Relative Frobenius algebras are groupoids

RELA TIVE FR OBENIUS ALGEBRAS ARE GR OUPOIDS CHRIS HEUNEN, IV AN CONTRERAS, AND ALBER TO S. CA TT ANEO Abstract. W e functorially c haracterize groupoids as special dagger F rob enius algebras in the category of sets and relations. This is then generalized to a non-unital setting, by establishing an adjunction b etw een H*-algebras in the category of sets and relations, and lo cally cancellativ e regular semigroup oids. Finally , w e study a univ ersal passage from the former setting to the latter. 1. Introduction Group oids generalize groups in tw o wa ys. They can b e regarded as groups with more than one ob ject, leading to the definition as (small) categories in which ev- ery morphism is inv ertible. Alternativ ely , they can b e regarded as groups whose m ultiplication is relaxed to a partial function. This article mak es the connection b et w een these t wo views precise b y detailing isomorphisms betw een the appropriate categories. The latter view is made rigorous by so-called sp ecial dagger F rob enius algebras in the category of sets and relations. Th us w e giv e a non-comm utativ e and functorial generalization of [6] in Section 2. These results are somewhat surprising from the point of view of quan tum groups, another generalization of the concept of group. Quan tum groups are usually for- malized as some sort of Hopf algebra. Ho w ever, this notion is at o dds with that of F rob enius algebra: if a multiplication carries both Hopf and F rob enius structures, then it m ust b e trivial. 1 Reform ulating group oids as relativ e F rob enius algebras has t w o adv antages. First, it yields several interesting new c hoices of morphisms b etw een group oids. Second, F rob enius algebras can b e interpreted in many categories without limits, whereas the usual formulation of a group oid as an in ternal category requires the am bient category to hav e finite limits. F or example, a commutativ e F rob enius algebra structure on a finite-dimensional Hilb ert space corresp onds to a choice of orthonormal basis for that space. F or this corresp ondence to hold in arbitrary dimension, the F rob enius algebra structure m ust b e relaxed to a so-called H*-algebra structure, basically dropping units [1]. Date : September 25, 2018. 2000 Mathematics Subje ct Classification. 18B40, 18D35, 20L05. Key words and phr ases. F rob enius algebra, H*-algebra, group oid, semigroupoid. Supported by ONR.. Partially supp orted by SNF Gran t 20-131813. 1 More precisely , in the language of Section 2, if morphisms m : X ⊗ X → X and m † : X → X ⊗ X in a monoidal category satisfy (F), (A), (U), and the v ariant of (U) for m † , and m † and u † are monoid homomorphisms, then X ∼ = I . This result holds fully abstractly; for a proof in the category of finite-dimensional vector spaces, see [4, Prop osition 2.4.10]. 1 2 CHRIS HEUNEN, IV AN CONTRERAS, AND ALBER TO S. CA TT ANEO Section 3 considers the relative version: it turns out that H*-algebras in the cate- gory of sets and relations corresp ond to so-called lo cally cancellative regular semi- group oids. The corresp ondence is functorial, but this time giv es adjunctions instead of isomorphisms of categories. Finally , Section 4 considers a universal passage from H*-algebras to F rob enius algebras. The generalization to H*-algebras is useful for an application to geometric quan- tization that will b e presented in subsequent w ork, where one is forced to work with semigroup oids instead of groupoids. Rather than in the category of sets and relations, this pla ys out in the smooth setting of symplectic manifolds and canonical relations, corresp onding to Lie group oids. One could imagine similar applications in a top ological or lo calic setting [7]. 2. Rela tive Frobenius algebras and groupoids A (small) category can b e defined as a category internal to the category Set of sets and functions, see [5, Section XI I.1]. This is a collection G 0 e / / G 1 s o o t o o G 1 × G 0 G 1 m o o of ob jects G 0 (ob jects) and G 1 (morphisms) and morphisms s (source), t (target), e (iden tit y), and m (comp osition). These data hav e to satisfy familiar algebraic form ulae, stating e.g. that comp osition m is asso ciative. A functor then is a pair of functions f i : G i → G 0 i that commute with the ab ov e structure. A category is a gr oup oid when there additionally is a morphism i : G 1 → G 1 (in verse) satisfying m ◦ (1 × i ) ◦ ∆ = e ◦ s : G 1 → G 1 . Notice that this form ulation requires the monoidal structure × to hav e diagonals ∆ : G 1 → G 1 × G 1 , and the category to ha ve pullbac ks. This section pro v es that groupoids correspond precisely to so-called relative F rob enius algebras. T o in tro duce the latter concept, we pass to the category Rel of sets and relations, where morphisms X → Y are relations r ⊆ X × Y , and s ◦ r = { ( x, z ) | ∃ y . ( x, y ) ∈ r, ( y, z ) ∈ s } . It carries a contra v arian t iden tit y-on-ob jects inv olution † : Rel op → Rel given by relational conv erse. It also has compatible monoidal structure, namely Cartesian pro duct of sets. This makes Rel into a so-called dagger symmetric monoidal cat- egory . T o distinguish Rel from its sub category Set , w e write morphisms in the former category as r : X  / / Y , and morphisms in the latter as f : X → Y . Definition 1. Consider the follo wing prop erties of a morphism m : X × X  / / X in Rel : (1 × m ) ◦ ( m † × 1) = m † ◦ m = ( m × 1) ◦ (1 × m † ) , (F) m ◦ m † = 1 , (M) m ◦ (1 × m ) = m ◦ ( m × 1) , (A) there is u : 1  / / X with m ◦ ( u × 1) = 1 = m ◦ (1 × u ) . (U) If u exists then it is automatically unique. An ob ject X together with such a morphism m is called a (unital) sp ecial dagger F rob enius algebra in Rel , or r elative F r ob enius algebr a for short. Notice that this definition requires neither pullbacks nor diagonals, and makes sense in an y dagger monoidal category . RELA TIVE FROBENIUS ALGEBRAS ARE GROUPOIDS 3 The defining conditions of F rob enius algebras can also b e presented graphically . Suc h string diagrams enco de comp osition by dra wing morphisms on top of eac h other, and the monoidal product becomes dra wing morphisms next to each other. The dagger b ecomes a v ertical reflection; we refer to [8] for more information. W e depict m : X × X  / / X as , and u : 1  / / X as . (F) = (F) = (M) = (A) = (U) = (U) = T o prev en t jumping bac k and forth b etw een formalisms, w e will mostly compute algebraically . But o ccasionally we will illustrate conditions graphically . 2.1. F rom relative F rob enius algebras to group oids. F or the rest of this subsection, fix a relative F rob enius algebra ( X , m ). Concretely , (M) implies that m is single-v alued. Therefore we may write f = hg instead of (( h, g ) , f ) ∈ m . Notice that this notation implies that hg is defined, whic h fact we denote by hg ↓ . W e will use this Kle ene e quality throughout this article, by reading x = y for x, y ∈ X as follows: if either side is defined, so is the other, and they are equal. The concrete meaning of (M) is completed by ∀ f ∈ X ∃ g , h ∈ X . f = hg . Concretely , (F) means that for all a, b, c, d ∈ X ab = cd ⇐ ⇒ ∃ e ∈ X . b = ed, c = ae ⇐ ⇒ ∃ e ∈ X. d = eb, a = ce. The condition (A) comes do wn to ( f g ) h = f ( g h ). Finally , iden tifying the morphism u : 1  / / X with a subset U ⊆ X , we find that (U) means ∀ f ∈ X ∃ u ∈ U. f u = f , ∀ f ∈ X ∃ u ∈ U. uf = f , ∀ f ∈ X ∀ u ∈ U. f u ↓ = ⇒ f u = f , ∀ f ∈ X ∀ u ∈ U. uf ↓ = ⇒ uf = f . Definition 2. Giv en a relative F rob enius algebra m , define the following ob jects and morphisms in Rel : G 1 = X, G 2 = { ( g , f ) ∈ X 2 | g f ↓} , G 0 = U, e = U × U : G 0  / / G 1 , s = { ( f , x ) ∈ G 1 × G 0 | f x ↓} : G 1  / / G 0 , t = { ( f , y ) ∈ G 1 × G 0 | y f ↓} : G 1  / / G 0 , i = { ( g , f ) ∈ G 2 | g f ∈ G 0 , f g ∈ G 0 } : G 1  / / G 1 . 4 CHRIS HEUNEN, IV AN CONTRERAS, AND ALBER TO S. CA TT ANEO W e will prov e that the collection G of these data is a group oid (in Set ). First, w e show that the relations s, t, i are in fact (graphs of ) functions, as is clearly the case for e . Lemma 3. F or f ∈ X and u, v ∈ U : (1) if f u ↓ then u 2 ↓ ; (2) if f u ↓ and f v ↓ then uv ↓ ; (3) if f u ↓ and f v ↓ then u = v . Henc e the r elation s is (the gr aph of ) a function. Similarly, t is (the gr aph of ) a function. Pr o of. If f u ↓ , then f u = f by (U), so that also ( f u ) u = f . By (A), this means in particular that u 2 ↓ , establishing (1). F or part (2), assume that f u ↓ and f v ↓ . Then f u = f = f v , and b y (F) we hav e u = ev for some e ∈ X , so that uv = ev 2 ↓ . Finally , for (3), notice that if f u ↓ and f v ↓ then u = uv = v by (U).  Lemma 4. The pul lb ack of s and t in Set is (isomorphic to) G 2 . Pr o of. The pullback of s and t is giv en b y P = { ( g , f ) ∈ X | s ( g ) = t ( f ) } . No w s ( g ) is the unique y ∈ U with g y ↓ , and t ( f ) is the unique y 0 ∈ U with y 0 f ↓ . So, if ( g , f ) ∈ P , then y = y 0 so that g y f ↓ , and by (U) also g f ↓ so ( g , f ) ∈ G 2 . Con versely , if ( g , f ) ∈ G 2 , then by (U) there exists y ∈ U suc h that g y f ↓ , and we ha ve s ( g ) = y = t ( f ).  Lemma 5. The fol lowing diagr am in Rel c ommutes. G 1 ∆   s / / G 0 e   G 1 × G 1 1 × i / / G 1 × G 1 m / / G 1 Her e, ∆ is (the gr aph of ) the diagonal function x 7→ ( x, x ) . Pr o of. First we compute e ◦ s = { ( f , g ) ∈ G 1 × G 1 | ∃ u ∈ U. g = u, f u ↓} = { ( f , u ) ∈ X × U | f u ↓} , and m ◦ (1 × i ) ◦ ∆ = { ( f , h ) ∈ G 2 1 | ∃ g ∈ G 1 . f g ∈ U, g f ∈ U, h = g f } = { ( f , g f ) ∈ G 2 1 | g ∈ G 1 , f g ∈ U 3 g f } . Clearly m ◦ (1 × i ) ◦ ∆ ⊆ e ◦ s . F or the conv erse, supp ose that ( f , u ) ∈ e ◦ s . Since f u ↓ we then hav e f u = f b y (U). Therefore, again by (U), there exists v ∈ U such that f u = v f . Now it follo ws from (F) that there exists g with u = g f and v = f g . Th us ( f , u ) = ( f , g f ) ∈ m ◦ (1 × i ) ◦ ∆.  Lemma 6. The r elation i is (the gr aph of ) a function. Pr o of. W e need to prov e that to each f ∈ X there is a unique g ∈ X with g f ∈ U 3 f g . W e already ha v e existence of suc h a g by Lemma 5, so it suffices to prov e unicit y . Supp ose that g f ∈ U 3 f g and g 0 f ∈ U 3 f g 0 . Then in particular f g ↓ and g f ↓ , so that by (A) also f g f ↓ and similarly f g 0 f ↓ . Now (U) implies f g f = f = f g 0 f , so that b y the previous conjecture g f = g 0 f . But then g = g f g = g 0 f g = g 0 .  RELA TIVE FROBENIUS ALGEBRAS ARE GROUPOIDS 5 Theorem 7. If m is a r elative F r ob enius algebr a, then G is a gr oup oid (in Set ). Pr o of. The proof consists of routine v erifications that the maps e, s, t, i indeed sat- isfy the axioms for a group oid. The most in teresting one has already b een dealt with in Lemma 5.  2.2. F rom group oids to relativ e F rob enius algebras. F or the rest of this subsection, fix a group oid G =    G 0 e / / G 1 i   s o o t o o G 1 × G 0 G 1 m o o    . Definition 8. F or a group oid G , define X = G 1 , and let m : G 1 × G 1  / / G 1 b e the graph of the function m . W e will pro v e that m is a relative F rob enius algebra. F or starters, it follows directly from asso ciativit y of comp osition in the group oid G that m satisfies (A). Lemma 9. The morphism m of Rel satisfies (U). Pr o of. Define a relation u : 1  / / X b y u = { ( ∗ , e ( x )) | x ∈ G 0 } . Then m ◦ ( u × 1) = { ( f , e ( x ) f ) | f ∈ G 1 , x = t ( f ) ∈ G 0 } = { ( f , et ( f ) f ) | f ∈ G 1 } = 1 . The symmetric condition also holds, and so (U) is satisfied.  Lemma 10. The morphism m of Rel satisfies (M). Pr o of. W e ha v e m ◦ m † = { ( f , f ) ∈ G 2 1 | ∃ g , h ∈ G 2 . s ( h ) = t ( g ) , f = hg } . Because w e can alwa ys take g = f and h = e ( t ( f )), this relation is equal to { ( f , f ) ∈ G 2 1 | f ∈ G 1 } = 1. Thus (M) is satisfied.  Lemma 11. The morphism m of Rel satisfies (F). Pr o of. First compute m † ◦ m = { (( a, b ) , ( c, d )) ∈ G 2 2 | ab = cd } , ( m × 1) ◦ (1 × m † ) = { (( a, b ) , ( c, d )) ∈ G 2 2 | ∃ e ∈ G 1 . ed = b, ae = c } . If ed = b and ae = c , then cd = aed = ab . Hence ( m × 1) ◦ (1 × m † ) ⊆ m † ◦ m . Con versely , suppose that (( a, b ) , ( c, d )) ∈ m † ◦ m . T aking e = bd − 1 , then ed = bdd − 1 = b , and ae = abd − 1 = cdd − 1 = c . Therefore m † ◦ m ⊆ (1 × m † ) ◦ ( m × 1). The symmetric condition is v erified analogously . Thus (F) is satisfied.  Theorem 12. If G is a gr oup oid, then m is a r elative F r ob enius algebr a.  6 CHRIS HEUNEN, IV AN CONTRERAS, AND ALBER TO S. CA TT ANEO 2.3. F unctorialit y. Notice that the constructions m 7→ G and G 7→ m of the previous t wo sections are each other’s inv erse. This subsection prov es that the assignmen ts extend to an isomorphism of categories under v arious choices of mor- phisms: one that is natural for group oids, one that is natural for relations, and one that is natural to F rob enius algebras. (See also [2].) W e start by considering a c hoice of morphisms that is natural from the point of view of relations: namely , morphisms b etw een group oids are subgroup oids of the pro duct. The category Rel is c omp act close d , i.e. allows morphisms η X : 1 → X × X satisfying ( η † × 1) ◦ (1 × η ) = 1 = (1 × η † ) ◦ ( η × 1). Drawing η as , this property graphically b ecomes the following. = = In fact, any relativ e F rob enius algebra induces such a compact structure on X , by η = m † ◦ u . There is a canonical choice of compact structure, that we use from no w on, namely η = { ( ∗ , ( x, x )) | x ∈ X } . It is induced by the relativ e F rob enius algebra corresp onding to the discrete group oid on X , and therefore has a univ ersal categorical characterization. It follows that morphisms r : X  / / Y in Rel ha v e transp oses p r q = (1 × r ) ◦ η = { ( ∗ , ( x, y )) | ( x, y ) ∈ r } : 1  / / X × Y . F urthermore, the dagger in Rel is compatible with the symmetric monoidal structure, giving a natural swap isomorphism σ : X × Y → Y × X with σ − 1 = σ † . Definition 13. The category F rob ( Rel ) rel has relative F rob enius algebras as ob- jects. A morphism ( X, m X ) → ( Y , m Y ) is a morphism r : X  / / Y in Rel satisfying ( m X × m Y ) ◦ (1 × σ × 1) ◦ ( p r q × p r q ) = p r q , (R) ( r × η † ) ◦ ( m † X × 1) ◦ ( u X × 1) = ( u † Y × 1) ◦ ( m Y × 1) ◦ ( r × η ) (I) These conditions translate into string diagrams as follows. r (R) = r r r (I) = r Prop osition 14. F rob ( Rel ) rel is a wel l-define d c ate gory. Pr o of. Clearly , identities 1 X = { ( x, x ) | x ∈ X } : X  / / X satisfy (R) and (I). Observ e that the comp osition (1 × η † ) ◦ ( m † × 1) ◦ ( u × 1) : X  / / X is the relation { ( x, x − 1 ) | x ∈ X } , where x − 1 is the inv erse of x when regarding X as the set of morphisms of a group oid as per Theorem 7. So (I) means that ( x, y ) ∈ r if and only if ( x − 1 , y − 1 ) ∈ r . No w, if r : X  / / Y and s : Y  / / Z satisfy (R) and (I), then so do es s ◦ r : p s ◦ r q = { ( ∗ , x 00 , z 00 ) ∈ 1 × X × Z | ∃ y 00 ∈ Y . ( x 00 , y 00 ) ∈ r , ( y 00 , z 00 ) ∈ s } = { ( ∗ , xx 0 , z z 0 ) | x, x 0 ∈ X, z , z 0 ∈ Z, ∃ y , y 0 ∈ Y . ( x, y ) ∈ r, ( x 0 , y 0 ) ∈ r , ( y , z ) ∈ s, ( y 0 , z 0 ) ∈ s } = ( m X × m Z ) ◦ (1 × σ × 1) ◦ ( p s ◦ r q × p s ◦ r q ) . Let us justify the second equation. If ( x 00 , y 00 ) ∈ r , then ( x 00− 1 , y 00− 1 ) ∈ r by (I), and hence (1 , 1) ∈ r by (R). Hence we ma y tak e x = x 00 , x 0 = 1, y = y 00 and y 0 = 1.  RELA TIVE FROBENIUS ALGEBRAS ARE GROUPOIDS 7 Definition 15. The c ategory Gp d rel has group oids as ob jects. Morphisms G → H are subgroup oids of G × H . That this is a well-defined category will follo w from the follo wing theorem. Iden- tities are the diagonal subgroup oids, and composition of subgroup oids R ⊆ G × G 0 and S ⊆ G 0 × G 00 is the group oid S 1 ◦ R 1 ⇒ S 0 ◦ R 0 . Theorem 16. Ther e is an isomorphism of c ate gories F rob ( Rel ) rel ∼ = Gp d rel . Pr o of. Let ( X, m X ) and ( Y , m y ) b e relativ e F rob enius algebras, inducing group oids G and H . First, notice that if r : X  / / Y satisfies (R), then m r = ( m X × m Y ) ◦ (1 × σ × 1) = { ((( a, b ) , ( c, d )) , ( ac, bd )) | a, b, c, d ∈ X } : r × r  / / r is a w ell-defined morphism in Rel . In fact, since ( X , m X ) and ( Y , m Y ) are relativ e F rob enius algebras, so is ( r, m r ): one readily v erifies that it satisfies (M), (A), and (F). Also, (U) is satisfied b y the in tersection 1  / / R of r with U X × U Y . Theorem 7 th us sho ws that r induces a group oid R . It is a subgroupoid of G × H : we hav e R 1 ⊆ ( G × H ) 1 b y construction, and if u ∈ U R , then u = u − 1 , so u ∈ ( G × H ) 0 . The structure maps of R are easily seen to b e restrictions of those of G × H . Con versely , if R is a subgroup oid of G × H , then clearly R 1 ⊆ X × Y is a morphism in Rel satisfying (R) and (I). It now suffices to observe that these con- structions are in verses.  Next, w e consider a c hoice of morphisms that is natural to group oids, namely functors. This entails dealing with functions. F ortunately , functions can b e c har- acterized among relations purely categorically . The category Rel is in fact a 2- category , where there is a single 2-cell r ⇒ s when r ⊆ s . Hence it makes sense to sp eak of adjoints of morphisms in Rel . A morphism has a righ t adjoint if and only if it is (the graph of ) a function. Definition 17. The category F rob ( Rel ) has relative F rob enius algebras as ob jects. Morphisms ( X, m X ) → ( Y , m Y ) are morphisms r : X  / / Y that satisfy (I) and preserv e the multiplication: r ◦ m X = m Y ◦ ( r × r ). r = r r W e write F rob ( Rel ) func for the sub category of all morphisms r that ha ve a right adjoin t and allow a 2-cell r ◦ u X ⇒ u Y . Lemma 18. Morphisms in F rob ( Rel ) satisfy (R). Henc e we have inclusions F rob ( Rel ) func  → F rob ( Rel )  → F rob ( Rel ) rel . 8 CHRIS HEUNEN, IV AN CONTRERAS, AND ALBER TO S. CA TT ANEO Pr o of. Let r : X  / / Y b e a morphism in F rob ( Rel ). Then ( m X × m Y ) ◦ (1 × σ × 1) ◦ ( p r q × p r q ) = ( m X × m Y ) ◦ (1 × r × r ) ◦ (1 × σ × 1) ◦ { ( ∗ , ( x, x, y , y )) | x, y ∈ X } = (1 × r ) ◦ ( m X × m X ) ◦ { ( ∗ , ( x, y , x, y )) | x, y ∈ X } = (1 × r ) ◦ { ( ∗ , ( xy , xy )) | xy ↓} = (1 × r ) ◦ { ( ∗ , ( z , z )) | z ∈ X } = p r q , b ecause we can alwa ys choose x = z and y = 1.  W rite Gp d for the category of group oids and functors. Theorem 19. Ther e is an isomorphism of c ate gories F rob ( Rel ) func ∼ = Gp d . Pr o of. Let ( X, m X ) and ( Y , m Y ) be relativ e F rob enius algebras, inducing groupoids G and H . Let r : m X → m Y b e a morphism in F rob ( Rel ) func . The condition that r has a right adjoint means it is in fact a function r : G 1 → H 1 . F urthermore, the condition r ◦ u X ⊆ u Y means precisely that it sends G 0 to H 0 . Finally , the condition that r preserve m ultiplication makes it functorial G → H , b ecause relational com- p osition of graphs coincides with comp osition of functions. Conv ersely , it is easy to see that a functor b etw een groupoids induces a morphism in F rob ( Rel ) func . Finally , these tw o constructions are inv erse to each other.  Corollary 20. The c ate gory Gp of gr oups and homomorphisms is isomorphic to the ful l sub c ate gory of F rob ( Rel ) func c onsisting of those r elative F r ob enius algebr as for which u has a right adjoint. Pr o of. The morphism u : 1  / / U has a right adjoint precisely when it is a function u : 1 → U and hence amounts to an elemen t of U . That is, the corresp onding group oid has a single iden tit y , i.e. is a group.  Finally , we can consider a choice of morphisms that is natural from the p oin t of view of F rob enius algebras, namely the category F rob ( Rel ). There is a category b et w een Gp d and Gpd rel , that corresp onds to the middle category in the sequence F rob ( Rel ) func  → F rob ( Rel )  → F rob ( Rel ) rel , as follo ws. Definition 21. A multi-value d functor G → H b etw een categories is a m ulti- v alued function F : G 1 → H 1 that preserves iden tities and comp osition: for g , f ∈ G 1 × G 0 G 1 : g ◦ f 3 h ⇒ F ( g ) ◦ F ( f ) 3 F ( h ) , for x ∈ G 0 : F ( e ( x )) 3 H 0 . W e denote the category of group oids and m ulti-v alued functors by Gp d mfunc . Theorem 22. Ther e is an isomorphism of c ate gories F rob ( Rel ) ∼ = Gp d mfunc . Pr o of. Let m X and m Y b e relative F rob enius algebras, inducing group oids G and H . Let r : m X → m Y b e a morphism in F rob ( Rel ); b y Theorem 16 it induces a subgroup oid of G × H . By the argument of the pro of of Theorem 19, r is a multi- v alued function G 1 → H 1 . But then it is precisely a multi-v alued functor.  RELA TIVE FROBENIUS ALGEBRAS ARE GROUPOIDS 9 In summary , we ha ve the following comm utativ e diagram of categories. F rob ( Rel ) func ∼ =     / / F rob ( Rel ) ∼ =     / / F rob ( Rel ) rel ∼ =   Gp d   / / Gp d mfunc   / / Gp d rel 3. Rela tive H*-algebras and semigr oupoids This section establishes a non-unital generalization of the corresp ondence of the previous section. F rob enius algebras are relaxed to the follo wing non-unital v ersion. Definition 23. A r elative H*-algebr a is a morphism m : X × X  / / X in Rel satisfying (M), (A), and (H) there is an inv olution ∗ : Rel (1 , X ) → Rel (1 , X ) such that m ◦ (1 × x ∗ ) = (1 × x † ) ◦ m † and m ◦ ( x ∗ × 1) = ( x † × 1) ◦ m † for all x : 1  / / X. x ∗ = x x ∗ = x In the presence of (U) and (A), condition (H) ab o ve is equiv alen t to (F) [1]. But in the absence of (U), it is stronger than (F), and means concretely that ∀ A ⊆ X ∀ x, y ∈ X  ∃ a ∈ A. xa = y ⇐ ⇒ ∃ a ∗ ∈ A ∗ . y a ∗ = x, ∃ a ∈ A. ax = y ⇐ ⇒ ∃ a ∗ ∈ A ∗ . a ∗ y = x. Equiv alently , ∀ a ∈ A ⊆ X ∀ x ∈ X  xa ↓ = ⇒ ∃ a ∗ ∈ A ∗ . xaa ∗ = x, ax ↓ = ⇒ ∃ a ∗ ∈ A ∗ . a ∗ ax = x. W e will prov e that relativ e H*-algebras are precisely semigroup oids that are regular and lo cally cancellative. Recall that a semigroup oid is a ‘category without iden tities’, just like a semigroup is a ‘non-unital monoid’ [3]. More precisely , a semigr oup oid consists of a diagram G 0 G 1 s o o t o o G 1 × G 0 G 1 m o o (in the category Set of sets and functions) such that m ◦ ( m × 1) = m ◦ (1 × m ). W e ma y also assume that s and t are jointly epic, i.e. G 0 = s ( G 1 ) ∪ t ( G 1 ). A semifunctor is then a ‘functor without preserv ation of iden tities’, i.e. a pair of functions f i : G i → G 0 i that commute with the ab o ve structure. A pseudoinverse of f ∈ G 1 is an element f ∗ ∈ G 1 satisfying ( s ( f ) = t ( f ∗ ) and t ( f ) = s ( f ∗ ) and) f = f f ∗ f and f ∗ = f ∗ f f ∗ . A semigroup oid is r e gular when ev ery f ∈ G 1 has a pseudoin v erse. Finally , a semigroup oid is lo c al ly c anc el lative when f hh ∗ = g h ∗ implies f h = g , and h ∗ hf = h ∗ g implies hf = g , for an y f , g , h ∈ G 1 and an y pseudoin v erse h ∗ of h . The following lemma shows that the asymmetry in the latter condition is only apparent. Lemma 24. A semigr oup oid is lo c al ly c anc el lative if and only if f h ∗ h = g h implies f h ∗ = g for any f , g , h ∈ G 1 and any pseudoinverse h ∗ of h . Pr o of. If h ∗ is a pseudoin verse of h , then h is a pseudoinv erse of h ∗ , to o.  10 CHRIS HEUNEN, IV AN CONTRERAS, AND ALBER TO S. CA TT ANEO Examples of lo cally cancellative semigroupoids are semigroup oids in whic h ev- ery morphism is b oth monic and epic. Clearly , groupoids are examples of lo cally cancellativ e regular semigroupoids. The following lemma gives a conv erse in the presence of iden tities. Lemma 25. If a lo c al ly c anc el lative r e gular semigr oup oids has identities, then it is a gr oup oid. Pr o of. Let f ∈ G 1 . By regularity , there is a pseudoinv erse f ∗ . In the definition of lo cal cancellativit y , take h = f ∗ and g = 1 t ( f ) , and h ∗ = f . Then f hh ∗ = f f ∗ f = f = 1 f = g h ∗ , and so f f ∗ = 1. By Lemma 24 similarly f ∗ f = 1. Thus f ∗ is an in verse of f , and therefore unique.  3.1. F rom relativ e H*-algebras to semigroup oids. F or the rest of this sub- section, fix a relative H*-algebra m : X × X  / / X . Definition 26. Define G by G 0 = { f ∈ X | m ( f , f ) = f } , G 1 = X, s = { ( f , f ∗ f ) | f ∗ is a pseudoinverse of f } : G 1  / / G 0 t = { ( f , f f ∗ ) | f ∗ is a pseudoinverse of f } : G 1  / / G 0 . Lemma 27. F or e ach element a in a r elative H*-algebr a ther e exists a ∗ ∈ { a } ∗ satisfying a ∗ aa ∗ = a ∗ and aa ∗ a = a . Pr o of. By (M), we ha ve ∀ y ∈ X ∃ a, x ∈ X. y = ax . Applying (H) with A = X gives ∀ a ∈ X ∃ x ∈ X . xa ↓ . No w let a ∈ X . If we substitute A = { a } , then (H) b ecomes ∀ x, y ∈ X  xa = y ⇐ ⇒ ∃ a ∗ ∈ { a } ∗ . y a ∗ = x  ∀ x, y ∈ X  ax = y ⇐ ⇒ ∃ a ∗ ∈ { a } ∗ . a ∗ y = x  As ab ov e, there exists x ∈ X with xa ↓ . So b y the first condition ab ov e, there is a 0 ∈ { a } ∗ with aa 0 ↓ . Hence, b y the second condition, there is a 00 ∈ { a } ∗ with a 00 aa 0 = a 0 . Applying the first condition again, now with x = a 0 and y = a 00 a , giv es a 0 a = a 00 a . Therefore w e hav e a ∗ = a ∗ aa ∗ for a ∗ = a 0 ∈ { a } ∗ . Finally , applying the first condition again, this time with x = aa ∗ and y = a , we find that also aa ∗ a = a .  Lemma 28. The data G form a wel l-define d semigr oup oid. Pr o of. By (A), the condition m ◦ ( m × 1) = m ◦ (1 × m ) is clearly satisfied. It remains to pro ve that m , s and t are well-defined functions. The former means that ( g, f ) ∈ G 1 × G 0 G 1 implies g f ↓ . Assume s ( g ) = t ( f ), i.e. g ∗ g = f f ∗ for some pseudoinv erses g ∗ and f ∗ of g and f . Because g ∗ g is idemp oten t, w e hav e g ∗ g f f ∗ = g ∗ g g ∗ g = g ∗ g ↓ , and therefore also g f ↓ . Hence m is well-defined. As for t , suppose that f ∗ and f 0 are b oth pseudoin verses of f , so that ( f , f f ∗ ) ∈ s and ( f , f f 0 ) ∈ s . Then f f ∗ f = f = f f 0 f . Set A = { f ∗ } , a = f ∗ , x = f , and y = f f 0 . By Lemma 27, we obtain f ∈ A ∗ , and so y a ∗ = x for a ∗ = f . No w it follo ws from (H) that f f ∗ = xa = y = f f 0 . Similarly , s is a w ell-defined function.  RELA TIVE FROBENIUS ALGEBRAS ARE GROUPOIDS 11 Theorem 29. If m is a r elative H*-algebr a, then G is a lo c al ly c anc el lative r e gular semigr oup oid. Pr o of. Regularit y is precisely Lemma 27. Supp ose that f hh ∗ = g h ∗ for a pseudoin- v erse h ∗ of h . Applying (H) to A = { h } , x = f hh ∗ , y = g , a = h and a ∗ = h ∗ yields f h = f hh ∗ h = xa ∗ = y = g . Hence G is lo cally cancellative.  3.2. F rom semigroup oids to relativ e H*-algebras. F or the rest of this sub- section, fix a semigroup oid G . Definition 30. Define X = G 1 , m = { ( g , f , g f ) | s ( g ) = t ( f ) } : G 1 × G 1  / / G 1 , A ∗ = { a ∗ ∈ X | a ∗ aa ∗ = a ∗ and aa ∗ a = a for all a ∈ A } . Theorem 31. If G is a lo c al ly c anc el lative r e gular semigr oup oid, then m is a r elative H*-algebr a. Pr o of. Clearly , (A) is satisfied. Because m † ◦ m = { ( f , f ) ∈ G 2 1 | ∃ ( g , h ) ∈ G 2 . f = hg } w e ha v e m † ◦ m ⊆ 1. Conv ersely , if f ∈ G 1 , setting g = f and h = f ∗ f for some pseudoin verse f ∗ of f , then f = g h . Hence (M) is satisfied. Finally , w e verify (H). Let A ⊆ X b e given, let a ∈ A and x ∈ X , and supp ose that xa ↓ . That means that s ( f ) = t ( a ). By regularit y , a has a pseudoinv erse a ∗ ∈ A ∗ , and we hav e xa = xaa ∗ a . Setting f = xa , g = x , h = a and h ∗ = a ∗ in the definition of local cancellativity yields xaa ∗ = x . The symmetric condition is v erified similarly . Hence (H) is satisfied.  3.3. F unctorialit y. This subsection prov es that the assignments m 7→ G and G 7→ m extend functorially to an adjunction. The following definitions give well-defined categories, just as in Subsection 2.3. Condition (I) has to b e adapted to the non- unital setting of H*-algebras, and b ecomes the following. y ◦ r ◦ x = y ∗ ◦ r ◦ x ∗ for all x : 1  / / X, y : 1  / / Y (I’) Concretely , this means that ( x, y ) ∈ r if and only if ( x ∗ , y ∗ ) ∈ r for any pseudoin- v erses x ∗ of x and y ∗ of y . Definition 32. Relativ e H*-algebras can be made into the ob jects of sev eral categories. A morphism ( X , m X ) → ( Y , m Y ) in Hstar ( Rel ) rel is a morphism r : X  / / Y in Rel satisfying (R) and (I’). A morphism in Hstar ( Rel ) is a mor- phism r : X  / / Y in Rel that satisfies (I’) and preserves m ultiplication: r ◦ m X = m Y ◦ ( r × r ). A morphism in Hstar ( Rel ) func is a morphism in Hstar ( Rel ) that additionally has a right adjoin t. Definition 33. Lo cally cancellative regular semigroup oids can be made in to the ob jects of sev eral categories. Morphisms in LRSgp d are semifunctors. Morphisms in G → H in LRSgp d mfunc are multi-v alued semifunctors, i.e. multi-v alued func- tions G i → H i satisfying only the first condition of Definition 21. Morphisms in LRSgp d rel are lo cally cancellativ e regular subsemigroup oids of G × H . 12 CHRIS HEUNEN, IV AN CONTRERAS, AND ALBER TO S. CA TT ANEO Prop osition 34. The assignments m 7→ G and G 7→ m extend to functors Hstar ( Rel ) rel  LRSgp d rel , Hstar ( Rel )  LRSgp d mfunc , Hstar ( Rel ) func  LRSgp d . Pr o of. W e pro ve the case Hstar ( Rel ) rel  LRSgp d rel . Let ( X , m X ) and ( Y , m Y ) b e relative H*-algebras, inducing lo cally cancellative regular semigroupoids G and H . Given r : m X → m Y , define m r : r × r  / / r as in the pro of of Theorem 16; it satisfies (A) and (M). It also satisfies (H), as w e no w v erify . F or A ⊆ r , take A ∗ = { ( x ∗ , y ∗ ) | ( x, y ) ∈ A, x ∗ ∈ { x } ∗ , y ∗ ∈ { y } ∗ } . (1 × A ) ◦ m † r = { (( x, y ) , ( a, b )) ∈ r × r | ∃ ( c, d ) ∈ A . y = bd, x = ac } (H) = { (( x, y ) , ( a, b )) ∈ r × r | ∃ ( c, d ) ∈ A, c ∗ ∈ { c } ∗ , d ∗ ∈ { d } ∗ . a = xc ∗ , b = y d ∗ } = m r ◦ (1 × A ∗ ) . Theorem 29 no w shows that m r induces a subsemigroup oid of G × H . Conv ersely , if R is a subsemigroupoid of G × H , then R 1 : G 1  / / H 1 clearly satisfies (R) and (I’). Finally , the identit y relation r : m X  / / m Y corresp onds to the diagonal sub- semigroup oid, which is indeed regular and lo cally cancellativ e. These constructions clearly restrict to the sub categories of the statement.  Theorem 35. The functors fr om the pr evious pr op osition form adjunctions. LRSgp d rel / / ⊥ Hstar ( Rel ) rel o o LRSgp d mfunc / / ⊥ Hstar ( Rel ) o o LRSgp d / / ⊥ Hstar ( Rel ) func o o Pr o of. Starting with a relative H*-algebra m : X × X  / / X , we end up with { ( g , f , g f ) | ∃ g ∗ ∈ { g } ∗ ∃ f ∗ ∈ { f } ∗ . g ∗ g = f f ∗ } : X × X  / / X. Clearly this is a subrelation of m , and the inclusion forms the unit of the adjunction. Starting with a lo cally cancellativ e regular semigroup oid G , w e end up with { f ∈ G 1 | f 2 = f } G 1 s 0 o o t 0 o o G 1 × s 0 ,t 0 G 1 m o o where s 0 ( f ) = f ∗ f and t 0 ( f ) = f f ∗ . Clearly , the original G maps into this, giv- ing the counit of the adjunction. Naturalit y and the triangle equations are easily c heck ed for LRSgp d rel  Hstar ( Rel ) rel . Because the unit and counit are func- tions, the statemen t also holds for the other sub categories.  Prop osition 36. The lar gest sub c ate gories making the adjunctions of The or em 35 into e quivalenc es ar e Gp d rel and F rob ( Rel ) rel , and their variations. In that c ase, the e quivalenc es ar e in fact isomorphisms. Pr o of. Consider the counit of the proof of Theorem 35. It is an isomorphism pre- cisely when G 0 coincides with the idemp otents of G 1 . But then the unique idemp o- ten t on x ∈ G 0 is an iden tity , and G is a group oid by Lemma 25. In other words, Gp d and its v ariations are the largest sub categories of LRSgp d and its v ariations turning the adjunctions into reflections. RELA TIVE FROBENIUS ALGEBRAS ARE GROUPOIDS 13 Next consider the unit of the adjunctions. It is an isomorphism when g f ↓ implies g ∗ g = f f ∗ for some pseudoinv erses f ∗ ∈ { f } ∗ and g ∗ ∈ { g } ∗ . In that case w e can define a unit for the H*-algebra ( X , m ) b y { u ∈ X | u = u ∗ u } , for if uf ↓ and u = u ∗ u then uf = u ∗ uf = f f ∗ f = f . But recall that unital relativ e H*-algebras are relative F rob enius algebras. In other w ords, F rob ( Rel ) and its v ariations are the largest sub categories of Hstar ( Rel ) and its v ariations turning the ab ov e adjunctions into reflections.  4. Groupoids and semigroupoids The forgetful functor Gp d → Cat has a left adjoint, that freely adds inv erses. Similarly , the forgetful functor Cat → Sgp d to the category of semigroup oids and semifunctors has a left adjoint, that freely adds identities. The image of the latter left adjoin t consists precisely of those categories in which the only isomorphisms are identities. Hence there is a functor Sgp d → Gp d giving the free groupoid on a semigroup oid. Restricting it giv es a functor that turns a locally cancellative regular semigroup oid into a group oid. LRSgp d / / ⊥ Gp d ? _ o o The morphisms in these categories are (semi)functors. This section establishes righ t adjoin ts to the inclusion Gpd rel  → LRSgp d rel and its v ariations with other choices of morphisms. This is then applied to obtain adjunctions betw een HStar ( Rel ) rel and F rob ( Rel ) rel (and their v ariations). The idea in building the righ t adjoin t is to iden tify all idemp oten ts: a group is a regular semigroup with a single idemp otent. Definition 37. F or a semigroupoid G , define ∼ as the congruence (see [5, Sec- tion I I.8]) generated by f ∼ g when s ( f ) = s ( g ) and f 2 = f and g 2 = g . Set G 0 0 = G 0 , s 0 ([ f ]) = s ( f ) , m 0 ([ g ] , [ f ]) = [ m ( g , f )] , G 0 1 = G 1 / ∼ , t 0 ([ f ]) = t ( f ) . Lemma 38. If G is a (lo c al ly c anc el lative r e gular) semigr oup oid, then G 0 =  G 0 0 G 0 1 s 0 o o t 0 o o G 0 1 × G 0 0 G 0 1 m 0 o o  is again a wel l-define d (lo c al ly c anc el lative r e gular) semigr oup oid. Pr o of. Because ∼ is a congruence, m 0 is associative [5, Prop osition II.8.1]). Because s and t are join tly epic, so are s 0 and t 0 . Hence G 0 is a semigroupoid. If G is regular, then [ f ∗ ] is a pseudoin verse for [ f ] ∈ G 0 1 , where f ∗ is any pseudoinv erse of f in G 1 , and so G 0 is regular. Finally , it is easy to see that G 0 inherits lo cal cancellativity from G using that ∼ is a congruence.  Lemma 39. If the semigr oup G is lo c al ly c anc el lative and r e gular, then F ( G ) =    G 0 0 e 0 / / G 0 1 i 0   s 0 o o t 0 o o G 0 1 × G 0 0 G 0 1 m 0 o o    is a wel l-define d gr oup oid, wher e e 0 ( s ( f )) = [ f ∗ f ] , e 0 ( t ( f )) = [ f f ∗ ] , i 0 ([ f ]) = [ f ∗ ] . 14 CHRIS HEUNEN, IV AN CONTRERAS, AND ALBER TO S. CA TT ANEO Pr o of. Because G is regular, it makes sense to sp eak about f ∗ . Because G 0 0 = Im( s 0 ) ∪ Im( t 0 ), the ab o ve prescription completely defines e 0 . Finally , e 0 is well- defined, for if s ( f ) = s ( g ), then f ∗ f ∼ g ∗ g b ecause both are idemp otent. Similarly , if s ( f ) = t ( g ), then f ∗ f ∼ g g ∗ . W e now sho w that i is a well-defined function. Suppose that g and h are pseudo- in verses of f . Then g f and hf are idemp otent. Also s ( gf ) = s ( hf ) = t ( g f ) = t ( hf ), so [ g f ] = [ hf ]. But then [ g ] = [ h ] by local cancellativity of G 0 . By construction, these data mak es F ( G ) into a group oid.  Prop osition 40. The assignment G 7→ F ( G ) of the pr evious lemma extends to functors F rel : LRSgp d rel → Gp d rel , F mfunc : LRSgp d mfunc → Gp d mfunc , and F : LRSgp d → Gp d . Pr o of. Let R be a morphism G → H in LRSgp d rel . Then it is subsemigroup oid of G × H , and hence a semigroupoid in its own right. Hence we can define F rel ( R ) as in the previous lemma. It is easy to see that F rel ( R ) is a subsemigroupoid of F rel ( G ) × F rel ( H ). Finally , it clear that F rel preserv es identities and comp osition, and restricts to give functors F mfunc and F .  Theorem 41. The inclusion Gpd  → LRSgp d has F as a right adjoint. The inclusion Gp d mfunc  → LRSgp d mfunc has F mfunc as a right adjoint. The inclusion Gp d rel  → LRSgp d rel has F rel as a right adjoint.. Pr o of. W e start by exhibiting the unit of the adjunctions. Let G b e a lo cally cancellativ e regular semigroupoid. Then G 0 = F ( G ) 0 , and there is a pro jection function G 1  ( G 1 / ∼ ) = F ( G ) 1 . By construction of s 0 , t 0 and m 0 , this induces a semifunctor G → F ( G ), and hence a subsemigroup oid of G × F ( G ). Because G is lo cally cancellative and regular itself, this subsemigroupoid is a w ell-defined morphism G → F ( G ) in LRSgp d rel as w ell as in LRSgp d mfunc and LRSgp d . It is easy to see that this is natural in G . As for the counit, notice that if G is a group oid, then G 1 ∼ = ( G 1 / ∼ ). So the subsemigroup oid of G × F ( G ) is in fact a groupoid, and hence gives a well-defined morphism F ( G ) → G in Gp d rel , Gp d mfunc and Gp d , that is natural in G . One readily v erifies that this unit and counit satisfy the triangle equations.  Th us the functor F pro vides a universal w ay to pass from lo cally regular semi- group oids to groupoids. Restriction to the one-ob ject case shows that collapsing all idemp otents turns a lo cally cancellative regular semigroup in to a group. Corollary 42. Ther e ar e adjunctions Hstar ( Rel ) func   / /   Hstar ( Rel )   / /   Hstar ( Rel ) rel   F rob ( Rel ) func   / / O O a F rob ( Rel )   / / O O a F rob ( Rel ) rel O O a Explicitly, the right adjoints map a r elative H*-algebr a ( X , m X ) to ( X / ∼ , m 0 ) , wher e ∼ is the e quivalenc e r elation gener ate d by f ∼ g if f 2 = f and g 2 = g and g f ↓ , and m 0 ([ g ] , [ f ]) = [ m ( g , f )] . RELA TIVE FROBENIUS ALGEBRAS ARE GROUPOIDS 15 Pr o of. Simply comp ose the following adjunctions, and similarly for the other choices of morphisms. F rob ( Rel ) rel / / ∼ = Gp d rel o o   / / ⊥ LRSgp d rel F rel o o / / ⊥ Hstar ( Rel ) rel o o Applying Definitions 26, 37, and 8 in order to a giv en relative H*-algebra results in the relative F rob enius algebra of the statement.  References [1] Samson Abramsky and Chris Heunen. H*-algebras and nonunital F rob enius algebras: first steps in infinite-dimensional categorical quan tum mechanics. Clifford Le ctur es, AMS Pr o c e e d- ings of Symp osia in Applie d Mathematics , 2011. [2] Bob Coeck e, ´ Eric P aquette, and Du ˇ sko Pa vlo vi´ c. Semantic te chniques in quantum c omputa- tion , c hapter Classical and quantum structuralism, pages 29–69. Cambridge Universit y Press, 2010. [3] John M. Ho wie. F undamentals of semigroup the ory . London Mathematical Society mono- graphs. Oxford Universit y Press, 1995. [4] Joachim Kock. F r ob enius algebras and 2-D T opolo gic al Quantum Field The ories . Number 59 in London Mathematical Society Student T exts. 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Dep ar tment of Computer Science, University of Oxf ord, Wolfson Building, P arks Rd, OX1 3QD, O xford, United Kingdom E-mail address : heunen@cs.ox.ac.uk Institut f ¨ ur Ma thema tik, Universit ¨ at Z ¨ urich, Winter thurerstrasse 190, CH-8057, Z ¨ urich, Switzerland E-mail address : ivan.contreras@math.uzh.ch E-mail address : alberto.cattaneo@math.uzh.ch