algebra mor phis m φ . The seco nd equality is a c heap for m of the c hange o f v aria bles formula. The measure φ ∗ ν , the pul lb ack of ν under φ , is simply φ ∗ ν : E 7− → ν ( φ ( E )) (5 .8) A Ba nach algebra structur e 38 can b e int r o duced on L ∞ (Ω) such that χ ( E ∩ F ) = χ ( E ) χ ( F ), that is, the a dditive map χ : Ω − → L ∞ (Ω) is sp e ctr al o r multiplic ative . The universal prop erty of theorem 5.4 can b e extended to yie ld a n is o metric iso mor- phism b etw een the spaces o f Banach a lgebra morphisms a nd the b ounded spectr al ﬁnitely a dditive mea sures 39 . Sp ectra l mea sures a re ess ent ia l in the catego riﬁed in- tegral theo ry as will b e seen later and b ecaus e o f this we insert the next theorem, which is a more o r less tr iv ial extensio n o f 5.4 to sp ectra l maps. Theorem 5. 5. L et A b e a Banach algebr a and ν : Ω − → A a b oun de d sp e ctr al map. Then ther e is a unique b ounde d alge br a morphism R d ν : L ∞ (Ω) − → A such that the diagr am in ﬁgur e 5.4 is c ommut ative. L ∞ (Ω) R d ν / / _ _ _ A Ω χ O O ν < < x x x x x x x x x x Figure 5. 4 . Universal pro p erty o f the Banach alg ebra L ∞ (Ω). 38 In fact, by theorem 5.10 b elow L ∞ (Ω) is a C ∗ -algebra and with suitable order completeness h yp othesis on Ω , ev en a V on-Neumann algebra. This operator algebra vi ewpoint is very f ruitful, probably essen tial, but will not be pursued here. 39 Spectral measures are hardly ev er σ -additive, although in many cases they are σ -additive for certain weak er, lo cally con vex li near topologies. F or example, the sp ectral measure of the sp ectral theorem is σ -additive for the strong op erator topology (th e analysts name f or the topology of point wise conv ergence). CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 43 The asso ciation ν 7− → R d ν establishes a natur al isometric isomorphi sm b etwe en the set of b oun de d sp e ctr al me asur es Ω − → A and the set of b ounde d algebr a m or- phisms L ∞ (Ω) − → A , BAS (Ω , A ) ∼ = BAlg ( L ∞ (Ω) , A ) (5.9) 5.2. The Bochner integral. In this subsection we r evisit the stro ng (or Bo chner) int eg ral and show how a ll the basic prop erties follow easily with e asy pro ofs. Since our o b jective is to supp ort our introductor y injunction that a go o d deal of conce p- tual cla riﬁcation is achiev ed if we c o nsider Bo olean algebras and mea sures on it, we consider only ﬁnitely additive measures . Let (Ω , µ ) b e a measure a lg ebra. In subsection 5.1 we hav e deﬁned the spa ce of simple elements S (Ω). W e extend the deﬁnition to measure algebra s by deﬁning: S (Ω , µ ) = def S (Ω / N ( µ )) (5.10) The no tation S (Ω , µ ) is sligh tly misle ading, because this spa c e only dep ends on N ( µ ) not on the sp eciﬁc v a lues of µ . On S (Ω , µ ) w e have the norm:      X n k n χ ( E n )      1 = def X n | k n | µ ( E n ) (5.11) It is not diﬃcult to see that the right-hand side of (5.11) is independent of the chosen repre s entation as a linea r combination of c har a cteristic e lement s. Finally , we deﬁne the sp ac e of simple elements with values in B to b e the linear tensor pro duct, S (Ω , µ, B ) = def S (Ω , µ, ) ⊗ B (5.12) and deﬁne a norm on it by      X n f n ⊗ b n      1 = def X n k f n k 1 k b n k (5.13) Obviously , the space S (Ω , µ, B ) r educes to S (Ω , µ ) when B is the real ﬁeld. The space L 1 (Ω , µ, B ) is the completio n of L 1 (Ω , µ, B ) under the nor m (5.13 ). Before deﬁning the integral ma p, we need a lemma giving a ca nonical for m for the elemen ts of S (Ω , µ, B ). Lemma 5.6. F or every non-zer o f ∈ S (Ω , µ, B ) , t her e exist non- z er o p airwise disjoint elements E n ∈ Ω and elements b n ∈ B such that f = X n χ ( E n ) ⊗ b n (5.14) Pr o of. This is a c o mbination of lemma 5.2 with the prop er ties of linear tens or pro ducts.  There is a map R d µ : S (Ω , µ, B ) − → B given by X n χ ( E n ) ⊗ b n 7− → X n µ ( E n ) b n This map is linear and contractive for the k , k 1 -norm, therefor e it extends uniquely to a linear con tr a ction L 1 (Ω , µ, B ) − → B . The nex t theo rem iden tiﬁes the s pa ce L 1 (Ω , µ, B ) with a pro jective tensor pro duct a nd, in particular , it will show that L 1 (Ω , µ, B ) is co co ntin uo us in the Banach spa ce v ariable. It will also pro ve that 44 G. RODRIGUES not only the vector in tegra l R d µ commutes with ev er y bounded linear map, but also that R d µ is the scalar integral tenso red with the identit y 1 B . Theorem 5. 7. The biline ar map S (Ω , µ ) × B − → L 1 (Ω , µ, B ) given by ( f , b ) 7− → f ⊗ b establishes a natur al iso metric isomorp hism, L 1 (Ω , µ, B ) ∼ = L 1 (Ω , µ ) ˆ ⊗ B (5.15) wher e ˆ ⊗ denotes the pr oje ctive tensor pr o duct of Ba n ach sp ac es. F urthermor e, diagr am 5.5 is c ommutative. L 1 (Ω , µ ) ˆ ⊗ A 1 L 1 (Ω ,µ ) ˆ ⊗ T / / ∼ =   R d µ ˆ ⊗ 1 A % % K K K K K K K K K K L 1 (Ω , µ ) ˆ ⊗ B ∼ =   R d µ ˆ ⊗ 1 B y y s s s s s s s s s s A T / / B L 1 (Ω , µ, A ) T ∗ / / R d µ 9 9 s s s s s s s s s s L 1 (Ω , µ, B ) R d µ e e K K K K K K K K K K K Figure 5. 5 . Naturality of R d µ . Pr o of. Given our deﬁnitions, the pro of consists in a simple chec k that the pro jec- tive no rm is equal to the norm (5.13 ) a nd a few trivia l c o mputations with simple functions.  Next, we treat the universal proper t y o f L 1 (Ω , µ, A ). By theor e m 5.7, we hav e the chain of iso morphisms, Ban ( L 1 (Ω , µ, A ) , B ) ∼ = Ban  L 1 (Ω , µ ) ˆ ⊗ A, B  ∼ = Ban ( L 1 (Ω , µ ) , Ban ( A, B ) ) (5.16) so that it is suﬃcient to discuss the universal prop erty o f L 1 (Ω , µ ). In o rder to do that, we introduce a nother cla ss of measures, the Lipschitz me asur es . Deﬁnition 5.8 . Let B b e a B a nach space. An additive map ν : Ω − → B is µ - Lipschitz (or simply Lipschitz if the scalar measure µ is ﬁxed throug hout) if there is a p o sitive co ns tant C such that: k ν ( E ) k ≤ C µ ( E ) , ∀ E ∈ Ω (5.17) It is easy to see that the measure ν is µ -Lipschitz iﬀ the quotient map Ω − → Ω / N ( ν ) is Lipschit z according to (5.2 ). If ν is µ -Lipsc hitz, then w e also say that ν is µ -dominate d and µ is a dominating m e asure . This terminolog y par allels the terminology for c ontr ol me asur es . If ν is µ -Lipschitz, we deﬁne the µ -Lipschitz norm by: k ν k L = def inf C ν (5.18) where C ν is the set of consta nt s C in the conditions o f (5.17). The Lipschitz norm is an upper b ound for the distortion ra tio k ν ( E ) k /µ ( E ). W e deﬁne LA (Ω , µ, B ) to CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 45 be the space of µ -Lipschitz a dditive maps Ω − → B and with norm the µ -Lipsc hitz norm (5.1 8). Theorem 5 .9. If ν : Ω − → B is a µ -Lipschi tz, ﬁnit ely additive map, then t her e is a unique b ounde d line ar map R d ν : L 1 (Ω , µ ) − → B such that triangle 5.6 is c ommu tative. L 1 (Ω , µ ) R d ν / / _ _ _ B Ω χ O O ν ; ; v v v v v v v v v v Figure 5. 6 . Universal prop erty of L 1 (Ω , µ ). The map ν 7− → R d ν est ablishes a natur al isometric isomorphism LA (Ω , µ, B ) ∼ = Ban ( L 1 (Ω , µ ) , B ) (5 .19) Theorem 5.9 is something of a blind alley and I know o f no s erious analytical uses of it as it simply trades o ne complication for another. Its p os itive a sp ects howev er, are fa r more imp ortant for us. F or one, it se ttles the co contin uity prop erties of the bifunctor LA (Ω , µ, B ), thereby giving us another family of Banach sheav es. More impor tant is its conceptual meaning. The univ er sal prop erty expressed in theorem 5.9 , c o mpletely a nalogo us to the univ er sal pro p erty o f L ∞ (Ω) in theorem 5.4, deﬁnes the spa ce L 1 (Ω , µ ), while the constructions prec eding it pr ove that it do es indeed exist, while also giving a particularly convenien t pr esentation fo r it. Other basic theorems follow eq ually , with relatively ea s y pro ofs. F or example, the pro of o f F ubini’s theorem star ts with the c o nstruction of the copr o duct of Bo olean algebras Ω ⊗ Σ and the identiﬁcation of the mea sures on Ω ⊗ Σ as certain biadditiv e functionals. In par ticular, if w e hav e ﬁnitely additiv e measures µ and ν on Ω and Σ resp ectively , then the map µ ⊗ ν : E ⊗ F 7− → µ ( E ) ν ( F ) yields a well-deﬁned measur e on Ω ⊗ Σ. The F ubini theo r em 40 is now the ass e rtion that there is a natural iso metric isomo rphism L 1 (Ω ⊗ Σ , µ ⊗ ν ) ∼ = L 1 (Ω , µ ) ˆ ⊗ L 1 (Σ , ν ) (5.20) Since by theorem 5.7 we ha ve the natura l isomor phism L 1 (Ω , µ ) ˆ ⊗ L 1 (Σ , ν ) ∼ = L 1 (Ω , µ, L 1 (Σ , ν )), the e quality of the iter ate d inte gr als follows from the F ubini isomorphism (5.20 ). A t this p o int , the reader may b e justly impatient with our dabbling with formal trivialities 41 . In a sense it is an unjust criticism, b ecaus e a string of trivialities may 40 F or σ -additive m easures on σ -complete Bo olean algebras the construction of this copro duct is more inv olve d, b ecause the copro duct Ω ⊗ Σ in the ca tegory Bo ol of Bo olean algebras is σ - complete only in trivial cases. 41 Judging by the times it is rep eated, “mere formali t y” may very well be the fa vorite expression of [ DU77]. The pragmatic distrust of “mere formalities” is a nat ural, ev en health y attitude on the part of an analyst. The publishing date of [DU77] is 1977, but the prejudices i t ec ho es are still around and one can stil l occasionally hear that category theory is “just a language,” a 46 G. RODRIGUES yield, a nd do e s yield, non trivial results. In another sense, the criticism is entirely justiﬁed. But it is precisely the trivial quality of these pro ofs that not only justiﬁes the scaﬀolding, but it a llows to categorify it, which after all, is our purp ose all along. 5.3. The Stone space o f a Bo o lean algebra and w eak integrals. This sub- section has t wo distinct pur p o ses. The ﬁr st is to elucidate the relation of a Bo olean algebra Ω with its Stone space S(Ω) and the impor tant consequences this ha s for measure theory . The seco nd is to fulﬁll our pr omise of subsection 4.1 o f basing Radon-Nikodym t yp e theorems on weak integrals. Strictly sp eaking, it will not be needed for our s ubsequent work but it is such a b e a utiful piece of mathematics that we have not resisted the temptation to include it here. There are t wo weak int eg rals, the Pettis in teg ral inv o lving the weak to po logy and the Gelfand integral inv olving the wk ∗ -top ology in dual spa ces. The Gelfand integral will b e the ob ject of in terest in this subsection. Bo th integrals ca n b e constructed in a purely ana- lytical manner with no measure theory in volved. F or the g eneral construction, we refer the re a der to [Rud91, chapter 3], a nd for its construction in the context of vector measure theory we r e fer the r eader to [DU77, chapter II, section 3]. Con- trary to the previous subsec tions, whe r e only the mos t basic functional analysis was needed, in this subsection sophistication is upp ed by a few notches and some knowledge of weak top olog ies in Banach spa ces is r equired. Our basic reference is [Con90, chapter V]. In o ur introduction to the wk ∗ -in tegral, we follow a somewhat devious r oute. Ther e is metho d to our madness thoug h, as it leads to the descrip- tion of the dua l catego ry of Ban c as the categor y of W albr¨ o e ck s paces. W e refer the re ader to [CLM7 9, chapter I, sectio n 2 ] for mo re infor ma tion. Recall that if Ω is a Bo olean alge bra, its Stone space S(Ω) is the set of (pr o p er) ultraﬁlters on Ω. These can b e identiﬁed with B o olean alg ebra morphisms Ω − → 2 where 2 is the Bo olean ﬁeld { 0 , 1 } . A topolo gy can be int r o duced in S(Ω) in the following way: to ea ch E ∈ Ω we asso ciate the s et o f ultraﬁlters containing E , that is: η ( E ) = def { x ∈ S(Ω) : x ( E ) = 1 } “useful or ganizational tool” or some such, usually said in a scornful tone b etray ing ignorance and misunderstanding. This attitude derives muc h from the i ndividual perceptions of what coun ts as deep in mathematics, but ev en on the face of it, describing category theory as “just a language,” as if it were derogat ory , is sur ely bizarr e, for isn’ t language one of the highest functions of the h uman brain? Language enhances our per ceptions and more than expands the reac h of our minds. Similarly , even i f category theory were just an “organizational tool , ” it would already serve a very useful purpose muc h in the same wa y as a hammer is a might y ﬁne to ol for organizing nails along walls and other simi lar hard surfaces like blo ckhea d skulls . But category theory is ﬁr st and foremost, a division of mathematics with its o wn sp eciﬁc, autonomous ﬁeld of act ivi ty and its o wn sp eciﬁc set of problems and concerns. A mathematician can s p end his whole career without once usi ng an y deep r esul t from say , group theory; and y et, w ere he to decry group theory as a “mere organizational tool for symmetry ,” such a comment would be met by man y of us with an ironical silence in the (probably misguided) notion that there are some p oint s of view about which the l ess said the b etter. In a given mathematical ﬁeld, categ ory theory may not contribute m uch more than a few trivial observ ations; in others it m a y be essent ial – witness algebraic geometry and algebraic top ology . The evocation of these examples is not without reason, since by the v ery essence of their disciplines, al gebraic geometers and top ologists, less aﬄicted by the narrow my opia of sp ecialization, hav e to cross the boundaries and smuggle the go o ds betw een al gebra and geomet r y , and category theory is just the right language to or ganize that sort of traﬃcking. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 47 Now, take the collection { η ( E ) : E ∈ Ω } a s a base for the top olog y of S(Ω). It can then be prov ed that ea ch η ( E ) is b oth closed and open (a clop en ) and that S(Ω) is a compac t Hausdorﬀ, tota lly disconnected spa c e. I f we denote by Bo ol the catego ry of Bo olea n algebra s and b y Bo ol T op the full subc ategory of compact Hausdorﬀ, tota lly disconnected spaces (or Bo ole an sp ac es ), then it is a fundamental result that there is an a djoint equiv alence (called Stone duality ), Bo ol (Ω , clo p en( Y )) ∼ = Bo olT op ∗ (S(Ω) , Y ) (5.21) with unit the Bo o lean alg ebra isomorphism η : Ω − → clopen(S(Ω)) g iven by E 7− → η ( E ). Stone duality implies that ev er y measur e-theoretical co ncept has a natural top o- logical mirr or image in the world of Bo o lean spaces. The remark able asp ect is that these top olog ical counterparts ar e usually b etter behaved than their Bo olean alge- bra siblings. A ﬁrst indication of this is the identiﬁcation of the elements o f L ∞ (Ω) with actua l c ontinuous functions . The cr ucial theor em in this direction is : Theorem 5. 10. The map Ω − → C (S(Ω)) given by E 7− → χ ( η ( E )) is a b ounde d sp e ctra l m ap. The unique lift to L ∞ (Ω) as in di agr am 5.7 is an isometric Banach algebr a isomorphism. L ∞ (Ω) / / _ _ _ C (S(Ω)) Ω χ O O 9 9 r r r r r r r r r r r Figure 5. 7 . The Banach algebra isomorphism L ∞ (Ω) ∼ = C (S(Ω)). Pr o of. The map E 7− → χ ( η ( E )) is clea rly a sp ectral meas ure Ω − → C (S (Ω)). By theorem 5.5, there is a unique Ba na ch alg ebra morphism clo sing triangle 5 .7 . A simple computation prov es that this morphism is an isometry . Since S(Ω) is totally disconnected, the characteristic functions of clop en sets are contin uous and sepa rate po ints. The Stone-W eier strass theorem ([Con90, chapter V, s ection 8]) now implies that the morphism is bijective.  In par ticular, the sp ectral map Ω − → C (S(Ω)) has the same universal prop erty of theore m 5.10. This theorem a llows us to embed measure theory in top olo g ical measure theory and entails a sleuth of imp ortant res ults, starting with the following corolla r y: Corollary 5.11. The functor Bo ol − → Ban given on obje ct s by Ω 7− → L ∞ (Ω) is c ont inu ous. Let Ω b e a Boo lean algebr a and µ : Ω − → K a b ounded additiv e map. Using the isomorphism unit ma p η : Ω − → clop en(S(Ω)) we can transfer µ to the Bo olean algebra clop en(S(Ω)) of clop ens of S(Ω) by: U 7− → µ ( η − 1 ( U )) (5.22) 48 G. RODRIGUES The ﬁnitely additive meas ur e (5.22) will also be denoted by µ . Now a mira - cle happens, as µ is σ -additive on clop en(S(Ω)): let ( U n ) be a countable disjoint sequence of clopens with U n = η ( E n ) and U = [ n U n with U clop e n. Since U is clos ed, it is compact, and fr om the countable cov er ( U n ) we can extr act a ﬁnite sub cov er { U i } with U = S i U i . But since the U n are pairwis e disjoint, it follo ws that only the U i are no n-empty and th us µ ( U ) = X i µ ( U i ) = X n µ ( U n ) Also note that s ince the Bo olean a lg ebra domain o f µ is clop en(S(Ω)), µ is automatically r e gular . A p eek at the Hahn-Ca ratheo dor y extension theorem (see [Hal74, chapter II I, section 13]) tells us that µ ex tends uniquely to a σ -a dditive, regular measur e on the σ -a lgebra ge nerated by clop en(S(Ω)). This σ -algebr a is the Bair e σ - algebr a Ba (S (Ω)) of S(Ω) and is eq ual to the σ - algebra gener ated by the contin uous functions 42 . If X is a c o mpact Hausdorﬀ space, denote b y RM ( X ) the space o f regular, σ - additive sca lar mea sures on the Baire σ - algebra o f X with the total v ar ia tion nor m. The previo us constructions yield a map BA (Ω) − → RM (S(Ω)) (5.23) and the follo wing theorem 43 . Theorem 5. 12. The map (5.23) is a natur al isometric isomorphism. Theorem 5 .12 tells us that if we gr ind measure s through the Stone spa ce machin- ery , they come out improv ed on the other side. The genera lization o f σ -additive measures to ﬁnitely additive o nes is not muc h of a g eneraliza tio n. W e ask the rea der to hav e these facts in mind as they will make understanda ble analo gous phenomena in the c ategoriﬁed setting to b e uncov ered in subsection 7.1. Next, we discuss an imp ortant adjunction that gives an inkling on how to rep- resent every b ounded linear map L 1 (Ω , µ ) − → B by a we ak inte gr al . The basic idea is that theo rem 5.1 0 encourag es to view “b ounded mea surable functions on Ω” as c ontinuous functions on the Stone space S (Ω). But giv en a Banach space B , there a re weaker topolo gies a t o ur disp osa l giving us w eaker notions of mea- surability . Thes e notions were alrea dy used in the P ettis theor em 4.7 where w ea k measurability w as found to b e as g o o d a s Bor el measura bility in characterizing the strongly measurable functions. In what follows, we will consider wk ∗ -measura bility of functions f w ith v alues in dual Banach spaces. F or domain of f we will tak e a 42 The Baire σ - algebra is the natural σ -algebra for discussing i nt egration in top ological spaces. In general, it is strictly con tained in the Borel σ -algebra B o ( X ) generated by the op en sets with equalit y happ ening iﬀ X is metrizable. Nevertheless, for X compact Hausdorﬀ but not necessarily metrizable, there is a well-kno wn tech nique to extend measures from B a ( X ) to B o ( X ) (see [F ol84, c hapter VI I]). W e wil l hav e no need of such an extension. F or an exhaustiv e treatment of the int eractions b et ween measures and topologies see [F re03]. 43 The theorem readily extends to v ector measures with v alues in Banac h spaces. Instead of the H ahn-Car atheodory extension theorem, we simply note that µ is unif ormly con tinuous on the Boolean algebra clop en(S(Ω)) with the measure (pseudo)metric and that clop en(S(Ω)) is dense in B a (S (Ω)) for this metric. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 49 general compact Hausdorﬀ space X , but the reader can mentally replac e it by S(Ω) and consider them as wk ∗ - measurable functions. Let X b e a compact Hausdor ﬀ space. Then a p oint x ∈ X deﬁnes a linear functional δ x : C ( X ) − → K by ev a lua tion: δ x : f 7− → f ( x ) The δ x are the Dirac measur es with singleton suppo rt. F or example, if X is the Stone space S(Ω) of a Bo olea n a lgebra Ω, then by the isomorphism (5.23 ), the δ x can be iden tiﬁed with the ultraﬁlters x of Ω. In fact, they are even sp e ctr al me asu re s . The next theore m contains tw o easily prov ed facts abo ut δ x . Theorem 5.13 . L et X b e a c omp act Hausdorﬀ sp ac e and ∆ : X − → C ( X ) ∗ the map x 7− → δ x . Then: (1) The set { δ x : x ∈ X } is line arly indep endent in C ( X ) ∗ . (2) The map ∆ is wk ∗ -c ontinu ou s . It is not diﬃcult to see that each δ x is an extreme point of ball( C ( X ) ∗ ). With a little more work, we can ev en character iz e these extreme p oints. Theorem 5.14. A p oint µ ∈ ba ll( RM ( X )) is an extr eme p oint iﬀ µ = ǫδ x with ǫ an element of the unit sphe r e of K . Pr o of. It is more or less clear that mea sures of the for m ǫδ x are extreme p oints. That all extr eme points are of this form, follo ws from a cle ver application of Urysohn’s lemma to prov e that every ex tr eme p o int must hav e singleton sup- po rt.  There is a universal pr op erty a s so ciated to the map ∆ : X − → C ( X ) ∗ with v ar ious nice consequences such as giving a representation of bounded linear maps B − → C ( X ) as certain wk ∗ -inte gr als . Let X b e a co mpact Hausdorﬀ space and B ∗ a dual Banach spa ce. Since the wk ∗ -top ology of B ∗ is the topolo gy generated by the ev aluation linear functionals b ∗ 7− → h b ∗ , b i for b ∈ B , it follows that f : X − → B ∗ is wk ∗ -con tinuous iﬀ x 7− → h f ( x ) , b i is con tinuous for ev er y b ∈ B . Let us denote b y C wk ∗ ( X, B ∗ ) the linear space of suc h functions with the norm k f k ∞ = def sup {kh f ( x ) , b ik ∞ : b ∈ ball( B ) } (5.24) Since a subset of B ∗ is bo unded iﬀ it is wk ∗ -b o unded, a n interc hang e of sup’s in (5.24) yields that k f k ∞ = sup {k f ( x ) k : x ∈ X } The following theorem no w follows easily . Theorem 5. 15. The n orme d sp ac e C wk ∗ ( X, B ∗ ) is c omplete. Next we show that the map ∆ : X − → C ( X ) ∗ is universal among wk ∗ -contin uous maps in the sense that if f : X − → B ∗ is a wk ∗ -co ntin uous ma p then there is a unique b ounded linear ma p b f : C ( X ) ∗ − → B ∗ such tha t the triangle 5.8 is commu- tative. Denote by V the linear span o f { δ x } in C ( X ) ∗ . The function f lifts to b f : V − → B ∗ by (1) of theorem 5.13 and putting for a linear combination P n k n δ x n of δ x , b f ( µ ) : b 7− → Z X h f ( x ) , b i d µ = * X n k n f ( x n ) , b + (5.25) 50 G. RODRIGUES RM ( X ) b f / / _ _ _ B ∗ X ∆ O O f : : v v v v v v v v v v Figure 5. 8 . The induced map R M ( X ) − → B ∗ . Combining the Banach-Alaoglu theorem ([Con9 0, chapter V, section 3]) with the Krein-Milman theor em ([Con90, c hapter V, section 7]), it follows that the linear span of { δ x } is wk ∗ -dense in RM ( X ), ther efore there is at most one wk ∗ -contin uous linear extension of f clos ing the triangle 5.8. But taking the hin t from (5.25), we deﬁne b f on the whole space RM ( X ) as b f ( µ ) : b 7− → Z X h f ( x ) , b i d µ (5.26) It is natural to denote the element b f ( µ ) of B ∗ by R X f d µ and view it as the wk ∗ -inte gr al of f . In inner pr o duct notation, deﬁnitio n (5.26) can b e formulated as  Z X f d µ, b  = Z X h f ( x ) , b i d µ, ∀ b ∈ B and the map b f is simply µ 7− → R X h f ( x ) , b i d µ . As noted ab ove, this ma p is wk ∗ -con tinuous. By the Banach-Alaoglu theor em, the imag e under this map o f the unit ball of B ∗ is wk ∗ -compact a nd there fore (5.26) is actually a bounded linear ma p o n RM ( X ). In fac t it can be prov ed by a combination of (5 .2 6) with the Hahn-Banach theo rem that w e ha ve, sup      Z X f d µ     : µ ∈ ba ll( RM ( X ))  = k f k ∞ This mea ns that the map f 7− → b f is a natural isometric embedding C wk ∗ ( X, B ∗ ) − → Ban ( RM ( X ) , B ∗ ) The image of this map is the space of wk ∗ -contin uous o pe rators and it is well known that these ar e precis ely the adjoints of b ounded linear maps B − → C ( X ). In this ca se, the preadjoint ca n b e computed with the aid of for mulas (5.7 ). Star t by noting that the function x 7− → h f ( x ) , b i is the function η ( b ) f with η the e mbedding B − → B ∗∗ in the bidua l. Now, we have  Z X f d µ, b  = Z X η ( b ) f d µ = h η ( b ) f , µ i Thu s, the preadjoin t of µ 7− → R X f d µ is b 7− → η ( b ) f . Combining the r esults up to now, we can state the follo wing theore m. Theorem 5. 16. The map f 7− → ( b 7− → η ( b ) f ) establishes a natur al isometric iso- morphism C wk ∗ ( X, B ∗ ) ∼ = Ban ( B , C ( X )) (5.27) CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 51 Theorem 5 .16 can b e turned int o an adjunction by taking unit balls a nd passing to the dual category . If we denote by CHaus the catego ry of compact Hausdo rﬀ spaces, then isomorphism (5.27 ) descends to an adjunction CHaus ( X , ball( B ∗ )) ∼ = Ban c ∗ ( C ( X ) , B ) (5.28 ) where ball( B ∗ ) is the unit ball o f B ∗ endow ed with the wk ∗ -top ology . Theorem 5.16 gives a complete description of the b ounded linear maps B − → C ( X ). W e refer the reader to [DS58, chapter VI, section 7 ] for these r esults. It can also serve as a ﬁrst stepping s tone to prove some Radon-Nik o dym theorems. 6. Banach 2 -sp aces Since ca tegoriﬁed meas ure theory needs a catego riﬁed analog ue o f a Banach space, we return to the ideas adumbrated in subse c tion 3.1. W e will b e very brief, since this will b e treated more fully in a fut ur e pap er [Ro da ]. Notation. F ro m no w o n we denote the pro jective tensor product o f Ba nach spaces by A ⊗ B instead of A ˆ ⊗ B . This will g ua rantee more s ymmetric formulas. W e ﬁr st expand table 3 .3 in order to include the basic Ba nach space categ oriﬁed analogues . This is done in table 6.1. Ordinary Ba nach spaces Categoriﬁed Ba nach spaces Base ﬁeld K Base r ing Ban Sum + Copro duct ⊕ Diﬀerence − Co kernels of maps Additiv e zero 0 Zero ob ject 0 Scalar multiplication × T ensor s ⊗ Cauch y completeness Existence o f ﬁltered colimits T able 6. 1. Catego riﬁed analogue s of Banach space theo r y The only a ddition to the categor iﬁed analo gues of linear algebra presented in [Bae97] is the last row. The need for inﬁnitary co limits was expr essed in principle s 2 and 5, and given the simila rities betw een the for mal prop erties of ﬁltered colimits and those of conv er gence o f nets in top olo gical spaces, the la st row of 6.1 is at leas t plausible. All the categoriﬁed analogues in table 6 .1 are c olimits and deﬁned by suitable universal prop er ties. In fact, all the conditions put together amount to re- quire co completeness of a Banach categor y , so we register the following deﬁnition 44 . Deﬁnition 6. 1 . A Banach 2 -sp ac e is a co complete Ba nach ca tegory 45 . Given deﬁnition 6.1 for the categor iﬁed analogue of a Banach space, it is clear what is the righ t notion of morphism betw een Ba na ch 2-spaces : a co co ntin uous functor T : A − → B . If S, T : A − → B are co contin uous functors , a morphism 44 There is a w eaker notion of completeness f or enric hed categories called Cauchy c ompleteness . The work of section 3 has sho wn that we need inﬁnitary colimi ts but Cauc hy completeness wi ll mak e its appearance later on. 45 The “Banach ” qualiﬁer is b eing used in a di ﬀeren t sense in the tw o nouns. In Banach c ate gory is used to mean relative or enriche d in Banach spaces, while in Banach 2 -sp ac e it is used in the sense of co completeness. It is an unfor tunate consequence of our terminology that the term c omplete Banach 2 -sp ac e is not a r edundancy . 52 G. RODRIGUES S − → T is simply a co nt r active natur al transforma tion. I n this w ay , we obtain the 2-catego ry 2– Ban of Banach 2-spaces . Some comments on thes e deﬁnitions are in or der. A ﬁrst question tha t ca n aris e in the reader’s mind is what is the categor iﬁed analo gue o f the metric. The ans wer, which w e happily embrace, is given on [La w0 2]. The metric, or the ob ject that “measures the distance betw een t wo ob jects a and b ”, is the Banach spa c e A ( a, b ) of mo r phisms a − → b . Perceptive rea ders will also notice the co nspicuous a bsence of any completenes s conditions in deﬁnition 6.1. Even tually , w e will need them, e.g. when we discuss sheav es in subsection 7.1 . Their presence als o insures that the dual categ ory A ∗ is also a Banach 2-spac e. In ma ny instances o ne can prove direc tly that a given Banach 2-spa ce is complete; this is the case with all the ex amples of this pa pe r. One should view deﬁnition 6.1 as a minimal set of pr op erties a Banach 2-space must have; further developmen ts ma y dictate stronger re q uirements. Re- quiring limits a lso ma kes some constr uctions with Bana ch 2-spaces more diﬃcult (for a hin t of this, see the rema rks at the end of subsection 6 .1). A rela ted question is if co contin uous functor is the right notio n of mo r phism b etw een Banach 2-spa ces. The adjoint functor theorem gua rantees that in most ca ses co contin uous functor s will hav e right adjoin ts 46 , so taking the hint fro m top os theory and its geometric morphisms, we are inclined to tak e adjunctions as the right notio n of morphism betw een Banach 2-s pa ces. In fac t a s hinted earlie r, there is an (uneasy) relations hip b etw een categoriﬁed measure theory a nd top os theo ry . This relation will be more a nd more agitated a s we pro ceed, until by the end o f the ne x t section it will have mounted to an a lmost obsessive pattern. There is of course, a lesso n to be learned here and one cannot help but w onder if there is not a notion o f “Banach top os” lurking b ehind a ll the analogies . I do not know the answer to this que s tion, but I refer the reader to [Str04], esp ecially its introductor y remar ks. As mentioned in the b eg inning we will not develop the theory of Banach 2- spaces. But we will need t wo simple, rela ted constructions on 2– Ban . If A and B are tw o Banach 2-spaces then the Bana ch catego ry 2– Ban ( A , B ) is co complete with colimits computed p oint wise. The problem is tha t without size restrictions on A , 2– Ban ( A , B ) do es not ha ve to be lo ca lly small and thus not a Banach categor y according to our deﬁnitions. F or muc h the same r easons, the bitensor pr o duct A ˆ ⊗ B o f Ba nach 2-spaces may no t ex ist. Recall that a bifunctor A ⊗ B − → C is bic o c ontinuous if it is co co ntin uous in eac h v aria ble. The bitensor pr o duct is then a bicoc o ntin uo us bifunctor A ⊗ B − → A ˆ ⊗ B univ er sal among all suc h bifunctors , that is, each bicoco ntin uo us A ⊗ B − → C factors through it via a co contin uous functor as in diagr am 6.1, and this factoriza tion is unique up to unique ca nonical isomorphism. It is mor e o r less clear that if the bitensor pr o duct exists, then we hav e an equiv ale nc e 2– Ban  A ˆ ⊗ B , C  ≃ 2– Ban ( A , 2– Ban ( B , C )) (6.1) 46 A particularly striki ng and easy to remember form of the adjoint functor that applies to man y , if not all of our examples of co cont i nuous functors, can be found i n [Kel05, chapt er 5, section 6, theorem 33]: every co con tinuous f unctor deﬁned on a co complete category wi th a s mall dense sub category has a right adjoint. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 53 A ˆ ⊗ B / / _ _ _ ∼ = C A ⊗ B O O ϕ < < z z z z z z z z z Figure 6. 1 . Universal pro pe r ty of A ⊗ B − → A ˆ ⊗ B . 6.1. Preshe af categorie s. In principle 6 it was stated that the in tegr al functor is a left Kan extensio n. Since left Ka n extensions are ess ential for ev erything that follows, we devote this subsection to review them. This will lead to the fundamental concepts of dense fun ctor , which ca n b e see n a s a ca teg oriﬁed a nalogue of top olo gical density , and (sm al l) pr oje ctive obje ct . W e will use [Kel05] as our main reference for enriched ca tegory theory . Chapter 6 of [Bor9 4b] is another pr oﬁtable so urce. Before the deﬁnition of left Ka n extensions we take a step ba ck and reca ll the construction o f fr e e Banach c ate gories . An important sp ecial case o f the isometric isomorphism (4.1 ) is obtained by letting B x = K for every x ∈ X . Denoting ℓ 1 ( X, K ) simply b y ℓ 1 ( X ) then (4.1) sp ec ia lizes to Ban ( ℓ 1 ( X ) , A ) ∼ = B ( X , A ) (6.2) The isometric isomorphism (6.2) is easily seen to be na tural in X ∈ Set , so that we ca n view ℓ 1 ( X ) as a sort of free Banach spa c e . In fact, X 7− → ℓ 1 ( X ) can be made into a left adjoin t. There is a functor ball : Ban c − → Set , the unit b al l functor , that to a B a nach space B as s o ciates its unit ball ball( B ). Isomo rphism (6.2) now reads as Ban c ( ℓ 1 ( X ) , A ) ∼ = Set ( X , ball( A )) (6.3) which is true b ecause a function X − → ball( A ) induces via the isomorphism (6.2) a unique linear contraction ℓ 1 ( X ) − → A . All told: Theorem 6. 2. The unit b al l fu n ctor ball : Ban c − → Set has a left adj oint 47 that on obj e cts is given by X 7− → ℓ 1 ( X ) . Coupling the pro jective tenso r pro duct with the ℓ 1 free functor yields, Ban c ( ℓ 1 ( X ) ⊗ ℓ 1 ( Y ) , A ) ∼ = Ban c ( ℓ 1 ( X ) , Ban c ( ℓ 1 ( Y ) , A )) ∼ = Ban c ( ℓ 1 ( X ) , Set ( Y , ball( A ))) ∼ = Set ( X , Set ( Y , ba ll( A ) )) ∼ = Set ( X × Y , ball( A ) ) ∼ = Ban c ( ℓ 1 ( X × Y ) , A ) and it follows from Y oneda’s lemma that it deter mines a natura l iso metric isomor- phism, ℓ 1 ( X × Y ) ∼ = ℓ 1 ( X ) ⊗ ℓ 1 ( Y ) (6.4) that can b e s e e n as F ubini’s the or em for absolutely summable families . It a ls o means that ℓ 1 is a s ymmetric monoidal functor w he n Set is given the cartesian closed sy mmetr ic monoidal structure ( X , Y ) 7− → X × Y . 47 The unit ball functor is a righ t adjoint , faithful and reﬂects isomorphisms, but it is not monadic. F or the proof, see [BW83, cha pter 4, section 3]. 54 G. RODRIGUES The fact tha t X 7− → ℓ 1 ( X ) is symmetric mo noidal implies tha t the adjunction (6.3) lifts to a 2-a djunction b etw een the 2- categor ies BanCat and Cat . T he right adjoint part is the forgetful 2-functor BanCat − → Cat that asso ciates to a Ba nach category A its unit b al l c ate gory ball( A ), whic h is the sub categ ory o f co nt r actions of A . In sym b ols: ball( A ) = def { f ∈ A : k f k ≤ 1 } (6.5) T o cons truct the left adjoint of the unit ball functor, let X b e a categor y and denote in matrix form X y x the set of morphisms x − → y . The Banach ca tegory ℓ 1 ( X ) has for ob jects the ob jects of X . The Banach space ℓ 1 ( X ) ( x, y ) of morphisms x − → y is the free Bana ch space ℓ 1 ( X y x ). The unit map is the map K − → ℓ 1 ( X x x ) given by 1 x ∈ X x x and the comp o sition map ℓ 1 ( X y x ) ⊗ ℓ 1 ( X z y ) − → ℓ 1 ( X z x ) is the composite, ℓ 1 ( X y x ) ⊗ ℓ 1 ( X z y ) ∼ = / / ℓ 1 ( X y x × X z y ) ℓ 1 ( ◦ x,y,z ) / / ℓ 1 ( X z x ) where ◦ x,y ,z is the compos ition map X y x × X z y − → X z x . Clearly , bo th the unit a nd comp osition ma ps are co nt r actions and r outine dia grammatic co mputations which we leave to the reader would prov e the asso ciative and unital laws. Putting it all together, we hav e: Theorem 6. 3. The 2 - functor X 7− → ℓ 1 ( X ) establishes a natu r al iso morphism BanCat ( ℓ 1 ( X ) , A ) ∼ = Cat ( X , ball( A )) (6.6) Notation. In what follows we will make no notational distinction b etw een a categor y X and the free Banac h categ ory ℓ 1 ( X ). In other words, in the enric hed context an ordinary categ o ry X is automatically pr omoted to the free Banach category ℓ 1 ( X ). Likewise, a contractive functor X − → A in to a Banach category will be iden tiﬁed with the induced contractive functor ℓ 1 ( X ) − → A . A functor F deﬁned on a sub categor y M ⊆ A can hav e from none to many extensions to the whole categ o ry . Among all the p o ssible extensions of F to the whole ca tegory A w e can sing le out a maximal and a minimal o ne and these can be characterized by simple universal prop er ties. Deﬁnition 6. 4. Let I : M − → A and F : M − → B b e tw o functors, with B a Banach 2-s pace. The left Kan exten sion of F along I is a pair ( L , η ) w ith L : A − → B and η : F − → LI universal as an ar row fro m I ∗ : BanCat ( A , B ) − → BanCat ( M , B ) to F – se e diagram 6.2. Deﬁnition 6 .4 of a left Kan extensio n is the one tha t can be found in [Lan71, chapter X, section 3] for ordinar y categorie s , but for rea s ons that need not co ncern us here it is not ent ir e ly satisfactory in the general enriched context. The co co m- pleteness hypothes is on the co do main categ o ry B ta ck ed on the deﬁnition forces (small) c o limits to b e p ointwise c olimits and po s sibilitates the simpler deﬁnition 6.4. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 55 A T ( ( L 6 6     K S σ B ⇑ η M I O O F D D Figure 6.2. ( L, η ) as a left K a n extension. W e will denote the pair ( L, η ) simply by Lan I F , leaving the universal cone η implicit. By universality , the assignment σ 7− → σ I η establishes an iso metric isomorphism BanCat  Lan I F, T  ∼ = BanCat ( F, T I ) (6.7) In par ticular, if Lan I F exists for ev er y F then the functor F 7− → Lan I F is left adjoint to T 7− → T I and therefore, co contin uous. The calculus of co ends yields a v ery b eautiful formula for Lan I F and s imple suﬃcient co nditio ns for its existence . Co ends are co limits over bifunctors that exhibit behavior a na logous to int eg rals – we r efer the reader to [L an71, chapter IX] bes ides the already cited references for enriched catego ry theo ry where (co)ends play no less than a n essential role. Theorem 6.5. L et I : M − → A and F : M − → B b e two functors, with B a Banach 2 -sp ac e. Then the left Kan extension Lan I F exists and is given by, Lan I F ( a ) ∼ = Z m ∈M A ( I m, a ) ⊗ F m ( 6 .8 ) on the hyp othesis t hat for every a ∈ A the c o end on t he r ight hand side exists. F u rthermor e, the universal map η : F − → Lan I F I is given by the c omp osite in diagr am 6.3. F n / / η n ( ( P P P P P P P P P P P P P A ( I n, I n ) ⊗ F n w n,I n   R m ∈M A ( I m, I n ) ⊗ F m Figure 6. 3 . The univ ers al map η . The vertic al arr ow w n,I n in 6.3 is the c olimiting we dge for t he c o end and the top arr ow is the c omp osite F n ∼ = / / K ⊗ F n i 1 F n ⊗ 1 F n / / A ( I n, I n ) ⊗ F n Pr o of. The pro of in [Lan7 1, c hapter X, section 4] can b e adapted in a straightfor- ward fashion to the B an c -enriched case.  56 G. RODRIGUES As a n immediate corolla r y , if M is small the co ends in (6.8 ) exist and therefor e so do es every left Ka n ex tension Lan I F . Another co rollary is that if I is fully-faithful then (Lan I F ) I n ∼ = Z m ∈M A ( I m, I n ) ⊗ F m ∼ = Z m ∈M M ( m, n ) ⊗ F m but the seco nd co end is just F n b y Y oneda. Thus, η induces an isometric isomor- phism (Lan I F ) I ∼ = F so that Lan I F is indeed an extension o f F . Two ques tio ns present themselves naturally . First, when is the extension e ssen- tially unique and seco nd, when is the extension L an I F coc o ntin uo us as a functor on A . The seco nd question ca n b e ans wered a lmost immediately . Loo king a t the co e nd formula (6.8) fo r a left Ka n extension, w e see that if the functor a 7− → A ( I m, a ) is co contin uous, then the interc hange of colimits theorem ([Kel05, chapter 3, section 3]) gua r antees that Lan I F is coc o ntin uo us. Since the co domain B is co complete, colimits of functors are computed p o int wise and th us the sa me interchange of col- imits theorem yields that this condition is not o nly suﬃcient but actually necessar y . This leads us to the next imp ortant concept. Deﬁnition 6.6. Let M be a small Ba nach category , A a Banach 2-spa c e a nd I : M − → A a functor . An o b ject m ∈ M is smal l I -pr oje ctive if the functor a 7− → A ( I m, a ) is co contin uous. If I is the inclus ion functor we just say that m is smal l pr oje ctive . The notion of pro jectiv e ob ject comes fro m homo logical alg e br a (see [W ei94, chapter 2, section 2]). The deﬁnition makes sense in any category: an ob ject a ∈ A is pr oje ctive if every arrow f : a − → b factors through every epi e : c − → b as in diagram 6.4 . c e   a f / / ? ?     b Figure 6. 4 . Lifting prop e rty of pro jective ob jects. This is equiv alent to require that if e is epi then e ∗ is epi. In the context o f ab elian categorie s (= homolo gical algebra ) this is eq uiv alent to the fact that the functor b 7− → A ( a, b ) is ﬁn it ely c o c ontinuous . Deﬁnition 6.6 is a c o nsiderable str engthening in that w e require full co co nt inuit y o f the represent a ble, but since we will consider no other notion of pr o jective ob ject, w e drop the “ small” qualiﬁer. Example 6.7. Let A be a small B a nach ca teg ory . By Y o neda’s lemma, w e hav e the iso metric isomorphism, BanCat (Y a , F ) ∼ = F ( a ) which implies that the functor F 7− → BanCat (Y a , F ) is isomorphic to the ev alu- ation functor ev a : F 7− → F ( a ). Since Ban is co complete and co limits in functor categorie s ar e computed p oint wise, the ev aluation functor is co contin uous and thus every repr esentable is pr o jective in BanCat ( A ∗ , Ban ). CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 57 In the context o f Ba nach spac es, the extension of a b ounded linear map M − → B with M ⊆ A is unique when M is a dense linear subspace of A . In the con text of enriched ca tegories, this leads us to the concept of dense functor . Deﬁnition 6.8 . A functor I : M − → A is dense if the identit y 1 I : I − → 1 A I exhibits 1 A as the left Kan extensio n of I alo ng I . If I is the inclusion of a sub c ategory M w e just say that M is dense in A . Recall that if T : A − → B is a functor, then we hav e na tural transfor mations T a,b : A ( a, b ) − → B ( T a, T b ) given by f 7− → T f . The concr eteness of the bas e category Ban c makes the pro of of natur ality a triviality , fo r the gener al enriched case follow the instructions in [K e l05, chapter 1, section 8]. The existence of tensors in B turns these natural transformations into wedges, T a,b : A ( a, b ) ⊗ T a − → T b and the str ong fo rm of Y oneda’s lemma ([Kel05, chapter 2, sectio n 4 ]) implies that these wedges are universal, that is, they induce an isomo rphism T b ∼ = Z a ∈A A ( a, b ) ⊗ T a (6.9) The case of T the iden tity functor w as used abov e in deriving the iso mo rphism (Lan I F ) I ∼ = F . In this case a lso, the univ ers al wedge g ives us a canonical map, ev a,b : A ( a, b ) ⊗ a − → b which ca n b e see n a s a sort of int er nal version of the ev a lua tion map (and thus the notation). The co end formula for a left Kan extension now mea ns that I is dense iﬀ the canonical map ev I m,b is a univ ers al wedge with, a ∼ = Z m ∈M A ( I m, a ) ⊗ I m (6.10) that is, every ob ject b is a colimit over I : M − → A in a c anonic al way . F ur ther- more, if T : A − → B is co c ontin uous then by (6.10), we have, T a ∼ = T Z m ∈M A ( I m, a ) ⊗ I m ! ∼ = Z m ∈M A ( I m, a ) ⊗ T I m (6 .1 1) which implies that if T I ∼ = S I then T ∼ = S , that is, co contin uous extensio ns ov er dense functors, if they ex ist, a re unique (as a lwa ys, up to unique iso mo rphism). W e now apply these results to the sp e c ial case of presheaf catego r ies. If A is a small Banach ca teg ory , the c a tegory of contractive functors A ∗ − → Ban will b e denoted by PSh v ( A ) and its ob jects are called Banach sp ac e-value d pr eshe aves , or simply presheaves if there is no ambiguit y ab out the codo main categor y b eing Ban . F ollowing the bundle ter minology of section 2, a general pre s heaf will b e denoted by ξ , ζ , etc. and morphisms by τ , φ , etc. By the usual p oint wise c o mputation of limits and colimits, PShv ( A ) is a c o mplete Banach 2 -space. The Y oneda functor Y : A − → PShv ( A ) as s o ciates to each a ∈ A , the representable A ∗ − → Ban given on ob jects b y b 7− → Y ( a ) b = A ( b, a ) 58 G. RODRIGUES Computing the left Ka n extension of Y a long Y we have, Lan Y Y ( ξ ) ∼ = Z a ∈A A (Y ( a ) , ξ ) ⊗ Y ( a ) ∼ = Z a ∈A ξ ( a ) ⊗ Y ( a ) ∼ = Z a ∈A Y( a ) ⊗ ξ ( a ) ∼ = ξ where we hav e used Y oneda’s lemma and isomo rphism (6.9 ). By [Kel05, chapter 5, theorem 5 .1] a functor F : M − → A is dense iﬀ there is some natural isomor phism 1 A ∼ = Lan F F . Therefor e: Theorem 6. 9. The Y one da emb e dding Y : A − → PShv ( A ) is dense. If F : A − → B is a con tra ctive functor with B a Banach 2-space, we can lift F to PShv ( A ) by taking the left Ka n extension La n Y F . Theor em 6 .5 coupled with Y oneda’s lemma yields: Lan Y F ( ξ ) ∼ = Z a ∈A PSh v (Y ( a ) , ξ ) ⊗ F a ∼ = Z a ∈A ξ ( a ) ⊗ F ( a ) (6.12) This le ads to the nex t very imp or tant theorem that characterizes the Y oneda embedding Y : A − → PShv ( A ) a s the fr e e c o c ompletion of A . Theorem 6.10. L et A b e a smal l Banac h c ate gory and F : A − → B a funct or into a Banach 2 -sp ac e. Then ther e is a c o c ontinuous functor b F : PShv ( A ) − → B such t hat b F Y ∼ = F . F urt hermor e, if T : PSh v ( A ) − → B is another s uch c o c ontin- uous functor, ther e is a unique isometric isomorphi sm closing the 2 -diagr am 6.5. PSh v ( A ) T ) ) b F 5 5     K S ∼ = B ⇑ ∼ = A Y O O F F F Figure 6.5. Universal prop erty of the Y oneda embedding. Theorem 6 .10 implies that the functor F 7− → b F establishes an eq uiv alence 2– Ban ( PSh v ( A ) , B ) ≃ BanCat ( A , B ) (6.13) CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 59 Note tha t in (6.13) we hav e a n eq uiv alence, not an isomorphis m. In other words, we hav e a biadjunction , not a 2-adjunction. This follows from the fact that the factorization o f 1-cells is only deﬁned up to (unique) isometric isomo rphism, the universal 2-cell η ﬁlling the triangle in diagr am 6.2 . As a conseq ue nce , it is not true that the free Banach 2-space c o nstruction PShv extends to a 2-functor such that the equiv alence (6.13) is natura l. Instead, since the 1 - cells are only deﬁned up to (unique) isometric 2-cells, PShv extends to a pseudo 2 -fu nctor , roug hly , a 2-functor respecting compo sition of 1- c e lls up to co he r ent isomorphism. F or the exact deﬁnition of pseudo 2-functor, biadjunction and related notio ns and results we refer the reader to [KS74] or the more recent [Fio06], [La c 07] a nd references therein. Although w e will not expatiate on these inherently bicategorial notions, we do note tha t constructions with Banach 2-spaces tend to yield biadjunctions and not 2-adjunctions. The smallness of A is necessar y for the ar guments of this section. If A is not small then its co completio n ca n still be constructed a nd is the category of ac c essible pr eshe aves (see [Kel05, chapter 4, section 8]). The completeness o f the co co mpletion is then a more delicate matter. W e refer the r eader to [DL06] for more information. 6.2. F ree Banac h 2 -spaces o v er s ets. A very sp ecial, and esp ecially imp ortant, class of Banach 2-spaces is the class of free Banach 2-spac e s ov er a set X . Their study in this and the ne x t subse c tio n has t wo aims: ﬁrst to sho w ho w some of the results in [Y et] on catego riﬁed linear alg ebra can b e recovered and second, as a toy mo del of categoriﬁed measure theor y . By virtue of X b eing discrete, tw o things happ en. First, X is isomor phic to its dua l X ∗ so that there is no diﬀer ence be tw een presheav es and their dua ls, precosheaves. Second, ev ery functor F : X − → Ban is automatically contractive. This leads us to identify the category PSh v ( X ) with the categor y of bund les of Banach sp ac es over X . The deﬁnitions simply rep eat those of section 2 by re pla cing Hilber t with Banach spaces. If w e deno te by Bun ( X ) the Banach catego r y of Banach bundles a nd b ounded natural transfor mations, the next theorem is almo st self-evident. Theorem 6.11. The functor that to a bund le ξ assigns the functor x 7− → ξ x is an isomorphi sm Bun ( X ) ∼ = PSh v ( X ) (6.14) The ho m-spaces of Bun ( X ) are e asily desc r ib ed. If A is small, the end formula ([Kel05, c hapter 2, sectio n 2]) for the space of natural transfo rmations F − → G says that w e have an isometric is omorphism B A ( F, G ) ∼ = Z a ∈A B ( F a, Ga ) (6.15) If ξ and ζ are bundles, then the end for m ula gives (dro pping the X fr om the hom-spaces): Bun ( ξ , ζ ) ∼ = Z x ∈ X Ban ( ξ x , ζ x ) ∼ = Y x ∈ X Ban ( ξ x , ζ x ) (6.16) Since X is discr e te, X ∼ = X ∗ and, in terms of bundles, a repr e s entable Y x : X − → Ban is just a Dira c δ - like bundle supp orted on a p o int (or in more measur e theoretic 60 G. RODRIGUES terms, the char acteristic bund le of a singleton set ): δ x ( y ) = def ( K if y = x , 0 otherwis e. Inside Bun ( X ), Y oneda’s lemma implies Schur’s lemma , tha t is: Bun ( X )( δ x , δ y ) ∼ = ( K if x = y , 0 otherwis e. (6.17) More generally , the ev aluation or ﬁber functors ev x : ξ 7− → ξ x are representable with Bun ( δ x , ξ ) ∼ = ξ x . Since the Y oneda embedding X − → Bun ( X ) is dense, the co end formula g ives the canonical decomp osition o f a bundle ξ as a sum o f δ -likes: ξ ∼ = Z x ∈ X ξ x ⊗ δ x ∼ = X x ∈ X ξ x ⊗ δ x (6.18) F ormula (6.18 ) has imp orta nt co nsequences for the categ ories Bun ( X ). First, w e can introduce a sy mmetric monoida l s tructure o n Bun ( X ) b y ta king the p oint wise monoidal pro duct: ξ ⊗ ζ ∼ = X x ∈ X ( ξ x ⊗ ζ x ) δ x But denoting in exp onential notation  ζ the bundle x 7− → Ban ( ζ x ,  x ), Bun ( ξ ⊗ ζ ,  ) ∼ = Y x ∈ X Ban ( ξ x ⊗ ζ x ,  x ) ∼ = Y x ∈ X Ban ( ξ x , Ban ( ζ x ,  x )) ∼ = Bun  ξ ,  ζ  and Bun ( X ) is closed. Theorem 6.12. The Banach c ate gory Bun ( X ) with the p ointwise tensor pr o duct ξ ⊗ ζ : x 7− → ξ x ⊗ ζ x is a symmetric monoidal close d c ate gory. Now, le t T : Bun ( X ) − → Bun ( Y ) b e a co co ntin uous functor. By the decomp o- sition formula (6.18) applied to b o th ca tegories we have T ξ ∼ = T X x ∈ X ξ x ⊗ δ x ! ∼ = X x ∈ X ξ x ⊗ T ( δ x ) ∼ = X x ∈ X ξ x ⊗   X y ∈ Y T x y ⊗ δ y   ∼ = X y ∈ Y X x ∈ X ξ x ⊗ T x y ! ⊗ δ y This computation shows that we ca n identify a co contin uous T : Bun ( X ) − → Bun ( Y ) with a matrix of Banach sp ac es T y x parameterize d b y ( x, y ) ∈ X × Y . In other words, we have the follo wing theor em. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 61 Theorem 6. 13. The fun ctor T 7− → T y x establishes an e quivalenc e 2– Ban ( Bun ( X ) , Bun ( Y ) ) ≃ Bun ( X × Y ) (6.19) Using the equiv alence (6.1 9), w e can identify the co mpo site of c o contin uous functors with matrix mu lt iplic ation . If S : Bun ( Y ) − → Bun ( Z ) is a co co ntin uous functor with matr ix r epresentation S z y , then S T is a co co ntin uo us functor with matrix r e pr esentation given by: ( S T ) z x ∼ = X y ∈ Y S z y ⊗ T y x (6.20) Note that while co mp o s ition o f co contin uous functors is a sso ciative o n the nose and yields a 2 - categor y , matrix co mpo sition given by (6.20 ) o nly yields a bic ate gory . The ident iﬁca tion of co contin uous functors Bun ( X ) − → Bun ( Y ) with bifunctors on X × Y is but a sp ecial ca se of a muc h more gener al phenomenon and leads to the impo rtant concept of distributor in J. B´ enab ou’s terminolo g y . Alternative names are pr ofunctor , bimo dule a nd even mo dule in case the rea der liv es upside down in the southern hemisphere and sp eak s Austr alian 48 . W e will get to these matters later in a more gener al context, so we pro ceed with other co ncerns. Another imp orta nt consequence of these re s ults is that the bitensor pro duct PSh v ( X ) ˆ ⊗ PSh v ( Y ) of free Bana ch 2-mo dules exists and is PSh v ( X × Y ). This comes from the bia djunction (6.13) and using the line of reas oning used to establish isomorphism (6.4). Theorem 6.14 (F ubini) . The bitensor pr o duct PSh v ( X ) ˆ ⊗ PSh v ( Y ) exists and ther e is an e qu ivalenc e PSh v ( X ) ˆ ⊗ PSh v ( Y ) ≃ PSh v ( X × Y ) (6.21) Coupling theor ems 6.13 and 6.14, w e hav e that the 2-ca tegory of free Banach 2-mo dules over sets, co contin uous functors and contractive natural transformations with bitensor pro duct (6.21) is biclose d 49 . 48 In the deﬁnitiv e catalogue of the v agari es of human madness that i s The Anatomy of Melan- choly , Rob ert Burton in one of his endless li sts of inv ectiv es notes of a l ov er’s blindness: The ma jor part of l ov ers are carried headlong like so m an y brute b easts; reason counsels one wa y , thy fri ends, fortunes, shame, disgr ace, danger, and an o cean of cares that will ce r tainly follow; yet this furious lust precipitat es, coun ter- pois eth, weighs do wn on the other; though it be their utter undoing, p erp etual infamy , loss, y et they will do it, and b ecome at last insensati , v oid of sense; degenerate in to dogs, hogs, asses, brutes; as Jupiter in to a bull, Apuleius an ass, Lycaon a wolf, T ereus a lapwing, Calli sto a b ear, El penor and Gryll us into swine b y Cir ce. This met hinks, is the surest scien tiﬁck explanation for why Australia features such a strong sc ho ol in category theory: by walking upside down, the blo o d tha t in an av erage health y male concen trates in the ﬁery r egion of the loins, is diverted to the l o wer parts of the brain, oxyge nating it, op ening the pores and allowing a b etter v entilation of the lustful humors, unobstipated vis ion, clear thinking, etc. 49 There is a p oten tial terminology clash here, as the term bic lose d as been used in the literature in a diﬀerent sense – basically , as the generalization of closedness to monoidal, possibl y non- symmetric categories. Since we will mak e no use of the l atter, there should b e no confusion. 62 G. RODRIGUES 6.3. Catego ri ﬁed measure theo ry: the discrete case. In this subsection, we tackle categor iﬁed measure theory in the simplest case of discrete Bo olean algebr as and show that all the basic r esults o f sectio n 5 for ordinar y integrals hav e stra ight- forward categor iﬁed analogues prov ed b y suitably ca tegorifying the pro ofs – the essential work was a lready done in subsections 6.1 and 6 .2 . But ﬁrst, let us r e view the bas ic deﬁnitions. Let Ω b e a Boo lean algebr a and A a Bana ch 2-s pace. A pr e c oshe af Ω − → A is a simply a contractiv e functor µ : Ω − → A . Let E b e a partition of E = sup E . Then there is a unique map closing the tria ngle 6.6 where µ E denotes the copro duct P F ∈E µ ( F ). µ E ε E / / _ _ _ _ µ ( E ) µ ( E n ) i µ ( E n ) O O µ E n ,E ; ; w w w w w w w w w Figure 6. 6 . The map ε E : µ E − → µ ( E ). A cosheaf is a precosheaf for which ε E is an isomorphism for ev ery partition E , or equiv alently , for whic h the cone ( µ E n ,E ) is a copro duct. More precisely : Deﬁnition 6.15. A precosheaf µ is ﬁnitely additive if ε E is an isometric iso mor- phism for every ﬁnite partition. It is σ -additive if ε E is an iso metr ic isomo rphism for every countable partition and c ompletely additive if ε E is an iso metric isomor phis m for every partition. In this subsection only , the unadorned term c oshe af will mean σ -additive pr e- c oshe af . The fact that the Bo o le an algebra 2 X is atomic will allow us to bypass the more delicate ana ly tical issues tha t arise when w e conside r σ -a dditive cosheav es. As r emarked in the introductio n, w e view coshea ves as c ate goriﬁe d me asur es . The category of A -v alued co sheav es will be deno ted by CoShv (Ω , A ). It can b e proved that µ is a co sheaf iﬀ it satisﬁes the dual p atching c ondition for coun table co vers. The co sheav es of mor e impo rtance a re the completely additive ones while the ones coming from co ncrete situations (e.g. the cosheav es E 7− → L 1 ( E , µ ) for a measure algebra (Ω , µ ) ) are σ -additive. The gap b etw e e n σ -additivity and complete additivity 50 is bridged by the same technical condition used in the or dinary measure- theoretic situatio n: the c ountable chain c ondition . Deﬁnition 6.16. A Bo olea n alg ebra Ω satisﬁes the c ountable chain c ondition (or has ccc for short) if ev ery partition E is countable 51 . The ccc condition can be seen as the pur ely Bo olean algebraic version of total σ - ﬁniteness. A Bo olean algebra that is σ -complete and has ccc is a utomatically order 50 Completely additiv e functionals are treated in [F r e02, c hapter 2, sec tion 6]. They are the functionals of im p ortance for the abstract Radon-Nik o dym theorem. 51 Pa r titions are anti-c hains, so it would probably b e more reasonable to call the condition i n 6.16, the c ountable antichain c ondition or cac. But the terminology is en trenc hed and b esides, it is not diﬃcult to prov e that if Ω is σ -complete then ccc is equiv alent to the fact that there are no uncoun table chains i n Ω. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 63 complete. If Ω is σ -complete and the measure µ is σ -additive a nd non-dege nerate then Ω has ccc, so that the ccc c o ndition is a uto matically satisﬁed in the most impo rtant cla s s of mea sure a lgebras. Now, let us consider Boo lean algebras o f the for m 2 X for a s e t X . Since we want the ccc co nditio n o n 2 X , X m ust be a coun table set. In this ca se, the Bo olean algebra 2 X is complete, ccc and purely atomic. W e ha ve an obvious isomor phis m, atoms ( 2 X ) ∼ = X ident ifying the ato ms o f 2 X with the elemen ts of X . This implies that for every E ⊆ X we hav e the equalit y E = sup  F ∈ atoms ( 2 X ) : F ⊆ E  The next theorem should no w b e clear. Theorem 6.17 (Radon-Nikodym) . The functor µ 7− → ( x 7− → µ ( x )) establishes an isomorphi sm CoSh v  2 X , A  ∼ = BanCat ( X , A ) (6.22) Using the biadjunction 6 .13, we have that, CoSh v  2 X , A  ≃ 2– Ban ( PSh v ( X ) , A ) (6.23) and taking in to ac count theore m 6.11, it follows that the universal cosheaf χ : 2 X − → PShv ( X ) is given b y x 7− → δ x and th us by equiv alence (6.23), χ is just the Y oneda embedding. If µ : 2 X − → A is a co sheaf, then it factor s via χ a s R d µ : PSh v ( X ) − → A by taking the left Kan extension Lan χ µ . These constructions identify the category of measur able 52 functors L ( 2 X ) a s the Banach 2-space PShv ( X ) ∼ = Bun ( X ). As exp ected, this Bana ch 2-s pa ce do es not depe nd on the cosheaf µ , only on the set X . The co end for mula (6.8 ) for left Ka n extensions implies that, Z X ξ d µ ∼ = X x ∈ X ξ x ⊗ µ ( x ) (6.2 4) which is what was to b e exp ected from the fact that in a tomic Bo olea n a lgebras , int eg rals are nothing but sums. In particular, the functor ξ 7− → R X ξ d µ is co c o n- tin uous . Now that we have constructed the integral functor ξ 7− → R X ξ d µ w e can talk ab out the indeﬁnite inte gr al . This is just the co sheaf 2 X − → A : E 7− → Z E ξ d µ = def X x ∈ E ξ x ⊗ µ ( x ) W e have co ns tructed the int eg ral of scalar (that is, Ban -v alued) bundles ag ainst vector measures. F ollowing the path taken with measure algebra s and the cons truc- tion o f v ecto r integrals, the v ecto r in tegra l should b e constructed via a bitensor pro duct. Deﬁne the c a tegory ℓ ( X , A ) to b e BanCat ( X , A ). If A is the category Ban of Banac h spaces this is just another notation for PSh v ( X ) ∼ = Bun ( X ). 52 Since ev ery measurable is inte grable in the categoriﬁed setting, we will use the t wo adjectiv es indistinguishably . 64 G. RODRIGUES Theorem 6. 18. The bifunctor ( ξ , a ) 7− → ξ ⊗ a : x 7− → ξ x ⊗ a (6.25) establishes a natur al e quivalenc e, ℓ ( X , A ) ≃ ℓ ( X ) ˆ ⊗ A (6.26) Pr o of. W e merely sk etch the pro o f. Star t by noticing that since tensors are co- contin uous in each v a riable, (6.25) is bico contin uo us. Now, s imilarly to the case of ℓ ( X ), we intro duce a subca tegory of delta-like bundles that is dense in ℓ ( X, A ). Let X ∈ X and a ∈ A . Deﬁne: δ x ⊗ a ( y ) = def ( a if y = x , 0 otherwise. (6.27) The ca nonical decomp ositio n in ℓ ( X , A ) tak es the form ξ ∼ = X x ∈ X δ x ⊗ ξ x (6.28) Since X is discrete, the end-formula for hom-spaces of functor catego ries y ields that ℓ ( ξ , ζ ) ∼ = Q x ∈ X A ( ξ x , ζ x ). These t wo facts put together , imply that the full sub c ategory of (6 .27) is dense in ℓ ( X , A ). No w, let ϕ : ℓ ( X ) ⊗ A − → B be a bico- contin uous bifunctor. Deﬁne the functor b ϕ : ℓ ( X , A ) − → B o n elementary tensors δ x ⊗ a by ϕ ( δ x , a ) and ex tend by density to the whole ℓ ( X , A ): b ϕ ( ξ ) ∼ = X x ∈ X ϕ ( δ x , ξ x ) (6 .2 9) Straightforw ar d colimit manipulations show that b ϕ is co co ntin uo us. It is also clear tha t b ϕi ∼ = ϕ and that any such functor w ould ha ve to be isomor phic (up to a unique isomor phism) to b ϕ by (6.29 ).  If µ : 2 X − → Ban is a cosheaf we deﬁne the in tegr al functor ℓ ( X , A ) − → A to be Z X ξ d µ = def X x ∈ X µ ( x ) ⊗ ξ x (6.30) W e hav e no w the 2-diagr am of ﬁgure 6.7 where T : A − → B is a co contin uous functor. ℓ ( X ) ˆ ⊗ A 1 ℓ ( X ) ˆ ⊗ T / / ≃   R X d µ ˆ ⊗ 1 A $ $ H H H H H H H H H ℓ ( X ) ˆ ⊗ B ≃   R X d µ ˆ ⊗ 1 B { { v v v v v v v v v A T / / B ℓ ( X , A ) T ∗ / / R X d µ : : v v v v v v v v v ℓ ( X , B ) R X d µ d d H H H H H H H H H Figure 6. 7 . Natura lit y of R X d µ . Commutativit y of 6.7 means that b etw een every pair of parallel paths there is a canonical 2 - cell isomorphism betw een them and that 6.7 ﬁlled with these 2-cells CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 65 is commutativ e. The pr o of of all this is a relatively straightforward but so mewhat tedious c hor e, and since we will (ev entually) g eneralize ev er ything to other Bo olean algebras Ω we concen trate our attention in a small piece o f 6.7 and leave the re st to b e handled by the interested reader. F or the most par t, they ar e the direct cate- goriﬁcatio n o f the computatio ns with simple functions needed to establish theor em 5.7. Let ξ ∈ ℓ ( X , A ) b e an A -v a lued bundle. Then b y (6.28), ξ ∼ = P x ∈ X δ x ⊗ ξ x and we hav e: T  Z X ξ d µ  ∼ = T Z X X x ∈ X δ x ⊗ ξ x ! d µ ! ∼ = T X x ∈ X µ ( x ) ⊗ ξ x ! ∼ = X x ∈ X µ ( x ) ⊗ T ξ x ∼ = Z X X x ∈ X δ x ⊗ T ξ x ! d µ ∼ = Z X T ∗ ( ξ ) d µ This chain o f isomor phis ms pr ovides the unique isometric isomo r phism ﬁlling the low er trap ezoid in 6.7 and shows that co contin uous functors commute with int eg rals. In other w o rds, the contortions of [Y et0 5 ] to deﬁne the r ight notion of functor b etw een categories of integrable functors are not needed. F ubini’s theorem now follows from the fact that ℓ is a left biadjoint to the unit ball 2-functor BanCat − → Cat . The argument is just the categor iﬁcation of the argument used ab ov e to es tablish the is ometric is omorphism ℓ 1 ( X ) ⊗ ℓ 1 ( Y ) ∼ = ℓ 1 ( X × Y ). The isomorphism betw een iterated in tegrals follo ws likewise. 7. Ca tegorified measures and integrals With the basic notions of Banach 2-spaces in hand and categor iﬁed measure theory for the discrete case clariﬁed, we justify in this ﬁnal section some of the principles adv anced in sectio n 3 for ca teg oriﬁed measure theory . F ull details will be given in a future paper ([Ro db]). W e will b e concerned not so m uch with the direct ca teg oriﬁcatio n of the r e sults of section 5 , but in the new phenomena that arise in the categoriﬁed setting. The former was alr eady covered in subsection 6.3 for discrete Bo olean algebra s 2 X and the details for the more general case do not diﬀer by muc h o nce the basic constructions ar e understo o d. W e will conv ey only their ﬂav or and p er force many things will be left unsaid. 7.1. Banac h shea ves. By principles 2 and 3, the ob jects that we put under the int eg ral sign are sheaves for a suitable Grothendieck topolo gy on Ω. The general deﬁnition of Gr othendieck top o logy can b e seen in [LM92, chapter I I I] or in [Bor 94c, chapter 3, section 2]. So me considerable simpliﬁcations are p os sible in the case of Bo olean algebras (more generally , lattices). Let Ω b e a Boo le an alge br a viewed as a categ ory with ob jects the element s of Ω and an arrow E − → F iﬀ E ⊆ F . Since there is a t most one arr ow betw een any tw o o b jects, sp ecifying a Grothendieck 66 G. RODRIGUES top ology in Ω is easy as a sieve in Ω is just a do wnw ard closed subset, that is, subsets I ⊆ Ω such that if E ∈ I and F ⊆ E then F ∈ I . But when computing suprema o f subsets in lattice, there is no loss o f g enerality if we pa ss fro m a s ubset U to the sieve S( U ) generated b y it, so that w e can spe cify the co vering sieves b y simply saying what a re the cov ering families. Deﬁnition 7. 1 . W e s ay that a family V of elements of Ω c overs E ∈ Ω if E = sup V (7.1) A family V that co vers E is, logically enoug h, called a c over o f E . F or (7.1) to make sense w e hav e to assume that the supremum o n the r ight-hand side exists. W e could do without suc h completeness h yp othes is if w e w er e more careful in wording our deﬁnitions. There is little to b e gained fr o m such extra generality howev er, so her e and elsewher e, whenever needed, we s imply ass ume the existence of all the required s upr ema. There are three Gr o thendieck top ologie s av a ilable, dep ending on the cardinality of the allo wed covers. Deﬁnition 7.2. The ﬁn ite Gr othendie ck top olo gy on a Bo o lean algebr a is the top ol- ogy generated by the ﬁnite covers. The c ount able or σ - Gr othendie ck top olo gy (on a σ - complete Boo lean a lgebra) is the top ology g enerated b y the co un table cov er s and the c omplete Gr othendie ck top olo gy (on a complete Bo olean algebr a) is the one generated by arbitrar y cardinality covers. As the reader may susp ect, for measure theory the topo lo gy of more interest on a Bo olean algebra is the σ - top ology . The complete top ology is the b est for categoria l reasons that should b e obvious; the categor y of sheaves for this top ology is the ca tegory of sheav es fo r Ω as a lo ca le . The gap b etw een the countable and the complete top olog ies is bridged by the co untable chain condition 6.16. The ﬁnite top olo gy , the top olo gy of interest in this section, ro ug hly cor resp onds to ﬁnite additivit y a nd will be used to build a universal home for categoriﬁed measure theory just as L ∞ (Ω) is the univ ersa l ho me for all measure s ; its impor tance will bec ome appar ent by the end of the subse ction. Before pro ce e ding, tw o other asp ects should be mentioned. W e will study sheav es lo cally or o ne Bo olean alg ebra at a time, but many of these sheaves (e.g. Ω 7− → L ∞ (Ω)) ar e sheav es on the site 53 of the distributive ca tegory Bo olT op of Bo olea n spaces with the standard, or op en cover Gr o thendieck top ology . This to o , is a well known theme of algebraic geo metry where it go e s under the name of “p etit” and “gros ” top oi. O n the other hand, a functor like the L 1 -functor of section 5.2 is a cosheaf not on Bo olT op but o n the site Me as of measur e algebras . W e ar e being stingy on the details, in par t bec a use b oth these asp ects are b es t under sto o d in the co nt ex t of ﬁbrations (in the catego rial sense), star ting with the ﬁbration Meas − → Bo ol given by (Ω , µ ) 7− → Ω. Fibr ations are a powerful to ol to tackle these and other constructions like the several Ba nach 2-spac e stacks ﬂoating around, but w e have b een do dging even men tioning them so as not to raise the catego r ial requirements bar for the pap er. F or the r eader int er ested in the theory of ﬁbratio ns, we refer him to [Bo r 94b, chapter 8] a nd [Str05]. 53 This site, as well as the sites of the sl ice categories Bo ol T op /X , are all non-small, so stri ctly speaking we would hav e to trim Bo olT op to a suitable small sub category . CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 67 Please keep in mind that functor, pr esheaf, natural transformation, colimit, etc. is always understo o d in the Ban c -enriched sense. Since we hav e a top ology o n Ω we can deﬁne sheaves with v alues in any catego ry A . More sp eciﬁcally , we make the following deﬁnition. Deﬁnition 7.3. Let A b e a ﬁnitely co mplete Banach 2 -space. A presheaf ξ : Ω − → A is a Banach she af if it is a sheaf for the ﬁnite top olog y . Sheav es for the other tw o top ologie s can a ls o b e deﬁned, but w e must require more completeness from the codo main A . The next theorem works for any of the three top ologie s of deﬁnition 7.2. The rea der can ﬁsh out the details of the pro of from [Rez98, section 10]. Theorem 7 .4 (Sheaf condition) . A pr eshe af ξ : Ω − → A is a she af iﬀ for every p artition E of E ∈ Ω , the c anonic al map ﬁl ling diagr am 7.1 is an isometric isomor- phism, in other wor ds, the c one ( p E ,F ) with F ∈ E , is a pr o duct. Q F ∈E ξ ( F ) p ξ ( F )   ξ ( E ) p E ,F / / 9 9 s s s s s ξ ( F ) Figure 7. 1 . The sheaf condition. The Stone equiv alence be tw een the categor y of Bo olean algebras and the dua l of the category of Boolea n spaces was proven very useful in subsection 5.3, so it is natural to try to ﬁnd the top o lo gical equiv a lent o f sheaves as deﬁned in 7.3. Let us inv estigate this matter more clo sely . Sheav es in a to po logical spa ce X with v alues in a catego ry A such as a Banach 2- space are deﬁned by simply copying the usual deﬁnition (se e [LM92, chapter I I, sec- tion 1]): they ar e presheav es ξ : open( X ) − → A satisfying the p atching c ondition , that is , for any op en cov er { U i } of U diagram 7.1 is an equalizer . The left ar row is the map s 7− → ( p U,U i ( s )) and the parallel pair on the right is ( s i ) 7− →  p U i ,U i ∩ U j ( s i )  and ( s i ) 7− →  p U j ,U i ∩ U j ( s j )  . ξ ( U ) / / Q i ξ ( U i ) / / / / Q i,j ξ ( U i ∩ U j ) Figure 7. 2 . Equalizer pa tchin g condition. The patching co nditio n for sheav es of Banach spaces on X tra nslates int o the following tw o r equirements for ev ery open c over { U i } of U ⊆ X : (1) Uniquene ss of patc hing: F or ev ery s ∈ ξ ( U ) we hav e the equality , k s k = sup {k s i k} with s i = p U,U i ( s ). 68 G. RODRIGUES (2) E xistence of patc hing for b ounded families : F or ev er y family ( s i ) , i ∈ I with s i ∈ ξ ( U i ) such that, p U j ,U j ∩ U k ( s j ) = p U k ,U j ∩ U k ( s k ) and s up {k s i k : i ∈ I } < ∞ , then there is s ∈ ξ ( U ) suc h that p U,U i ( s ) = s i . The conditions (1) and (2) fo r a s heaf of Banach spaces ar e s imila r to the patching conditions for sheaves of sets and such algebr aic ob jects as groups o r linea r spa ces. W e must not be fo oled by the similarities how ever, bec a use the presence of the bo undedness hypo thesis in (2) implies fo r example, that if we a pply the underlying functor Ban − → V ect we do not o btain a sheaf of linear spaces b ecause the underlying functor is not contin uous. More directly relev ant to o ur purpo ses, is the fact that if we analyze the construction of the ´ etale space o f a s heaf, the et aliﬁc ation , and rer un it in the case o f Banach-space v alued presheav es, then it will not pr o duce the asso ciated sheaf or she aﬁﬁc ation functor. This ha s led N. Auspitz in [Aus75] (see also [Ban77]) to str engthen the patching condition to the so -called appr oximation c ondition . If ξ is a presheaf on X then the stalk of ξ a t x ∈ X is the Banach spa ce, ξ x = def Colim U ∈F x ξ ( U ) (7.2 ) where F x is the op en neighbor ho o d ﬁlter base of x ∈ X . If ι U,x : ξ ( U ) − → ξ x is the universal c o ne and s ∈ ξ ( U ) then w e will deno te by s x , the germ of s at x , the element ι U,x ( s ). W e construct a pr esheaf e ξ by putting, U 7− → e ξ ( U ) = def Y x ∈ U ξ x (7.3) with the obvious restriction maps. There is a natur al pres heaf map η ξ : ξ − → e ξ given by s ∈ ξ ( U ) 7− → e s = ( s x ). Note that Q x ∈ U ξ x is the b o unded section spa ce of the bundle ` x ∈ U ξ x − → U and in fact, we will see below that it almost (but no t quite) gives the etaliﬁcatio n of a pr esheaf of Banach spaces. Deﬁnition 7.5 . Let X be a top olo gical space which we will assume compact Haus - dorﬀ. A preshea f ξ of Banach spaces is a Banach she af if it satisﬁes the following t wo conditions for every op en cov er { U i } of U ⊆ X : (1) Uniquene ss of patc hing: F or ev ery s ∈ ξ ( U ) we hav e the equality , k s k = sup {k s i k} with s i = p U,U i ( s ). (2) Approximation condition: If t ∈ e ξ ( U ) is such that for every ǫ > 0 and every x ∈ U there is an open neigh b or ho o d V ⊆ U of x and an s ∈ ξ ( V ) such that k p U,V ( t ) − e s k < ǫ then t = e σ for some σ ∈ ξ ( U ). If ( s i ) is a family satisfying the h yp othesis o f the patch ing co ndition (2) and we put ǫ = 1 /n in the approximation c ondition (2), then taking the limit n → ∞ we see that a Banach sheaf satisﬁes the b ounded patching co ndition (2). The a pproxi- mation condition is a completeness (in the Cauch y metric space sense) r equirement, stating that if a family ( s i ) ca n b e lo c al ly appr ox imate d than it c a n b e pa tched up. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 69 In [Aus75] it is shown that the presheaf of b o unded analytic functions on the unit disk of the complex plane satisﬁes the uniqueness and bo unded patching conditions but not the approximation c o ndition. F or the pro of o f the next theore m w e re fer the reader to [Ban7 9]. Theorem 7. 6. L et X b e a c omp act Haus dorﬀ sp ac e and ( B x ) a family of Bana ch sp ac es. Then the pr eshe af U 7− → Y x ∈ U B x (7.4) is a Banach she af. Theorem 7.6 implies that the presheaf (7.2) is a Bana ch sheaf. It almost gives the co rrect sheaf of sections (a nd thus the correc t notio n of “´ etale bundle of Ba na ch spaces”). In order to ge t it, we hav e to cut down the mona d ξ 7− → e ξ o n the categor y of presheaves to its idempotent par t by tak ing the equalizer of the parallel pa ir η e ξ , e η ξ : e ξ − → e e ξ The existence of the functor ξ 7− → e ξ implies that the categor y Shv ( X ) of Ba nach sheav es on X is r eﬂective in PSh v ( X ) and thus a complete Banach 2-space. Ob vi- ously enough, the reﬂecto r PSh v ( X ) − → Shv ( X ) w ill b e called the s heaﬁﬁcation functor. The c a tegory o f Banach sheaves has been given several alterna tive descrip- tions. Starting with the etaliﬁcation 7 .3, it can b e sho wn that it is e quiv alent to a certain catego ry o f bundles o f Banach spa ces. Another impor tant descriptio n takes as starting point the following theorem (also lifted from [Aus75]), that describ es the shea ﬁﬁcation of constan t pr esheav es. Theorem 7. 7 . L et B b e a Banach sp ac e. The she aﬁﬁc ation of the c onstant pr eshe af U 7− → B is the pr eshe af, U 7− → C b ( U, B ) (7.5) wher e C b ( U, B ) is the Banach sp ac e of b ounde d c ontinuous functions U − → B . Note that for a co mpa ct Haus do rﬀ s pace X , the g lobal se ction s pace of (7.5) is just C ( X, B ). This latter space is isometr ically isomorphic to the inje ctive tensor pr o duct 54 : C ( X , B ) ∼ = C ( X ) ˘ ⊗ B (7.6) The catego ry Sh v ( X ) is a symmetr ic monoidal clo sed categor y . The tenso r pro duct is given b y the sheaﬁﬁcation of the p oint wise tensor pro duct structur e in PSh v ( X ). This means that the sheaﬁﬁcatio n is a monoidal functor and th us takes monoids in to mono ids. Since the pres heaf U 7− → R is a monoid in PShv ( X ), and the initia l monoid at that, it follows that its s heaﬁﬁcation U 7− → C b ( U ) is a monoid in Sh v ( X ) that acts in a canonical way on every shea f. This leads to the description of Banach sheaves as sheaf mo dules over this sheaf algebr a sa tisfying a lo c al c onvexity c ondition . These t wo equiv alences (and more) can be found in [HK 7 9]. How ever, the most striking result is the equiv alence of Sh v ( X ) with the category of internal Banach sp ac es in the top os of (set- value d) she aves on X for a suitable, intuitionistically v alid deﬁnition of Banach space (see [BM79]). This is not the place to dwell on 54 See [ Rya 02 , ch apter 3] f or the deﬁnition and basic prop erties of the i njectiv e tensor pro duct of Banach spaces. 70 G. RODRIGUES these matters; the next theor em ([Ban77, prop o sition 7 ] in new ga rbs) sho uld s uﬃce to underline the imp ortance of thes e remarks . Theorem 7.8. L et Ω b e a Bo ole an algebr a and X = S(Ω) its Stone sp ac e. If ξ is a pr eshe af in S(Ω) then η ∗ ( ξ ) : E 7− → ξ ( η ( E )) (7.7) is a pr eshe af on Ω . The fun ctor ξ 7− → η ∗ ( ξ ) establishes an e quivalenc e b etwe en the c ate gory Sh v ( X ) of Banach s he aves on S(Ω) and the c ate gory Shv (Ω) of she aves on Ω for the ﬁnite t op olo gy. Pr o of. Since η is a Bo olea n algebra isomor phism it is clear that (7.7 ) is a shea f for the ﬁnite topo lo gy . W e br ie ﬂy des crib e the w eak inv ers e : if ξ is a sheaf on Ω fo r the ﬁnite topo logy , since η is a Bo olean algebra isomorphism Ω ∼ = clop en( X ), w e hav e a preshea f clop en( X ) − → Ban given by E 7− → ξ ( η − 1 ( E )) This presheaf satis ﬁe s the ﬁnite patc hing condition. T o see that it satisﬁes the bo unded pa tchin g condition note that if { η ( E i ) } is an op en cover of η ( E ), by compactness it ha s a ﬁnite op en s ubc over  η ( E i j )  . The fact that it is an o pe n sub c ov er implies that the ideal base  η ( E i j )  is coﬁna l in the ideal base { η ( E i ) } and thus ξ ( η − 1 ( E )) ∼ = Lim j ξ ( η − 1 ( E i j )) ∼ = Lim i ( η − 1 ( E i )) In fact, the same compactness a r gument also yields that E 7− → ξ ( η − 1 ( E )) sat- isﬁes the a pproximation condition (2). By the well-kno wn r esult tha t a shea f is uniquely determined by its v a lues on a base fo r the top ology (se e [LM92, chapter 2,, sectio n 1, theorem 3 ] – the result readily extends to Banach sheav es), we obtain a Bana ch sheaf on S(Ω).  Theorem 7.8 together with the remarks preceding it, places the sheaf-half of cat- egoriﬁed meas ure theory within well-tro d territory : internal Ba nach space theory in shea f top oi. W e have av aila ble powerful results s uch as Gelfand-Naimark du- ality (see [BM0 6 a]), the beginning s of measure theory (see the delightf ul [Ja c06]), etc. Things get even more in teresting if we rea lize that if Ω is order complete (for example, if it is a σ -mea sure algebra s atisfying the ccc condition) then the top os Sh v (Ω) s a tisﬁes the internal version of the a xiom of c hoice (see [BW83, c hapter 7, theorem 7 .3]). By a theorem o f Dia c o nescu, the topos Sh v (Ω) is Boo le an and therefore classic al analysis is valid in Shv (Ω). The internal language of a top os, and in general the logical asp ects of top os theory , is fearsome black wizar dry that I a m not comp etent to deal with, so in the rest of the pap er we will have to proce e d using cruder, more pr imitive metho ds. T o ﬁnish oﬀ this subsection, here is a n example of a B anach shea f that should give w a rm fuzzy feelings to all Banach-space theorists out there. Recall that the forgetful functor CHaus − → Set that to a compa ct Hausdorﬀ space asso ciates the underlying set has a left adjoint F given on X ∈ Set by the Stone-Cech compa ctiﬁ- cation of X endow ed with the discrete top olo gy . Its universal prop erty is depicted in diagra m 7.3 for the case of b ounded functions f : X − → K . On the other hand there is a map X − → S( 2 X ) CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 71 F( X ) b f / / _ _ _ K X O O f < < z z z z z z z z z Figure 7. 3 . The free compact Ha usdorﬀ spa c e F( X ). that to x ∈ X asso ciates the principal ﬁlter g enerated b y the singleton atom { x } ∈ 2 X . By discre tenes s of X , this map is cont inuous a nd th us there is a unique map closing the triangle 7.4. It can b e shown that the da shed map is bijective and thus an homeomo rphism. F( X ) / / _ _ _ S( 2 X ) X O O : : v v v v v v v v v Figure 7. 4 . The homeomorphism F( X ) ∼ = S( 2 X ). Since the das he d map of 7.4 is an homeomor phis m, the map X − → S( 2 X ) satisﬁes the s ame universal prop erty of diagram 7.3. Coupling theorem 5 .1 2 with the Riesz representation theorem, w e have the isometric is omorphism BA ( 2 X ) ∼ = C (S( X )) ∗ The r ig ht-hand side of this isomorphism, tells us that we can obtain a larg e collection o f mea sures on 2 X by taking p o int mass Dira c mea sures δ b x for b x a p oint of S( 2 X ), or an ultraﬁlter of 2 X . By deﬁnition of the induced map on Stone-Cech compactiﬁcations, we hav e, δ b x b f = b f ( b x ) = lim b x f where the right-hand side is the limit of f with r esp ect to the ultraﬁlter b x . Since δ b x is a measure we ca n spe ak of a.e. e quality o f functions and from, Z F( X )    b f    d δ b x = δ b x    b f    = lim b x | f | we hav e that b f = a.e. 0 iﬀ lim b x | f | = 0. Let us gener alize. Since X with the disc rete top olo gy is completely regular, it follows that the the inclusion X − → F( X ) is an op en map with dense range and th us it establishes a n equiv alence Sh v (S( 2 X )) ≃ Shv ( X ) (7.8) But by discreteness of X the B a nach 2-space on the right-hand side is isomorphic to Bun ( X ). This means that a Banach sheaf ξ on S( 2 X ) amo unt s to a n X -family ( B x ) of Ba nach spaces. Given a Banach sheaf ξ = ( B x ) on X , the stalk ξ b x at an ultraﬁlter p o int b x ∈ S( 2 X ) is the colimit Colim U ∈U b x Q x ∈ U ∩ X B x but this colimit 72 G. RODRIGUES is pr ecisely the subspace of sections f ∈ Q x ∈ X B x of the global section space suc h that lim b x k f k = 0 Fixing an ultraﬁlter b x ∈ S( 2 X ), the ult r apr o duct Q b x B x of the family ξ = ( B x ) is the quotien t Y b x B x = def Y x ∈ X B x ! / ξ b x (7.9) The ultr apro duct (7.9) could b e further descr ib e d in ter ms of direct images, but the preceding commen ts should indicate that this imp ortant to o l of B anach space theory is but a sp ecia l ca se of a sheaf-theoretic co nstruction. 7.2. Co s hea v es and inv erters. Cosheaves on Boo lean algebras w er e deﬁned in subsection 6 .3. Co sheav es have b een studied in connection with Borel- Mo ore ho- mology ([Bre 97, c ha pter V]). The link with distributions on topoi in the s ense of F. W. Lawvere was uncovered in [Pit85]. I ﬁrst stum bled upon this t yp e of distri- butions when plo dding through the ﬁrs t chapter o f [BF06] and refer the reader to it for the references to the origina l F. W. Lawv ere’s pa p e rs. I was not able to access them, but F. W. Lawvere’s ideas on space and quantit y a re men tioned brieﬂy in [Law05 , section 7]. This pa p er , in conjunction with [La w0 2], is a v eritable gold mine of ideas ready for the plunder and a highly re c o mmended reading 55 . Recall theor em 5.9 express ing the universal prop erty o f L 1 (Ω , µ ). F or every Lipschitz ﬁnitely additive map ν : Ω − → B there is a unique bounded linear map L 1 (Ω , µ ) − → B such tha t tria ngle 7.5 is comm utative. L 1 (Ω , µ ) R d ν / / _ _ _ B Ω χ O O ν ; ; v v v v v v v v v v Figure 7. 5 . Universal prop erty of L 1 (Ω , µ ). Categorifying , we want a Ba na ch 2-s pace L (Ω) s uch that the functor ν 7− → R d ν establishes an equiv alence CoSh v (Ω , A ) ≃ 2– Ban ( L (Ω) , A ) (7.10) In order to c onstruct such a gadget, let us go back to the construction o f the space S (Ω) of simple functions on Ω depicted diagra mmatically in 7.6 . Diagram 7.6 tells us that S (Ω) is obtained by taking the free linear space F(Ω) and then the c o e qualizer that kills the subs pa ce ι ( E ∪ F ) − ι ( E ) − ι ( F ) for all pairwise disjoint E , F ∈ Ω. An a ppropriate mo diﬁcation of this construction inside the ca teg ory of B anach spaces produce s L ∞ (Ω) directly . Now catego rify . Let µ : Ω − → A be a cosheaf (to ﬁx ideas, fo r the ﬁnite topo logy). As seen in s ubsection 6.1 , the fre e B anach 2 -space on Ω is PShv (Ω) and the induced map co rresp o nding to the top right arrow of 7.6 is the left Kan extension Lan Y µ . 55 It w ould not b e too far oﬀ the mark, i f I had titled this pap er “T aking seriousl y “T aking categories seri ously””. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 73 F(Ω) / / _ _ _ _ _ _ _ _ π # # F F F F F F F F A S (Ω) = = | | | | Ω ι Z Z 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 χ O O µ J J Figure 7. 6 . Constr uction of S (Ω). The category of integrable bundles L (Ω) will be a suitable quotient of PSh v (Ω) that when composed with Y oneda yields the univ ersa l c osheaf χ . In order to gues s what this quotient is let ( E n ) be a partition of E ∈ Ω. The Y oneda functor would be a cosheaf if the cone Y( E n ) − → Y ( E ) was a copr o duct, that is, Y( E ) ∼ = P n Y( E n ). But o n this h yp othesis, by Y one da lemma we hav e for every pres he a f ξ ξ ( E ) ∼ = PSh v (Y ( E ) , ξ ) ∼ = PSh v X n Y( E n ) , ξ ! ∼ = Y n PSh v (Y ( E n ) , ξ ) ∼ = Y n ξ ( E n ) By theor em 7.4 this mea ns that ξ is a s heaf so that o ur quotient is the she aﬁ- ﬁc ation functor . Ano ther wa y to look at the sheaﬁﬁcation and that shows more clearly in what sense it is a categoriﬁcatio n of the co equalize r F(Ω) − → S (Ω) is to note that there is paralle l pa ir of functors and a 2 -cell that we depict as in dia gram 7.7. A Ω ∗ Ψ , , Φ 2 2       η A parts (Ω) ∗ Figure 7. 7 . The 2-ce ll η E : ξ (sup E ) − → Q F ∈E ξ ( E ). The functor Ψ se nds the preshea f ξ : Ω ∗ − → A to the functor E 7− → ξ (sup E ) and the functor Φ assig ns to ξ the functor E 7− → Q F ∈E ξ ( E ). The inner 2-cell η is the map closing triangle 7.8. T o force every presheaf ξ to b ecome a sheaf we hav e to mak e the natural map η ξ an isomorphism, or in other w or ds, we have to invert t he 2 -c el l η . Note the categoriﬁc a tion pattern: instead o f imp osing an equa lity (tak e the co eq ua lizer), we inv ert a 2 -cell. Quo tien ts of ca teg ories that inv ert g iven 2-cells are known as c oinverters and are o ne of a sp ecial class of ﬁnite colimits p eculiar to 2-categorie s . 74 G. RODRIGUES Q n ξ ( E n )   ξ ( E ) / / η ξ : : t t t t t ξ ( E n ) Figure 7. 8 . The natura l map η E . W e simply state the deﬁnition next and refer the rea der to [K e l89, section 4] for more information on this t yp e of 2-categor ial colimits. Deﬁnition 7.9 . Let A be a 2 -categor y and η : f − → g : a − → b a 2-ce ll. The coinv erter of η is a 1- cell π : b − → c such that pη is an iso morphism and π is universal among such 1-cells . That is , every 1-cell h : b − → d such that hη is an isomorphism facto r s uniquely through π as in diagram 7.9 , and the map h 7− → b h a f $ $ g : :       η b π / / h   = = = = = = = c b h      d Figure 7. 9 . Universal pro p erty of the coinv erter π . establishes an isomorph ism b etwe en the c ate gories of su ch 1 -c el ls . Coinv erters or lo c alizations 56 are hard to construct 57 , but it is w ell known that lo calizations of preshea f ca tegories PSh v ( A ) cor resp ond to Grothendieck to p o logies on A , so that s heaf categ ories should b e the solution for the bir epresentation (7.10). Denote by r the sheaﬁﬁca tion reﬂection PSh v (Ω) − → Shv (Ω). Preco mpo sing with Y oneda we obtain a functor χ : Ω − → Shv (Ω). Theorem 7.10. Pr e c omp osition with χ : Ω − → Shv (Ω) establishes a natu r al e quiv- alenc e 2– Ban ( Sh v (Ω) , A ) ≃ CoShv (Ω , A ) (7.11) Pr o of. The crucial r e s ult is that µ : Ω − → A is a cosheaf iﬀ the left Kan extension Lan Y µ fac tors uniq ue ly (up to inner is omorphism 2- cells) through the shea ﬁﬁcation r as in diagram 7.10. 56 Localization usually means s omething str onger: a ﬁnitely c ontinuous coin verter. Contrary to the category of sets or categories of algebraic ob jects li ke V ect , ﬁnite limits do not comm ute with ﬁltered colim its in Ban c . What do es happen is that Ban c is ω 1 -lo cally presentable with ω 1 the ﬁrst uncounta ble ordinal (see [Bor94b, c hapter 5], esp ecially example 5.2.2.e). It is al so true that ﬁltered colimits commut e with ﬁnite pro ducts, a r esult that is useful for Banac h sheav es by theorem 7.8. This is prop osition 5 of [Bor82]. 57 The most famous example of a coinv er ter may ve ry well b e the coin version of the homotop y functor by the weak equiv alences i n homotopy theory (see [GZ67]). In a s ense, the whole tec hnol- ogy ranging from the calculus of fractions to Quillen mo del categories, has come to l ife precisely to handle coinv erter constructions. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 75 PSh v (Ω) Lan Y µ / / r & & L L L L L L L L L L A Sh v (Ω) < < x x x x x Ω Y ] ] ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; χ O O µ I I Figure 7. 1 0. F actoriza tion of co sheav es via Shv (Ω). Once that is done, equiv alence 7.11 follo ws fro m the re s ults of subsection 6 .1 , theorem 6.10 in particular. T he remarks ab ov e already hinted at how to construct the factoriza tion; for mor e details, the r eader is advis e d the cons ult the refences given at the b eginning o f the subsection.  Theorem proves that χ : Ω − → Sh v (Ω) is the univ er sal cosheaf: every other cosheaf fa ctors uniquely throug h it. Although χ is a cos heaf, for ea ch E ∈ Ω, χ ( E ) is a sheaf. Thoug h limits Sh v (Ω) are formed p oint wise, colimits in Shv (Ω) are not . Instead, one forms the colimit in PSh v (Ω) a nd she a ﬁﬁes it. Because of this, the sheav es χ ( E ) ar e not pro jective. On the o ther hand, since sheaﬁﬁca tio n is a left adjo int, it is co co nt inuous and it is w ell- known that the co mp o sition o f a dense functor with a left adjoin t is dense. The following co end c a lculation pr ov es this: ξ ∼ = ri ( ξ ) ∼ = r Z E ∈ Ω PSh v (Y ( E ) , iξ ) ⊗ Y ( E ) ! ∼ = r Z E ∈ Ω Sh v ( χ ( E ) , ξ ) ⊗ Y( E ) ! ∼ = Z E ∈ Ω Sh v ( χ ( E ) , ξ ) ⊗ χ ( E ) Since the las t term is pr ecisely the left Kan extension La n χ χ by the co end formula, density of χ follows from the already cited theorem [Kel0 5, chapter 5, theorem 5.1 ]. Given the resemblance of the univ ers al prop erties of L ∞ (Ω) and L 1 (Ω , µ ) one could ask what exactly hav e we catego riﬁed. In the ordinary decategor iﬁed setting, the diﬀerence b etw een L ∞ (Ω) and L 1 (Ω , µ ) arises fro m the fact that b oundedness and summabilit y ar e non-v a cuous concepts. Thes e distinctions disapp ear in the categoriﬁe d setting since al l small co limits exist, but there is still one s ubstantial diﬀerence b etw e e n L ∞ (Ω) and L 1 (Ω , µ ) worth noticing. The former is a Bana ch algebra and the cor resp onding universal measur e χ is spectr a l. W e will now see that the univ ersal cosheaf χ provided by theorem 7.10 is sp ectral. 76 G. RODRIGUES In the ﬁrs t plac e , intersection ∩ makes of Ω a monoidal categor y 58 . The preshea f category PSh v (Ω) is monoidal with the p oint wise tensor pro duct 59 and it is easy to see that Y( E ∩ F ) ∼ = Y( E ) ⊗ Y ( F ). Since the mono idal structure in Sh v (Ω) is the shea ﬁﬁcation of the po int wis e tensor product it follows tha t: χ ( E ∩ F ) ∼ = χ ( E ) ⊗ χ ( F ) (7 .1 2) Theorem 7 .8 ca n now b e seen as the categor iﬁcation of theo rem 5.1 0. The pro of of 7 .8 do es no t lo ok like a categor iﬁcation of the pro of of 5.10 , but we c an ca tegorify this pr o of ! Let X b e the Stone space of Ω a nd for a clop en η ( E ) of X consider the presheaf on open( X ) given by U 7− → Y x ∈ η ( E ) ∩ U K (7.13 ) By theorem 7.6 it is a Banach shea f. More impo rtantly , we can re a dily recognize it as the shea ﬁﬁcation of the repr esentable of E ∈ clop en( X ) ⊆ op en( X ) and isomorphic to U 7− → C b ( U ∩ E ). Since the representables are dense and the clop ens form a basis o f X , the she aves (7 .13) form a dense sub catego ry of Shv (S(Ω)). This is the ca tegoriﬁca tion of the Stone-W eiers trass theorem. Theor em 7.8 now dro ps out by noting that E 7− → ( U 7− → C b ( U ∩ E )) is a sp ectral cosheaf, taking its direct int eg ral via theor em 7.10, and then using co end calculus to prov e that it is fully- faithful and densit y of the U 7− → C b ( U ∩ E ) to prov e essential s urjectiveness. By theore m 7.10 a coshea f µ will factor uniquely via the full sub categ o ry of the characteristic sheaves χ ( E ). Deﬁnition 7.11 . Deﬁne the ca tegory S (Ω) of simple she aves to b e the full sub- category of χ ( E ) with E ∈ Ω. The universal cosheaf χ : Ω − → Sh v (Ω) factor s uniquely thro ugh the inclusio n S (Ω) − → Sh v (Ω), therefor e no confusion s hould arise if we denote the inclusion Ω − → S (Ω) also by χ . Density of S (Ω) together with the unique factor ization of 7.10 yields the following impo r tant theor em. Theorem 7. 12. Consider the left Kan extens ions Lan χ µ as in diagr am 7.11. The functor µ 7− → Lan χ µ establishes an e quivalenc e b etwe en Banach 2 -sp ac es BanCat ( S (Ω) , A ) ≃ CoSh v (Ω , A ) (7.14 ) In par ticular, CoShv (Ω , A ) is complete if A is. In view o f theorem 7.12 it pays oﬀ to hav e an idea of what sor t o f category is S (Ω). The ﬁrst thing to notice is that S (Ω) is self-dual. T o see this, we ﬁr st need to rec all the adjunction b etw e e n presheav es a nd precosheaves. If A is a small Ba nach category , there is a functor L : PSh v ( A ) − → PCoSh v ( A ) ∗ given o n ob jects b y ξ 7− → ( a 7− → PSh v ( ξ , Y a )) (7.15) 58 Ev en more, since Ω i s a Bo olean algeb r a, it is cartesian closed with pro ducts giv en by ∩ . The cartesian closed structure of Ω is also goo d in the sense that it m ak es of Ω a c omp act close d category – for example, see [ FY89]. This goo d dualit y structure is another thing that gets l ost in the passage f rom ﬁnite to inﬁnite-dimensional spaces. 59 F or the knowledgeable reader, we note that the Day conv olution on PSh v (Ω) reduces to the point wise tensor pro duct. One l ess monoidal structure to w orry about, whic h is alwa ys a goo d thing. This follows from the fact that Ω i s a l attice with only one arrow b et ween any tw o ob jects. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 77 Sh v (Ω) R d µ / / A S (Ω) i I I Lan χ µ < < x x x x x Ω χ O O µ E E                 Figure 7. 1 1. The factor ization La n χ µ . Similarly , there is a functor R : PCoShv ( A ) ∗ − → PShv ( A ) given o n ob jects by µ 7− → ( a 7− → PCoShv ( µ, Y a )) (7.16) The next theo rem shows that this pair of functors is pa rt of an a djunction. This adjunction is called Isbell conjugation in [Law05, se ction 7]. Since the pap er do es not supply a pro of we do. Theorem 7.1 3 (Isb ell conjugatio n) . L et A b e a smal l c ate gory. Ther e is a natu r al isomorphi sm: PCoSh v ∗ (L ξ , µ ) ∼ = PSh v ( ξ , R µ ) (7.17) Pr o of. The calc ulus of ends a ﬀords a nifty pro of of (7.17). Starting with the righ t- hand side, using the end formula 6 .15, coco ntin uity , in ter change of ends and sym- metry of ⊗ we have: PSh v ( ξ , R µ ) ∼ = Z a ∈A Ban ( ξ ( a ) , R µ ( a )) ∼ = Z a ∈A Ban ( ξ ( a ) , PCoSh v ( µ, Y a )) ∼ = Z a ∈A Ban  ξ ( a ) , Z b ∈A Ban ( µ ( b ) , A ( a, b ))  ∼ = Z a ∈A Z b ∈A Ban ( ξ ( a ) , Ban ( µ ( b ) , A ( a, b ) )) ∼ = Z a ∈A Z b ∈A Ban ( ξ ( a ) ⊗ µ ( b ) , A ( a, b )) ∼ = Z a ∈A Z b ∈A Ban ( µ ( b ) ⊗ ξ ( a ) , A ( a, b )) ∼ = Z a ∈A Z b ∈A Ban ( µ ( b ) , Ban ( ξ ( a ) , A ( a, b ))) ∼ = Z b ∈A Z a ∈A Ban ( µ ( b ) , Ban ( ξ ( a ) , A ( a, b ))) ∼ = Z b ∈A Ban  µ ( b ) , Z a ∈A Ban ( ξ ( a ) , A ( a, b ))  ∼ = Z b ∈A Ban  µ ( b ) , PSh v  ξ , Y b  78 G. RODRIGUES ∼ = Z b ∈A Ban ( µ ( b ) , L ξ ( b )) ∼ = PCoSh v ( µ, L ξ )  F rom Isbell conjugacy , the next theo r em follows. Theorem 7. 14. The Isb el l adjunction (7.17) desc ends to an e quivalenc e S (Ω ∗ ) ≃ S (Ω) ∗ (7.18) The ﬁnal ingredient w e need is that the complemen tation map c : Ω − → Ω ∗ is a Bo olean algebr a isomorphism betw een Ω and its dual Bo o le an algebra Ω ∗ . By taking the left K an extensio n a s in squa r e 7.1 2 we obtain an equiv a lence PSh v (Ω) ≃ PCoSh v (Ω). PSh v (Ω) Lan Y Ω Y Ω ∗ c / / _ _ _ _ _ _ PCoSh v (Ω) Ω Y Ω O O c / / Ω ∗ Y Ω ∗ O O Figure 7. 1 2. The equiv alence PSh v (Ω) ≃ PCoSh v (Ω). It is now easy to see that this equiv alence des c e nds to an equiv ale nce S (Ω) ≃ S (Ω ∗ ) (7.19) Next, we describ e the sheav es χ ( E ). Since χ ( E ) is the sheaﬁﬁca tion of Y( E ) and by de ﬁnitio n we have the isomorphism 60 , Ω( F, E ) ∼ = ( K if F ⊆ E , 0 otherwis e. the re presentable Y ( E ) is iso mo rphic the sheaﬁﬁcation 61 of E 7− → K precomp ose d with the Boo lean algebra map F 7− → E ∩ F . Theorem 7. 15. F or e ach E ∈ Ω , the she af χ ( E ) is isomorphic to F 7− → L ∞ ( E ∩ F ) Pr o of. Co m bine the preceding o bserv ations with theorems 5.10, 7.7 and 7.8.  60 Recall our notational pr actice of iden tifyi ng ordinary categories with their fr ee Banac h categories. 61 A word of caution i s necessary here. By theorem 7. 7 and the isometric isomorphism (7.6) the sheaﬁﬁc ation of the constan t presheaf E 7− → B is E 7− → L ∞ ( E ) ˘ ⊗ B . This Banac h space is strictly smal ler than the Banach space L ∞ ( E , B ) considered i n subsection 4.1 if B is inﬁnite- dimensional, since the functions in L ∞ ( E ) ˘ ⊗ B ha ve c omp act r ange . A t ypical example of a strongly measurable function not in L ∞ ( E ) ˘ ⊗ B is buil t by pi cking a b ounded sequence with no con vergen t subsequences. A more sophisticated one go es like this: pick a separable r eﬂexiv e space B . Than its unit ball ball( B ) is co mpact f or the w eak top ology; tak e the corresp onding Borel structure. Since the inclusi on ball( B ) − → B is weakly con tinuo us, by the Pe ttis theorem 4.7 i t is strongly measurable. But of course, i ts range i s very far fr om b eing compact. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 79 The next step is the computation of morphism spaces Sh v ( ξ , ζ ) where ξ a nd ζ are constant shea ves. This is perfectly doable, but the more impor tant cas e is contained in the next theorem a nd its pro of alrea dy contains the basic ingredients of the more general ca se. Theorem 7. 16. If E , F ∈ Ω then the she af Shv ( χ ( E ) , χ ( F )) is isomorphi c to G 7− → L ∞ ( E ∩ F ∩ G ) Pr o of. The pro of pro ceeds by pa s sing to the Stone spa ce S(Ω) a nd then using (a v ar iation of ) the Bana ch-Stone theorem (see [Con90, chapter 6, section 2]).  Theorems 7.15 and 7.16 give us a fairly accura te picture of S (Ω) as little more than a categoria l disguise of the Banach a lgebra L ∞ (Ω). The morphism spa ces S ( E , F ) ar e the sheaves L ∞ ( E ∩ F ) a nd co mpo sition is given by multiplication in the Ba nach algebra. Note tha t theorem 7.16 implies the symmetry of S ( E , F ) in its arg umen ts. Theorem 7.12 tells us tha t µ : Ω − → A is a coshea f iﬀ it factors through χ : Ω − → S (Ω). This factorization has a striking description in terms of integration against a sp ectral meas ure – this will b e descr ib ed in subsectio n 7.4. F or now, we just no te that theorem 7.12 lands us in a situa tio n a nalogous to that o bserved at the end of subsection 6.2 with sets X replaced b y the Bana ch categ ories S (Ω). In particular , we ar e a ble to prov e a F ubini theorem, c haracter ize the coco ntin uo us functors L (Ω) − → L (Σ) either as distr ibutors S (Ω) ∗ ⊗ S (Σ) − → Ban or as cosheav es o n Ω ⊗ Σ, etc. It a ls o allows us to explain why D. Y etter’s deﬁnition of mea surable functors in [Y e t05] a s a certain measurable bundle over the pro duct X × Y is essentially correct. This is done in full in [Ro db], but one la s t theorem is worth men tioning. Recall tha t a category is Cauchy c omplete if ev er y idempo tent splits. Theorem 7. 17. The c ate gory S (Ω) is Cauchy c omplete. Pr o of. O nc e aga in this is prov able by passag e to the Stone space S(Ω). Retr acts o f Banach s pa ces a re co mplemen ted subspaces and complemented subspaces o f L ∞ (Ω) are of the form L ∞ ( E ) for E ⊆ Ω.  Cauch y completeness is just one strand in a web of connections inv olv ing dis- tributors, Morita equiv alence and T annak a-Kr ein reconstruction via the cen ter of categorie s 62 . W e refer the reader to [Kel05, c hapter 5, section 8], [Bor9 4a, chapter 7, sections 8 a nd 9 ] and ab ove all, to [Lin74] that con tains many impo rtant results in the context o f enr iched ca tegories that a re dir ectly useful for categor iﬁed measur e theory . F or T a nnak a -Krein reconstr uctio n, see [J S91]. 7.3. The cosheaﬁﬁcation functor. It is easy to see that CoShv (Ω , A ) is a Ba- nach 2- space with colimits co mputed p oint wise, that is, the inclusion i : CoSh v (Ω , A ) − → A Ω (7.20) is co co ntin uous . In the cas e where A is the base category Ban , CoSh v (Ω) has a small dense s ubc a tegory and by the adjoin t functor theorem the inclusion has a 62 The c enter of a categ ory A is the commuta tive monoid of natural transformations 1 A − → 1 A . Since χ is dense, the center of Shv (Ω) i s isomorphic to the cente r of S (Ω) and by theorem 7.16, isomorphic to L ∞ (Ω). 80 G. RODRIGUES right adjoint called the c oshe aﬁﬁ c ation functor . In this section, we construct the cosheaﬁﬁcatio n mo re directly , by taking appropriate limits o f Ba nach spaces 63 . Recall that the variation of an additive map µ is g iven by the (pos sibly inﬁnite) quantit y k µ k ( E ) = sup ( X F ∈E k ν ( F ) k : E ∈ parts ( E ) ) (7.21) The cosheaﬁﬁcatio n functor can b e seen a categor iﬁcation of (7.21 ). More pre- cisely , for ev er y preco sheaf µ , deﬁne µ ∗ ( E ) as the limit of the functor parts ( E ) − → A , that is E 7− → Lim E ∈ parts ( E ) X F ∈E µ ( F ) (7.22) In other w ords, an element θ ∈ µ ∗ ( E ) is a c o germ , or a co mpatible c hoice of elements θ E ∈ P F ∈E µ ( F ). Theorem 7. 18. The fun ctor (7.2 2) is a righ t adjo int to the inclusion functor. Pr o of. The fact that (7.22) indeed gives the coshea ﬁﬁcation functor, is a straig ht- forward dualization of the fa ct that E 7− → Colim E ∈ parts ( E ) Y F ∈E µ ( F ) yields the sheaﬁﬁca tion functor. F or the latter, the details can b e g limpsed from the alr eady cited [Rez98, se ction 1 0].  The univ er sal prop erty of the cosheaﬁﬁcation µ ∗ is depicted in dia gram 7.13. Equiv a lently , w e have the na tur al isomo rphism, PCoSh v ( i ( ν ) , µ ) ∼ = CoSh v ( ν, µ ∗ ) (7.23) µ ∗ ε   ν τ / / > > } } } } µ Figure 7.13. Universal prop erty of the cosheaﬁﬁcation. Since Ω has a top element, also denoted by Ω, it follows that the colimit functor Colim E ∈ Ω µ ( E ) ∼ = µ (Ω) is just the ev aluation of µ at Ω. By deﬁnition of c o limit, this yields the adjunction, Ban ( µ (Ω) , B ) ∼ = PCoSh v ( µ, ∆ B ) (7.24) where ∆ is the dia gonal functor a s so ciating to ea ch B anach spa ce B the corr esp ond- ing cons tant pr ecosheaf E 7− → B . Comp osing the a djunctions (7.2 4 ) and (7.23 ) we obtain the importa nt result that ev aluatio n is left a djoint to the constant cosheaf: Ban ( µ (Ω) , B ) ∼ = CoSh v ( µ, B ∗ ) (7.25) 63 On the assumption that A is complete, the limi t con struction of the cosheaﬁﬁc ation also yields a r ight adjoint to (7.20). CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 81 If µ is the co sheaf and w e tak e the cosheaf E 7− →  Z ξ d µ  E = def Z E ξ d µ ∼ = Z Ω χ ( E ) ⊗ ξ d µ given by the indeﬁnite integral of a sheaf, then its ev aluation at Ω is simply its tota l direct integral and adjunction (7.25) rea ds as: Ban  Z Ω ξ d µ, B  ∼ = CoSh v  Z ξ d µ, B ∗  (7.26) Given these adjunctions and the fact that for a measure µ and a co ns tant sheaf χ ( E ), R Ω ξ d µ is the space L 1 ( E , µ ), it is imp ortant to iden tify the class of co nstant cosheav es. This is done in the next subsection. 7.4. The sp ectral measure o f a cosheaf. In this subsec tio n, we describ e the fundamen tal construction of the sp e ctr al me asur e of a c oshe af . The co ns truction depe nds strongly o n the fact that Banach 2-spaces as w e have deﬁned them ha ve linear s pa ces of mor phisms and the arguments with dir ect sums in additive cat- egories go through to some extent . W e will restrict ours elves to ﬁnitely additiv e precosheaves, that is, cosheaves for the ﬁnite Grothendieck top o logy . All the es- sential ideas are alrea dy co ntained in this cas e including the description of the factorization S (Ω) − → A as the integral against the sp ectra l mea sure. The σ - additive case needs mor e analysis and measure theory and will b e trea ted in the forthcoming [Ro db]. Let µ b e a cosheaf Ω − → A and F ⊆ E an element of Ω. Since { F, E \ F } is a par tition of E , the co ne  µ F, E , µ E \ F ,E  is a copro duct. W e can now deﬁne the c oshe af pr oje ctions as the unique arr ows ﬁlling dia g ram 7.14 . µ ( E \ F ) µ ( F ) µ F,E / / 0 : : t t t t t t t t t 1 µ ( F ) $ $ J J J J J J J J J µ ( E ) p E ,E \ F O O    p E ,F      µ ( E \ F ) µ E \ F,E o o 0 x x r r r r r r r r r r 1 µ ( E \ F ) f f L L L L L L L L L L µ ( F ) Figure 7. 1 4. Cos heaf pro jections. It can b e pr ov ed that p E ,F is a pr e s heaf on Ω. In fact, putting (note the capi- talization), P E = def µ E , 1 p 1 ,E we hav e: Theorem 7.19. The map E 7− → P E is a sp e ctr al me asur e on µ (Ω) . The induc e d Banach algebr a morphism, L ∞ (Ω) − → Ban ( µ (Ω) , µ (Ω)) pr ovide d by the or em 5.5 is an isometry. 82 G. RODRIGUES Pr o of. The pro of is a series of routine dia gram manipulatio ns with copro ducts in Banach 2-s paces.  Since A is co complete it has tensors, and the Ba nach alg ebra morphism induced by the spectra l measure is equiv alent to an a rrow µ : L ∞ (Ω) ⊗ µ (Ω) − → µ (Ω) (7.27) satisfying the a x ioms of a monoid action , that is, we ha ve the commutativ e dia- grams 7.15 and 7.16 for the action map (7.27), wher e the s ymbol · denotes the m ultiplicatio n map in L ∞ (Ω). K ⊗ µ (Ω) ∼ = / / 1 ⊗ 1 µ (Ω)   µ (Ω) 1 µ (Ω)   L ∞ (Ω) ⊗ µ (Ω) µ / / µ (Ω) Figure 7. 1 5. Unital la w for the ac tion µ : L ∞ (Ω) ⊗ µ (Ω) − → µ (Ω). ( L ∞ (Ω) ⊗ L ∞ (Ω)) ⊗ µ (Ω) ∼ = / / ·⊗ 1 µ (Ω)   L ∞ (Ω) ⊗ ( L ∞ (Ω) ⊗ µ (Ω)) 1 L ∞ (Ω) ⊗ µ   L ∞ (Ω) ⊗ µ (Ω) µ ( ( R R R R R R R R R R R R R R L ∞ (Ω) ⊗ µ (Ω) µ v v l l l l l l l l l l l l l l µ (Ω) Figure 7. 1 6. Asso ciative law for the action µ : L ∞ (Ω) ⊗ µ (Ω) − → µ (Ω). Theorem 7 .19 places very s trong constraints on the L ∞ (Ω)-actions c o ming from cosheav es. This is not the place to pr ovide a deta iled exp osition, suﬃce to say that the precise characteriza tio n of these actions is tied to a notion dual to that of lo c al c onvexity , a notion br ieﬂy mentioned in subsection 7.1 in connection with the characteriza tion of Banach sheav es as certa in C ( X )-Banach mo dules. It ca n b e seen as a genera lization to C ( X )-Banach mo dules o f the Bana ch la ttice conditions inv olved in the Riesz-K akutani duality . Ries z-Kakuta ni dua lit y is disc ussed on a ny bo ok on Ba nach lattices. Possible references are [Lac7 4], [F re02, chapter 5], [LZ 71] and [Zaa 83]. W e need a more co nceptual w ay to lo ok at the spectr al mea s ure. By theorem 7.19 it corresp onds to a n algebr a morphism L ∞ (Ω) − → Ban ( µ (Ω) , µ (Ω)) (7.28) The s heaf E 7− → L ∞ ( E ) is a mono id in Shv (Ω). The presence of Ω on b oth the domain and the codo main of the spec tral measure map, leads us to susp ect that the right-hand side of (7.28) can b e made into a sheaf in suc h a way that (7.28) is a monoid sheaf map. Let µ a nd ν b e tw o cosheaves a nd τ : µ − → ν a cosheaf map. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 83 If E ∈ Ω and E is a partition of E we hav e the comm utative diagr am 7.17 for each E n ∈ E . µ E τ E / / ε E $ $ I I I I I I I I I ν E ε E z z v v v v v v v v v µ ( E ) τ E / / ν ( E ) µ ( E n ) τ E n / / i µ ( E n ) O O µ E n ,E ; ; w w w w w w w w w ν ( E n ) i ν ( E n ) O O ν E n ,E c c G G G G G G G G G Figure 7. 1 7. Induced cosheaf maps. Since the maps ε E are is o morphisms, it follo ws that, k τ E k = max {k τ E n k : E n ∈ E } and thus, the map τ E 7− → ( τ E n ) is an isometric isomo rphism. In other words, diagram 7.1 7 provides us with pro jections, Ban ( µ ( E ) , ν ( E ) ) − → Ban ( µ ( E n ) , ν ( E n )) giving a sheaf E 7− → Ban ( µ ( E ) , ν ( E ) ). A more a bstract w ay of se e ing this is by using the end formula (6.1 5) and noting that CoSh v ( µ, ν ) ∼ = PCoSh v ( iµ, iν ) ∼ = Z E ∈ Ω Ban ( iµ ( E ) , iν ( E )) By the preceding ar gument, this preshea f is a ctually a sheaf (for the appropr iate Grothendieck top olog y) and s ince the comp osition map CoSh v ( µ, ν ) ⊗ CoShv ( ν, λ ) − → CoShv ( µ, λ ) is a pr esheaf map, w e ha ve: Theorem 7.20. L et A b e a Banach 2 - sp ac e. The c ate gory CoShv (Ω , A ) is a Sh v (Ω) -enriche d c ate gory. Since L ∞ is the shea ﬁﬁcation of the co nstant preshea f E 7− → K , it follows that L ∞ is initial in the category of mo noids in Sh v (Ω). By theor e m 7.20, for each cosheaf µ , the sheaf CoSh v ( µ, µ ) is a monoid in Sh v (Ω) and th us ther e is a uniq ue monoid map, L ∞ − → CoS hv ( µ, µ ) which of course, is just the spe ctral meas ure map of theor em 7 .19. Whenever we hav e a categor y A e nr iched in a nother catego ry B , the ﬁrst ques tion to answer is if A has B -tensors . The next theor em could b e called the fundamental theorem o f ca tegoriﬁed measure theory and says that CoSh v (Ω , A ) ha s all Shv (Ω)- tensors (and thus it is co complete as a Sh v (Ω)-enriched category ) and that these are just the indeﬁnite integral co sheav es. It provides the correct for mu la tion of the abstract Radon-Nikodym prop er ty characterizing direc t integrals via a universal prop erty . 84 G. RODRIGUES Theorem 7. 21. Ther e is a natur al isometric isomorphi sm CoSh v  Z ξ d µ, ν  ∼ = Sh v ( ξ , CoSh v ( µ, ν )) (7.29) W e end this s ubs ection with tw o bas ic applications of the sp ectral measur e con- struction: the description of the factorization S (Ω) − → A for a cosheaf µ and the characterization o f the c onstant co sheav es. Let µ b e a cos heaf of Banach spaces. W e w ant to deﬁne a functor o n the ca tegory S (Ω) and with v alues on Ban that o n ob jects is given by χ ( E ) 7− → µ ( E ) 64 . In order to understa nd wha t is µ ( f ) for a function f ∈ L ∞ ( E ∩ F ) : E − → F let us start by taking µ to be the L 1 ( E , µ )-coshea f induced b y a ﬁnite, positive measure µ . Thus, µ ( f ) is now a map µ ( f ) : L 1 ( E ) − → L 1 ( F ) W e can no w hazard a gues s for µ ( f ) as a co mp o s ite o f the form display ed in diagram 7.1 8. L 1 ( E ) / / _ _ _ _ _ p E ,E ∩ F   L 1 ( F ) L 1 ( E ∩ F ) / / L 1 ( E ∩ F ) µ E ∩ F,F O O Figure 7. 1 8. The map µ ( f ). The outer terms just ﬁddle with the domain of the functions of L 1 ( E ) in order to make them land in L 1 ( F ). The inner ter m is given by comp osing g ∈ L 1 ( E ∩ F ) with a ma p K − → K obtained b y some sort of integral R E ∩ F f . Let f b e a simple function E ∩ F − → K of the form, f = X n χ ( E n ) k n with ( E n ) a ﬁnite partition of E ∩ F a nd k n base ﬁeld elements. Let g ∈ L 1 ( E ∩ F ) be a simple function of the form P m χ ( F m ) l m with ( F m ) a ﬁnite partition of E ∩ F and l m ∈ K . Since we exp ect that R f ( g ) = R f  g we hav e  Z f  g = Z f ( g ) = X n,m χ ( E n ∩ F m ) k n ( l m ) = X n,m P E n ( χ ( F m )) k n ( l m ) = X n P E n k n ! X m χ ( F m ) l m ! But the term P n P E n k n is just the int eg ral of f ag ainst the sp ectral cosheaf mea - sure. Plugg ing in the outer morphisms of diagra m 7.18 , w e arrive a t the following deﬁnition. 64 The construct i on can b e generalized in a straightforw ard fashion to a cosheaf Ω − → A . CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 85 Deﬁnition 7.2 2 . Let µ b e a cosheaf of Ba nach spaces and f : E ∩ F − → K a simple function of the form P n χ ( E n ) k n (or a simple morphi sm χ ( E ) − → χ ( F )). Then the induced map R f d µ is Z f d µ = def X n k n ( µ E n ,F p E ,E n ) (7.30) That deﬁnition 7.22 is indeed functorial is the essen tial conten t o f the next theorem. Theorem 7.23. L et µ b e a c oshe af of Banach sp ac es and f : χ ( E ) − → χ ( F ) a simple morphi sm of char acteristic she aves. If R f d µ is deﬁne d as in (7.3 0) then:     Z f d µ     = k f k ∞ F u rthermor e, if g is a simple morphism χ ( F ) − → χ ( G ) then: Z g f d µ = Z g d µ Z f d µ Z 1 χ ( E ) d µ = 1 µ ( E ) Pr o of. The pr o of is a se r ies of routine manipula tions with the asso ciated sp ectral measure of a cosheaf. F ull details in [Rodb].  Since the simple fun ctio ns a re dense in L ∞ , the functor S (Ω) − → Ban can now be constructed in the obvious manner. On to the tas k of ch a racterizing the co nstant c osheav es. The p oset of par titions parts ( E ) is ﬁltered and b y construction o f limits of Banac h spaces, an ele ment of Lim E ∈ parts ( E ) P F ∈E µ ( F ) is a compa tible family λ E ∈ P F ∈E B with: sup ( X F ∈E k λ F k : E ∈ parts ( E ) ) < ∞ This s uggests tha t the cosheaﬁﬁca tion of B is the spac e of measur e s of b ounde d variation . It is indeed, but let us deﬁne the term “bounded v a riation” ﬁr st. Deﬁnition 7.24. The map ν has b ounde d variatio n if for every E ∈ Ω, its v aria tion k ν k ( E ) as deﬁned in (7.21) is ﬁnite. If ν has bo unded v aria tion, then its v aria tio n k ν k ( E ) is a p ositive, ﬁnitely addi - tive measure on Ω that dominates ν with Lipschitz nor m 1 . Deno te b y BV A (Ω , B ) the space of additiv e maps Ω − → B o f bounded v ariation with the total variation norm k ν k (Ω). Theorem 7. 25. The sp ac e BV A (Ω , B ) is c omplete. Pr o of. Stra ightforw ard, since o nly ﬁnite additivit y is inv olved.  T o understand functoriality of BV A on Ω start by identifying each E ∈ Ω with the principa l ideal I ( E ) it generates. Ea ch I ( E ) is a Boolea n algebr a with unit E and the inclusion I ( E ) − → Ω is a ring morphism that do es n ot pr e s erve the unit (and th us, it is not a Bo olean algebra mor phis m). If E ⊆ F , there is a Boo le an algebra pro jection, p F, E : I ( F ) − → I ( E ) 86 G. RODRIGUES given b y intersection: G ⊆ F 7− → G ∩ E . The presheaf E 7− → I ( E ) is a sheaf o f Bo olean alg ebras. The map BV A ( E , B ) − → BV A ( F , B ) is now given by pullback, that is, ν 7− → ( p F, E ∗ ν : G 7− → ν ( E ∩ G )). Theorem 7.2 6. L et Ω b e a Bo ole an algebr a and B a Banach sp ac e. Then the pr e c oshe af E 7− → B V A ( E , B ) is a c oshe af for the ﬁnite Gr othendie ck t op olo gy. Pr o of. Fix a ﬁnite partition ( E n ) of E ∈ Ω. The inclusions E n ⊆ E induce inclusion maps A ( E n , B ) − → A ( E , B ). If ν : E − → B is an additiv e map and F ⊆ E , then the pullback of ν via the pro jectio n is, ν n ( F ) = ν ( F ∩ E n ) and we see that ν ( F ) = P n ν n ( F ) so that the induced map M n A ( E n , B ) − → A ( E , B ) is a linear isomorphism. T o prov e that this map descends to an isometric isomor- phism on the spaces o f bounded v ariation maps, let F n be a ﬁnite partition of E n . Then F = S n F n is a ﬁnite partition of E and X n X F ∈F n k ν n ( F ) k = X F ∈F k ν ( F ) k so that P n k ν n k ( E n ) ≤ k ν k ( E ). On the other hand, if F is a partition of E that reﬁnes E , putting F n = { F ∈ F : F ⊆ E n } , then F n is a partition of E n and X F ∈F k ν ( F ) k = X n X F ∈F n k ν n ( F ) k ≤ X n k ν n k ( E n ) Since the set of ﬁnite partitions reﬁning E is coﬁnal in parts ( E ), we hav e that k ν k ( E ) ≤ P n k ν n k ( E n ), so that the induced map X n BV A ( E n , B ) − → BV A ( E , B ) is an isometric isomorphism a nd E 7− → BV A ( E , B ) is a n additiv e precoshea f o n Ω.  Theorem 7.26 implies that not every cosheaf is of the form L 1 (Ω , µ, B ) for a measure algebra (Ω , µ ) and a Ba nach space B . Jus t tak e BV A (Ω , B ) with B a Banach space without the Radon-Nikodym prop er ty (e.g. L 1 (Ω , µ ) for (Ω , µ ) an atomless σ -a dditive mea sure algebr a). Finally , we can state the character iz a tion of the constant c osheav es a s the s pa ces of bo unded v a riation meas ur es. In essence, it is an applicatio n of the sp ectr a l measure constr uc tio n. Theorem 7.27 . L et θ b e a c oshe af and τ : θ − → B b e a map b etwe en θ and the c onst ant pr e c oshe af E 7− → B . Then ther e is a unique map closing t he triangle 7.19. CA TEGORIFYING MEASURE THEOR Y: A R OADMAP 87 BV A ( E , B ) ev E   θ ( E ) τ E / / 9 9 r r r r r B Figure 7. 1 9. Universal prop erty of BV A (Ω , B ). Pr o of. Let us sketc h the construction of the asso ciated map. Since θ is a cosheaf, by theorem 7.19 ther e is an as s o ciated spectra l mea s ure E 7− → P E . Let µ ∈ θ ( E ). Then we hav e a ﬁnitely additive map given by: F 7− → τ F ( P F µ ) (7.31) The ma p tha t to µ asso cia tes (7.31) is easily seen to be a precoshea f contractiv e map. 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