Bicategories of spans as cartesian bicategories

BICA TEGORIES OF SP ANS AS CAR TESIAN BICA TEGORIES STEPHEN LA CK, R.F.C W AL TERS, AND R.J. W OOD A BSTRACT . Bicategor ies of spans are characterize d as cartesian bicateg o ries in which every comonad ha s a n Eilenber g -Mo ore ob ject and every left adjoint ar row is comona dic . 1. In tro duction Let E b e a category with finite limits. F or the bicategory Span E , the lo cally full subbi- category MapSpan E determined b y the left a djoin t arro ws is essen tially lo cally discrete, meaning that eac h hom category Map Span E ( X , A ) is an equiv alence relation, and so is equiv alen t to a discrete category . Indeed, a span x : X o o S / / A : a has a right a djoin t if and only if x : S / / X is inv ertible. The functors MapSpan E ( X , A ) / / E ( X , A ) giv en b y ( x, a ) 7→ ax − 1 pro vide equiv alences of categories whic h are the effects on homs f o r a biequiv a lence MapSpan E / / E . Since E ha s finite pro ducts, MapSpan E has finite pro ducts as a bic ate gory . W e refer the reader to [CKWW] for a thorough treatment of bicategor ies with finite pro ducts. Eac h hom cat ego ry Span E ( X , A ) is iso morphic to t he s lice category E / ( X × A ) whic h has bina r y pro ducts giv en by pullbac k in E and terminal ob ject 1 : X × A / / X × A . Th us Span E is a pr e c artesian bic ate gory in the sense of [CKWW]. The canonical lax monoidal structure Span E × Span E / / Span E o o 1 for this precartesian bicategory is seen to hav e its bina r y asp ect give n on arrows b y ( X x ← − S y − → A , Y y ← − T b − → B ) 7→ ( X × Y x × y ← − − S × T a × b − − → A × B ) , and its n ullary aspect pro vided b y 1 1 ← − 1 1 − → 1 , the terminal ob ject of Span E (1 , 1) . Both of these lax functors are readily seen to be pseudofunctors so that Span E is a c artesian bic ate gory as in [ CKWW ]. The authors gra tefully ac knowledge financial supp ort fr o m the Australian AR C, the Italian CNR and the Canadian NSERC. Diagrams t yp eset using M. Ba rr’s diagram pack age, diagxy .tex. 2000 Mathematics Sub ject Classificatio n: 1 8 A25. Key words and phrases: bicatego ry , finite pro ducts, discrete, comonad, Eilenber g-Mo ore ob ject. c  Stephen Lack, R.F.C W a lters, and R.J. W o o d, 20 09. Permission to cop y for priv ate use granted. 1 2 The purp ose o f this pap er is to c haracterize those cartesian bicategories B whic h are biequiv alen t to Span E , fo r some catego ry E with finite limits. Certain asp ects of a solution to the problem are immediate. A biequiv a lence B ∼ Span E pro vide s Map B ∼ MapSpan E ∼ E so that w e mus t ensure firstly tha t Map B is essen tially lo cally discrete. F ro m the c harac- terization of bicategories of relations as lo cally ordered cartesian bicategor ies in [C&W ] one suspects that the f ollo wing axiom will figure prominen tly in pro viding essen tia l lo cal discretenes s for Map B . 1.1. Axiom. F ro b enius: A c artesian bic ate gory B is said to satisfy the F rob enius axiom if, for e ach A in B , A is F r ob enius. Indeed F rob enius ob jects in cartesian bicategories w ere defined and studied in [W&W] where amongst other things it is sho wn that if A is F ro b enius in cartesian B then, for all X , Map B ( X , A ) is a group oid. (This theorem w as generalized considerably in [LSW] whic h explained further asp ects of the F rob enius concept.) How eve r, essen tial lo cal discreteness for Map B r equires also that the Map B ( X , A ) b e ordered sets (whic h is a utomatic fo r lo cally or dered B ). Here w e study also sep ar abl e ob j ects in cartesian bicategor ies for which w e are able to sho w that if A is separable in cartesian B then, for all X , Map B ( X , A ) is an ordered set and a candidate axiom is: 1.2. Axiom. Separable: A c artesian bic ate gory B is said to satisfy the Separable axiom if, for e ach A in B , A is sep ar able. In addition to essen tial lo cal discreteness , it is clear tha t w e will need an a xiom whic h pro vides tabulation o f each arro w o f B b y a span of maps. Sin ce existence of Eilen berg- Mo ore ob jects is a basic 2 -dimensional limit concept, we will express tabulation in terms of t his requiremen t; w e note that existence of pullbac ks in Map B follows easily from tabulation. In the bicategor y Span E , the comonads G : A / / A are precisely the symmetric spans g : A o o X / / A : g ; the map g : X / / A together with g η g : g / / g g ∗ g provide s an Eilen b erg - Mo ore coalgebra for g : A o o X / / A : g . W e will p osit: 1.3. Axiom. Eilen b erg - Mo ore for Comonads: Each c om onad ( A, G ) in B has an Eilenb er g-Mo or e obje ct. Con v ersely , a n y map (left adjoin t) g : X / / A in Span E pro vides an Eilen berg- Mo ore ob ject for the comonad g g ∗ . W e further p osit: 1.4. Axiom. Maps are Comonadic: Each lef t adjoint g : X / / A in B is c omonadic. from whic h, in our con text, w e can also deduc e the F rob enius and Separable axioms. In fact we shall also giv e, in Prop osition 3.1 b elow, a straig htforw ard pro of that Map B is lo cally essen tially discrete whenev er Axiom 1.4 holds. But due t o the imp ortance of the F ro b enius and separabilit y conditions in other con texts, w e ha v e c hosen to analyze them in their o wn righ t. 3 2. Preli m inar i es W e recall from [CKWW ] tha t a bicategory B (alw a ys, for con v enience, assumed to b e normal) is said to b e c artesian if the subbicategory of maps (b y which w e mean left adjo int arro ws), M = Map B , has finite pro ducts − × − and 1; eac h hom-category B ( B , C ) has finite pro ducts − ∧ − and ⊤ ; and a certain deriv ed tensor pro duct − ⊗ − a nd I on B , extending the pro duct structure of M , is f unctoria l. As in [CKWW], we write p and r for the first and second pro jections at the globa l lev el, and similarly π and ρ f o r the pro jections at the lo cal lev el. If f is a map of B — an arrow of M — w e will write η f , ǫ f : f ⊣ f ∗ for a chose n adjunction in B that mak es it so. It was show n that the deriv ed tensor pro duct of a cartesian bicategory underlies a symmetric monoidal bicategory structure. W e recall to o that in [W&W] F rob enius ob j ects in a general cartesian bicategory w ere defined and studied. W e will need the central results of that paper to o. Throughout this pap er, B is assumed to b e a cartesian bicatego r y . As in [CKWW] w e write M M G = Gro B M ∂ 0                G = Gro B M ∂ 1   ? ? ? ? ? ? ? ? ? ? ? ? ? for the Grothendiec k span corresponding to M op × M i op × i / / B op × B B ( − , − ) / / CA T where i : M / / B is the inclusion. A typical arr o w of G , ( f , α , u ) : ( X , R, A ) / / ( Y , S, B ) can b e depicted b y a square A B u / / X A R   X Y f / / Y B S   α / / (1) and suc h arrow s are comp osed b y pasting. A 2 -cell ( φ, ψ ) : ( f , α , u ) / / ( g , β , v ) in G is a pair of 2-cells φ : f / / g , ψ : u / / v in M whic h satisfy the obv ious equation. The (strict) pseudofunctors ∂ 0 and ∂ 1 should b e regarded as domain and c o domain resp ectiv ely . Th us, applied to (1), ∂ 0 giv es f and ∂ 1 giv es u . The bicategory G also has finite pro ducts, whic h are giv en on ob jects b y − ⊗ − and I ; these are preserv ed b y ∂ 0 and ∂ 1 . The Grothendiec k span can also b e thoug h t of as giving a double category (of a suitably w eak fla v our), although w e shall not emphasize that po int of view. 2.1. The arrows of G are particularly we ll suited to relating the v arious pro duct structures in a cartesian bicategory . In 3.3 1 o f [CKWW ] it was shown tha t the lo cal 4 binary pro duct, for R, S : X / / / / A , can b e recov ered to within isomorphism fr o m the defined tensor pro duct b y R ∧ S ∼ = d ∗ A ( R ⊗ S ) d X A sligh tly more precis e v ersion of this is that the mate of t he isomorphism ab ov e, with resp ect to the single adjunction d A ⊣ d ∗ A , defines an arro w in G A A ⊗ A d A / / X A R ∧ S   X X ⊗ X d X / / X ⊗ X A ⊗ A R ⊗ S   / / whic h when comp osed with the pro jections of G , recov ers the lo cal pro jections as in A A ⊗ A d A / / X A R ∧ S   X X ⊗ X d X / / X ⊗ X A ⊗ A R ⊗ S   / / A ⊗ A A p A,A / / X ⊗ X A ⊗ A R ⊗ S   X ⊗ X X p X,X / / X A R   ˜ p R,S / / ∼ = A A 1 A / / X A R ∧ S   X X 1 X / / X A R   π / / for the first pro jection, a nd similarly for the second. The unsp ecified ∼ = in G is giv en b y a pair of con v enien t isomorphisms p X,X d X ∼ = 1 X and p A,A d A ∼ = 1 A in M . Similarly , when R ∧ S / / R ⊗ S is comp osed with ( r X,X , ˜ r R,S , r A,A ) the r esult is (1 X , ρ, 1 A ) : R ∧ S / / S . 2.2. Quite generally , an a r r ow of G as giv en b y the square (1) will b e called a c ommutative square if α is inv ertible. An arrow of G will b e said to satisfy the Be ck c o n dition if the mate of α under the adjunctions f ⊣ f ∗ and u ⊣ u ∗ , as give n in the square b elo w (no longer an arro w of G ), is inv ertible. A B o o u ∗ X A R   X Y o o f ∗ Y B S   α ∗ / / Th us Prop osition 4.7 of [CKWW] sa ys that pro jection squares of the form ˜ p R, 1 Y and ˜ r 1 X ,S are commutativ e while Prop osition 4.8 of [CKWW] say s that these same squares satisfy the Bec k condition. If R and S a re also maps and α is inv ertible then α − 1 giv es rise to another arro w of G , from f to u with reference to the square ab o v e, whic h may or may not satisfy the Bec k condition. The p o in t here is that a comm utativ e square of ma ps giv es rise to t w o, generally distinct, Bec k conditions. It is w ell kno wn that, for bicategories of the form Span E and Rel E , all pullbac k squares of maps satisfy b oth Bec k conditions. A 5 category with finite pro ducts has automatically a num b er of pullbac ks whic h w e might call pr o duct-absolute pullbac ks b ecause they are preserv ed by all functors whic h preserv e pro ducts. In [W&W] t he Bec k conditions for the pro duct-absolute pullbac k squares of the form A A × A d / / A × A A O O d A × A A × A × A d × A / / A × A × A A × A O O A × d w ere in v estigated. (In fact, in this case it was sho wn that either Bec k condition implies the other.) The ob jects for whic h these conditions are met are called F r ob enius ob jects. 2.3. Proposition. F or a c artesian bic ate gory, the ax iom Maps are Comonadic implies the ax i o m F rob enius . Proo f. It suffices to sho w that the 2-cell δ 1 b elo w is in v ertible: A ⊗ A A ⊗ ( A ⊗ A ) o o 1 ⊗ d ∗ A A ⊗ A d   A A ⊗ A o o d ∗ A ⊗ A A ⊗ ( A ⊗ A ) A ⊗ A ( A ⊗ A ) ⊗ A d ⊗ 1   ( A ⊗ A ) ⊗ A A ⊗ ( A ⊗ A ) a   δ 1 / / A A ⊗ A o o d ∗ A ⊗ A A r   A ⊗ A A ⊗ ( A ⊗ A ) o o 1 ⊗ d ∗ A ⊗ ( A ⊗ A ) A ⊗ A r   ˜ r 1 A ,d ∗ / / The paste comp osite of the squares is in v ertible (b eing essen tially the identit y 2- cell on d ∗ ). The lo w er 2- cell is in v ertible b y Prop osition 4 .7 of [CKWW] so that the whisk er comp osite r δ 1 is in v ertible. Since r is a map it reflects isomorphisms, b y Maps are Comonadic, and hence δ 1 is in v ertible. 2.4. Remark. It w as sho wn in [W&W] that, in a cartesian bicategory , the F rob enius ob jects are closed under finite pro ducts. It follows that the full subbicategory of a cartesian bicategory determined by the F rob enius ob jects is a cartesian bicategory whic h satisfies the F rob enius axiom. 3. Separabl e Ob ject s and Discret e Ob ject s i n Cart esi an Bica tegor i es In this section w e lo ok at separability fo r ob jects of cartesian bicategories. Since fo r an ob ject A whic h is b oth separable and F rob enius, the hom-category Map B ( X , A ) is essen tia lly discrete, for all X , we shall t hen b e able to sho w that Map B is essen tially discrete by show ing that all ob jects in B are separable and F rob enius. But first w e record the follo wing direct argumen t: 6 3.1. Proposition. If B is a bic ate gory in which al l m aps ar e c omonadic and Map B has a terminal obje ct, then Map B is lo c al ly e ssential ly disc r ete. Proo f. W e m ust show that for all ob jects X and A , the hom-category Map B ( X , A ) is essen tia lly discrete. As usual, w e write 1 fo r t he terminal ob j ect of Map B and t A : A / / 1 for the ess en tially unique map, whic h b y assumption is comonadic. L et f , g : X / / A b e maps from X to A . If α : f / / g is an y 2-cell, then t A α is in v ertible, since 1 is terminal in Map B . But since t A is comonadic, it reflects isomorphisms, and so α is in v ertible. F urthermore, if β : f / / g is another 2-cell, then t A α = t A β by the univ ersal prop ert y of 1 once a gain, and now α = β since t A is faithful. Thus t here is at most o ne 2 - cell f r om f to g , and an y suc h 2-cell is in v ertible. In a n y (bi)category with finite pro ducts the diagonal ar r o ws d A : A / / A × A are (split) monomorphisms so that in the bicategory M the f o llo wing square is a pro duct-absolute pullbac k A A ⊗ A d A / / A A 1 A   A A 1 A / / A A ⊗ A d A   that giv es rise to a single G arrow . 3.2. Definition. A n obje ct A in a c artesia n bic ate gory is said to b e separable if the G arr ow ab ove satisfies the Be ck c ondition. Of course the in v ertible mate condition here sa ys precisely that the unit η d A : 1 A / / d ∗ A d A for the adjunction d A ⊣ d ∗ A is in v ertible. Thu s Axiom 1.2, as stated in the In tro duction, sa ys that, for all A in B , η d A is in v ertible. 3.3. Remark. F or a map f it mak es sense t o define f is ful ly faithful to mean that η f is in v ertible. F or a c ate gory A the diagonal d A is fully faithful if a nd only if A is an ordered set. 3.4. Proposition. F or an obje ct A in a c a rtesia n bic ate gory, the fol lowin g ar e e quiva l e n t: i ) A is s ep ar able; ii ) for al l f : X / / A in M , the diagr am f o o f / / f is a pr o duct in B ( X , A ) ; iii ) 1 A o o 1 A / / 1 A is a pr o duct in B ( A, A ) ; iv ) 1 A / / ⊤ A,A is a monomorphi s m in B ( A, A ) ; v ) for al l G / / 1 A in B ( A, A ) , the diagr am G o o G / / 1 A is a pr o duct in B ( A, A ) . 7 Proo f. [ i ) = ⇒ ii )] A lo cal pro duct of maps is not generally a map but here w e ha v e: f ∧ f ∼ = d ∗ A ( f ⊗ f ) d X ∼ = d ∗ A ( f × f ) d X ∼ = d ∗ A d A f ∼ = f [ ii ) = ⇒ iii )] is trivial. [ iii ) = ⇒ i )] Note the use of pseud o-functoria lit y of ⊗ : d ∗ A d A ∼ = d ∗ A 1 A ⊗ A d A ∼ = d ∗ A (1 A ⊗ 1 A ) d A ∼ = 1 A ∧ 1 A ∼ = 1 A [ iii ) = ⇒ iv )] T o sa y that 1 A o o 1 A / / 1 A is a pro duct in B ( A, A ) is precisely to say that 1 A ⊤ A,A / / 1 A 1 A 1 1 A   1 A 1 A 1 1 A / / 1 A ⊤ A,A   is a pullbac k in B ( A, A ) whic h in turn is precisely to sa y t hat 1 A / / ⊤ A,A is a monomor- phism in B ( A, A ) [ iv ) = ⇒ v )] It is a generalit y that if an ob ject S in a category is subterminal then for an y G / / S , necessarily unique, G o o G / / S is a pro duct diagram. [ v ) = ⇒ iii )] is trivial. 3.5. Coro llar y. [Of iv )] F or a c artesian bic ate gory, the axiom Maps a re Comonadic implies the axiom Separable . Proo f. W e hav e ⊤ A,A = t ∗ A t A for the map t A : A / / 1. It fo llo ws that the unique 1 A / / t ∗ A t A is η t A . Since t A is comonadic, η t A is the equalize r shown : 1 A η t A / / t ∗ A t A t ∗ A t A η t A / / η t A t ∗ A t A / / t ∗ A t A t ∗ A t A and hence a monomorphism. 3.6. Coro llar y. [Of iv )] F or sep ar able A in c artesian B , an arr ow G : A / / A ad m its at most on e c op oint G / / 1 A dep ending up on whether the unique arr ow G / / ⊤ A,A factors thr ough 1 A / / / / ⊤ A,A . 3.7. Proposition. In a c artesian bic ate gory, the sep ar able ob j e cts ar e close d under finite pr o ducts. Proo f. If A and B are separable ob jects then applying the homomor phism ⊗ : B × B / / B w e hav e an adjunction d A × d B ⊣ d ∗ A ⊗ d ∗ B with unit η d A ⊗ η d B whic h b eing an isomorph of the adjunction d A ⊗ B ⊣ d ∗ A ⊗ B with unit η d A ⊗ B (via middle-four in terc hange) shows that the separable ob jects are closed under binar y pro ducts. O n the other hand, d I is an equiv alence so that I is also separable. 8 3.8. Coro llar y. F or a c artesian bic ate gory, the ful l subbic ate gory determine d by the sep ar able obj e cts is a c artesian bic ate gory which sa tisfi e s the axiom Separable . 3.9. Proposition. If A is a sep a r able o b je ct in a c artesian bic ate gory B , then, for al l X in B , the hom-c ate go ry M ( X , A ) is an o r der e d s et, me aning that the c ate gory structur e forms a r eflexive, tr ans i tive r elation. Proo f. Supp o se that w e ha v e arro ws α, β : g / / / / f in M ( X, A ). In B ( X , A ) w e hav e f f g f α                 g f β   ? ? ? ? ? ? ? ? ? ? ? ? ? ? g f ∧ f γ   f ∧ f f π o o f ∧ f f ρ / / By Prop osition 3.4 w e can tak e f ∧ f = f and π = 1 f = ρ so that w e ha v e α = γ = β . It follo ws that M ( X, A ) is an ordered set. 3.10. Definition. A n obje ct A in a c artesian bic ate go ry is sa i d to b e discrete if it is b oth F r ob enius and sep ar able. W e write Dis B for the ful l s ubbic ate gory of B determine d by the discr ete obje cts. 3.11. Remark. Bew ar e that this is quite differen t to the notion of discreteness in a bicatego ry . An ob ject A of a bicategory is discrete if each hom-category B ( X , A ) is discrete; A is essen tially discrete if eac h B ( X , A ) is equiv alen t to a discrete category . The notion of discreteness for cartesian bicategories defined ab ov e turns out to mean that A is esse n tially discrete in the bicategory Map B . F ro m Prop osition 3.7 ab ov e and Prop osition 3.4 of [W&W] w e immediately ha v e 3.12. Proposition. F or a c artesian bic ate gory B , the ful l subbic ate gory Dis B of discr ete obje cts is a c artesian bic ate gory in which every obj e ct is discr ete. And from Prop osition 3.9 ab ov e and Theorem 3.13 of [W&W] w e hav e 3.13. Proposition. If A is a disc r ete obje c t in a c artesian b i c ate go ry B then, for al l X in B , the hom c a te gory M ( X , A ) i s an e quivalenc e r elation. If b oth the F r ob enius axiom o f [W&W ] and the Sep ar able axiom of this pap er hold for our cartesian bicategory B , then ev ery ob ject of B is discrete. In this case, b ecause M is a bicategory , the equiv alence relatio ns M ( X , A ) a re stable under comp osition f r om b oth sides. Th us writing | M ( X , A ) | for the set of o b jects of M ( X , A ) w e hav e a mere category , E whose ob jects are tho se of M (and hence also tho se of B ) a nd whose ho m sets are the quotien ts | M ( X , A ) | / M ( X , A ). If the E ( X, A ) are regarded as discrete categories, so that E is a lo cally discrete bicategory then the functors M ( X , A ) / / | M ( X , A ) | / M ( X , A ) constitute t he effect on homs functors for an identit y on ob jects biequiv alence M / / E . T o summarize 3.14. Theorem. If a c artesian bic ate go ry B satisfies b oth the F r ob enius and S e p ar a ble axioms then the bic ate gory of maps M is bie quivalent to the lo c al ly discr ete b i c ate go ry E . 9 In the following lemma w e show that any cop ointed endomorphism of a discrete ob ject can b e made in to a comonad; later on, w e shall see that this comonad structure is in fact unique. 3.15. Lemma. If A i s a discr ete obje ct in a c artesian bic ate gory B then, for any c op ointe d endom o rphism arr ow ǫ : G / / 1 A : A / / A , ther e is a 2-c el l δ = δ G : G / / GG satisfying G G G G 1                G G 1   ? ? ? ? ? ? ? ? ? ? ? ? ? G GG δ   GG G Gǫ o o GG G ǫG / / and if b oth G, H : A / / / / A ar e c op ointe d, so that GH : A / / A is also c op ointe d, and φ : G / / H is any 2-c el l, then the δ ’s satisfy GG H H φφ / / G GG δ   G H φ / / H H H δ   and GH GH 1 / / GH GH GH ? ? δ              GH GH GH ( Gǫ )( ǫH )   ? ? ? ? ? ? ? ? ? ? ? ? ? Proo f. W e define δ = δ G to b e the pasting comp osite A AA d / / A AAA d 3 ? ? ? ? ?   ? ? ? ? ? AA AAA 1 d   AAA AAA G 11 / / AA AAA 1 d   AA AA G 1 / / AA AAA 1 d   AAA AA d ∗ 1 / / AA AAA 1 d   AA A d ∗ / / A AA d   AAA AAA G 11 / / AAA AAA GGG   ? ? ? ? ? ? ? ? ? ? ? ? ? AAA AAA 11 G   AAA AA d ∗ 1 / / AAA AAA 11 G   AAA AA d ∗ 1 / / AA AA 1 G   AAA AA d ∗ 1 / / AAA A d ∗ 3 ? ? ? ? ? ?   ? ? ? ? ? ? AA A d ∗   A A G ∧ G ∧ G , , A A G 2 2 A A G # # A A G   GǫG   ? ? δ 3   ? ? 1 2 3 4 5 6 wherein ⊗ has b een abbreviated by j uxtap osition a nd all subregions not explicitly inhab- ited b y a 2-cell are deemed to b e inhabited b y the obvious inv ertible 2-cell. A reference n um b er has b een assigned t o those in v ertible 2-cells whic h arise fro m the h ypo theses. As 10 in [W&W], d 3 ’s denote 3-fold diagonal maps a nd, similarly , w e write δ 3 for a lo cal 3- f old diagonal. The in v ertible 2- cell la b elled by ‘1’ is that defining A to b e F rob enius. The 3- fold comp osite of ar r ows in t he region lab elled by ‘2 ’ is G ∧ 1 A and, similarly , in that labelled b y ‘3’ w e hav e 1 A ∧ G . Eac h of these is isomorphic to G b ecause A is separable and G is cop o inted. The isomorphisms in ‘4’ and ‘5 ’ express the pseudo-functoriality of ⊗ in the cartesian bicategory B . Finally ‘6’ expresses the ternar y lo cal pro duct in terms of the ternary ⊗ a s in [W&W ]. Demonstration of the equations is effected easily by pasting comp osition calculations. 3.16. Theorem. If G and H ar e c op ointe d endom orphisms o n a discr ete A i n a c artesia n B then G o o Gǫ GH ǫH / / H is a pr o duct di a g r am in B ( A, A ) . Proo f. If we are giv en α : K / / G and β : K / / H then K is also cop ointed and w e hav e K δ / / K K αβ / / GH as a candidate pairing. That this candidate satisfies the unive rsal prop ert y follo ws fro m the equations of Lemma 3.15 whic h are precisely those in the equational desc ription o f binary pro ducts. W e remark that the ‘naturality ’ equations for the pro jections f ollo w immediately from uniqueness of copoints. 3.17. Coro llar y. I f A is d i s cr ete in a c artesian B , then an endo-arr o w G : A / / A admits a c omonad s tructur e if and on ly if G ha s the c o p ointe d pr op erty, and any such c om onad structur e is unique. Proo f. The Theorem sho ws that the a rro w δ : G / / GG constructed in Lemma 3 .15 is the pro duct dia gonal on G in the categor y B ( A, A ) and, giv en ǫ : G / / 1 A , this is the only comonad com ultiplication on G . 3.18. Remark. It is clear that 1 A is terminal with resp ect to the cop oin ted ob jects in B ( A, A ). 3.19. Proposition. If an obj e ct B in a bic ate gory B has 1 B subterminal in B ( B , B ) then, for any map f : A / / B , f is subterminal in B ( A, B ) an d f ∗ is subterminal in B ( B , A ) . In p articular, in a c artesian bic a te gory in which every obje ct is sep ar a b le, every a d joint arr ow is subterminal. Proo f. Precomp o sition with a map preserv es terminal ob jects and monomorphisms, as do es p ostcomp osition with a righ t adjoin t. 11 4. Bicateg ories o f Comonads The starting p oin t of this section is the observ atio n, made in the in tro duction, that a comonad in the bicategory Span E is precisely a span of the fo rm A g ← − X g − → A in whic h b oth legs are eq ual. W e will write C = Com B fo r the bicategory of comonads in B , Com b eing one o f the duals of Street’s construction Mnd in [ST]. Th us C has ob jects given b y the comonads ( A, G ) of B . The structure 2-cells for comonads will b e denoted ǫ = ǫ G , f or counit and δ = δ G , for comultiplication. An arrow in C from ( A, G ) to ( B , H ) is a pair ( F , φ ) as sho wn in A B F / / A A G   A B F / / B B H   φ / / satisfying F 1 A 1 B F = / / F G F 1 A F ǫ   F G H F φ / / H F 1 B F ǫF   and F GG H F G φG / / F G F GG F δ   F G H F G H F G H H F H φ / / H F G H F H F H H F δF   F G H F φ / / (2) (where, as often, we hav e suppressed the asso ciativity constrain ts of our normal, cartesian, bicategory B ). A 2- cell τ : ( F , φ ) / / ( F ′ , φ ′ ) : ( A, G ) / / ( B , H ) in C is a 2- cell τ : F / / F ′ in B satisfying F ′ G H F ′ φ ′ / / F G F ′ G τ G   F G H F φ / / H F H F ′ H τ   (3) There is a pseudofunctor I : B / / C giv en b y I ( τ : F / / F ′ : A / / B ) = τ : ( F , 1 F ) / / ( F ′ , 1 F ′ ) : ( A, 1 A ) / / ( B , 1 B ) F ro m [ST] it is we ll know n that a bicategory B has Eilen b erg - Mo ore ob jects fo r comonads if and o nly if I : B / / C has a rig h t biadjoint, whic h w e will denote by E : C / / B . W e write 12 E ( A, G ) = A G and the counit fo r I ⊣ E is denoted by A G A g G / / A G A G 1 A G   A G A g G / / A A G   γ G / / A G A g G ' ' O O O O O O O O O O A A G 7 7 g G o o o o o o o o o o A A G   or, using normalit y of B , b etter b y γ G / / with ( g G , γ G ) abbreviated to ( g , γ ) when there is no danger of confusion. It is standard that eac h g = g G is necessarily a map (whence o ur low er case notatio n) and the mat e g g ∗ / / G of γ is an isomorphism whic h iden tifies ǫ g and ǫ G . W e will write D for the lo cally full subbicategory of C determined by all the ob jects and those arrow s of the form ( f , φ ), where f is a map, a nd write j : D / / C for the inclusion. It is clear that the pseu dofunctor I : B / / C restricts to giv e a pseudofunctor J : M / / D . W e sa y that the bicategory B has Eilenb er g-Mo or e obje c ts fo r c omonads, as se en by M , if J : M / / D has a righ t biadjoin t. (In general, this prop ert y do es not fo llow from that o f Eilen b erg - Mo ore ob jects for comonads.) 4.1. Remark. In the case B = Span E , a comonad in B can, as we hav e seen , b e iden tified with a morphism in E. This can b e made into the ob ject part of a biequiv alence b et w een the bicategory D and the category E 2 of arro ws in E . If we further identify M with E , then the inclusion j : D / / C b ecomes the diago nal E / / E 2 ; of course this do es ha v e a righ t adjoin t, giv en b y the domain functor. 4.2. Theorem. If B is a c artesian bic ate gory in which every obje ct is d i s cr ete, the bic a te gory D = D ( B ) admits the fol lowing simpler de scription: i ) An obje ct is a p air ( A, G ) wher e A i s an ob j e ct of B and G : A / / A admits a c op oint; ii ) An arr ow ( f , φ ) : ( A, G ) / / ( B , H ) is a map f : A / / B and a 2-c el l φ : f G / / H f ; iii ) A 2-c el l τ : ( f , φ ) / / ( f ′ , φ ′ ) : ( A, G ) / / ( B , H ) is a 2-c el l sa tisfying τ : f / / f ′ satisfying e quation (3). Proo f. W e hav e i) by Corollary 3 .17 while iii) is precisely the description of a 2 - cell in D , mo dulo the description of the domain and co domain arrows. So , w e hav e only to sho w ii), whic h is to sho w that the equations (2) hold automatically under the h yp otheses. F o r the first equation of (2) w e hav e uniqueness of an y 2-cell f G / / f b ecause f is subterminal b y Prop osition 3.19. F or t he second, observ e that the terminating vertex , H H f , is the pro duct H f ∧ H f in M ( A, B ) b ecause H H is the pro duct H ∧ H in M ( B , B ) b y Theorem 3.16 and precomp osition with a map preserv es all limits. F or H H f seen as a pro duct, the pro jections are, ag a in b y Theorem 3.16, H ǫf and ǫH f . Th us, it suffices to sho w tha t the diagr a m for the second equation comm utes when comp osed with b ot h 13 H ǫf and ǫH f . W e ha v e f GG H f G φG / / f G f GG f δ   f G H f G H f G H H f H φ / / H f G H f H f H H f δf   f G H f φ / / H f G H H f H φ / / H f G H f H f ǫ   ? ? ? ? ? ? ? ? ? ? ? ? H H f H f H ǫf   f G H f φ / / f GG f G f Gǫ   f GG H f G φG / / f G f GG f δ   f G H f G H f G H H f H φ / / H f G H f H f H H f δf   f G H f φ / / f GG H f G φG / / f GG f G f ǫG   H f G f G ǫf G               f G H f φ / / H H f H f ǫH f   in whic h eac h of the low er triangles comm utes b y the first equation of (2) already estab- lished. Using comonad equations for G a nd H , it is ob vious that eac h comp o site is φ . Finally , le t us note that D is a subbicategory , neither full no r lo cally full, of the Grothendiec k bicategory G a nd write K : D / / G for the inclusion. W e also write ι : M / / G for the comp o site pseudofunctor K J . Summarizing, w e ha v e in tro duced the following comm utativ e diagram of bicategories and pseudofunctors B C I / / M B i   M D J / / D C j   D G K / / M G ι $ $ ; note also that in our main case of in terest B = Span E , each o f M , D , and G is biequiv alen t to a mere category . Ultimately , w e are inte rested in ha ving a right biadjoin t, say τ , of ι . F or suc h a biadjunction ι ⊣ τ the counit at an ob ject R : X / / A in G will tak e the f o rm τ R A v R ' ' O O O O O O O O O O X τ R 7 7 u R o o o o o o o o o X A R   ω R / / (4) (where, as for a biadjunction I ⊣ E : C / / B , a triangle rather than a square can b e tak en as the b oundary of the 2 -cell b y the norma lity of B ). In fact, we are in terested in the case where w e ha v e ι ⊣ τ and moreov er the counit comp onents ω R : v R / / Ru R enjo y the prop erty that their mates v R u ∗ R / / R with respect to the adjunction u R ⊣ u ∗ R are in v ertible. In this w a y we represen t a general arrow of B in terms of a span of maps. Since biadjunctions comp ose w e will conside r adjunctions J ⊣ F and K ⊣ G and we b egin with the second of these. 14 4.3. Theorem. F or a c artesian bic ate gory B in which ev ery obje ct i s discr ete, ther e is an adjunction K ⊣ G : G / / D wher e, for R : X / / A in G , the c omonad G ( R ) a nd its witnes s ing c o p oint ǫ : G ( R ) / / 1 X A ar e given by the left d iagr a m b elow and the c ounit µ : K G ( R ) / / R is given by the right diagr am b elow, al l in notation suppr essin g ⊗ : X A X A 1 X A / / X A X X A dA   X A X X A ? ? p 1 , 3              ≃ / / X X A X AA X RA   X A X A 1 X A   ˜ p 1 , 3 / / X A X A 1 X A / / X AA X A X d ∗   X AA X A p 1 , 3   ? ? ? ? ? ? ? ? ? ? ? ? ? / / X A X A G ( R )   X A X p / / X A X X A dA   X X X A ? ? p 2              X X A X AA X RA   X A R   X A A r / / X AA X A X d ∗   X AA A p 2   ? ? ? ? ? ? ? ? ? ? ? ? ? ≃ / / ˜ p 2 / / / / X A X A G ( R )   Mor e over, the mate r G ( R ) p ∗ / / R of the c ounit µ i s invertible. In the left diagr am, the p 1 , 3 c ol le c tivel y denote pr oje ction fr om the thr e e-fold pr o duct in G to the pr o duct of the fi rst and thir d factors. In the right diagr am, the p 2 c ol le c tivel y denote pr o je ction fr om the thr e e-fold pr o duct in G to the se c ond factor. T h e upp er triangles of the two diagr ams ar e the c anonic al isomorphisms. The lower left triangle is the mate of the c anonic a l isomorphism 1 ≃ / / p 1 , 3 ( X d ) . The lower right triangle is the mate o f the c an o nic a l isomorphism r ≃ / / p 2 ( X d ) . Proo f. Given a comonad H : T / / T and an arrow T A a / / T T H   T X x / / X A R   ψ / / in G , w e verify the adjunction claim by sho wing that there is a unique arrow T X A f / / T T H   T X A f / / X A X A G ( R )   φ / / 15 in D , whose comp osite with the putative counit µ is ( x, ψ , a ). It is immediately clear that the unique solution for f is ( x, a ) and to giv e φ : ( x, a ) H / / X d ∗ ( X RA ) d A ( x, a ) is to give the mate X d ( x, a ) H / / ( X RA ) d A ( x, a ) whic h is ( x, a, a ) H / / ( X RA )( x, x, a ) and can b e seen as a G ar r ow: T X AA ( x,a,a ) / / T T H   T X X A ( x,x,a ) / / X X A X AA X RA   ( α,β ,γ ) / / where w e exploit the description o f pro ducts in G . F rom this description it is clear, since ˜ p 2 ( α, β , γ ) = β a s a comp osite in G , that the unique solution f o r β is ψ . W e ha v e seen in Theorem 3.17 that the conditions (2) hold auto matically in D under the assumptions of the Theorem. F ro m the first of these w e hav e: T X AA ( x,a,a ) / / T T H   T X X A ( x,x,a ) / / X X A X AA X RA   ( α,β ,γ ) / / X AA X A p 1 , 3 / / X X A X AA X RA   X X A X A p 1 , 3 / / X A X A 1 X A   p 1 , 3 / / = T X A ( x,a ) / / T T 1 T   T X A ( x,a ) / / X A X A 1 X A   κ ( x,a ) / / T T H ( ( ǫ H / / So, with a mild abuse of not ation, w e hav e ( α, γ ) = (1 x ǫ H , 1 a ǫ H ), uniquely , and th us the unique solutions for α and γ are 1 x ǫ H and 1 a ǫ H resp ectiv ely . This show s t hat φ is necessarily the mate under the a dj unctions considered o f (1 x ǫ H , ψ , 1 a ǫ H , ). Since D and G are essen tially lo cally discrete this suffices to complete the claim that K ⊣ G . It only remains to sho w that the mate r G ( R ) p ∗ / / R of the counit µ is in v ertible. In the three middle squares of the dia gram X A X A G ( R )   X X A X X o o p ∗ X A X X A dA   X A X o o p ∗ X X X d   ˜ p ∗ d, 1 A / / X AA X A o o p ∗ X X A X AA X RA   X X A X X o o p ∗ X X X A X R   ˜ p ∗ X R, 1 A / / X A A r / / X AA X A X d ∗   X AA X A o o p ∗ X A A r   ≃ / / X A A r / / X X X A X R   X X X r / / X A R   r 1 X ,R / / X X 1 X   ? ? ? ? ? ? ? ? ? ? ? ? ? A A 1 A                16 the top tw o are in v ertible 2-cells b y Prop osition 4.18 of [CKWW] while the low er one is the obv ious inv ertible 2-cell constructed from X d ∗ p ∗ ∼ = 1 X,A . The r igh t square is an in v ertible 2- cell b y Prop osition 4.1 7 o f [CKWW]. This sho ws that the mate r G ( R ) p ∗ / / R of µ is inv ertible. 4.4. Remark. It now follows that the unit o f the adjunction K ⊣ G is given (in notation suppressin g ⊗ ) b y: T T T d / / T T T T d 3   ? ? ? ? ? ? ? ? ? ? ? ? ? T T T T T dT   T T T d / / T T T T ? ? d 3              T T T T T T d ∗   T T H   T T T T T T H H H   T T T T T T T H T x x ≃ / / ˜ d 3 / / ǫH ǫ / / ≃ / / where the d 3 collectiv ely denote 3- fold diagonalization (1 , 1 , 1 ) in G . The top t r ia ngle is a canonical isomorphism while the lo w er triangle is the mate of the canonical isomorphism ( T ⊗ d ) d ≃ / / d 3 and is itself in v ertible, b y separabilit y of T . Before turning to the que stion of an adjunction J ⊣ F , w e note: 4.5. Lemma. In a c artesian bic ate gory in which Maps ar e Comonadic, i f g F ∼ = h with g and h maps, then F is also a map. Proo f. By Theorem 3.11 of [W&W] it suffice s to sho w that F is a comonoid homo- morphism, whic h is to sho w that the cano nical 2-cells ˜ t F : tF / / t and ˜ d F : dF / / ( F ⊗ F ) d are in v ertible. F or the first w e hav e: tF ∼ = tg F ∼ = th ∼ = t Simple diagrams sho w tha t w e do get the right isomorphism in this case and also for the next: ( g ⊗ g )( dF ) ∼ = dg F ∼ = dh ∼ = ( h ⊗ h ) d ∼ = ( g ⊗ g )( F ⊗ F ) d whic h giv es dF ∼ = ( F ⊗ F ) d since the map g ⊗ g reflects isomorphisms . 17 4.6. Theorem. If B is a c artesian bic ate gory which has Eile nb er g-Mo or e obje cts for Comonads and for which Maps ar e C omonadic then B h as Eilenb er g-Mo or e obje cts for Comonads as Se en by M , which is to say that J : M / / D has a righ t adjoint. Mor e ove r, the c ounit for the ad junction, s ay J F / / 1 D , ne c es sarily ha ving c o mp o nents of the form γ : g / / Gg with g a map, has g g ∗ / / G in vertible. Proo f. It suffices to show that the a djunction I ⊣ E : C / / B r estricts to J ⊣ F : D / / M . F or this it suffices to sho w tha t , giv en ( h, θ ) : J T / / ( A, G ), the F : T / / A G with g F ∼ = h whic h can b e found using I ⊣ E has F a map. This follows f r o m Lemma 4.5 . 4.7. Theorem. A c a rtesia n bic ate gory which has Eilenb er g-Mo or e obje cts fo r Comon- ads and for whi ch Maps ar e Comonadic ha s tabulation in the sense that the in c lusion ι : M / / G has a right adjoint τ and the c ounit c omp onents ω R : v R / / Ru R as in (4) have the pr op erty that the mates v R u ∗ R / / R , with r esp e ct to the adjunctions u R ⊣ u ∗ R , ar e invertible. Proo f. Using Theorems 4.3 and 4.6 we can construct the adjunction ι ⊣ τ by comp osing J ⊣ F with K ⊣ G . Moreov er, the counit f or ι ⊣ τ is the pasting composite: T X ⊗ A ( u,v ) ' ' O O O O O O O O X ⊗ A T 7 7 ( u,v ) o o o o o o o o o X ⊗ A X ⊗ A G ( R )   T X u ! ! T A v = = γ O O X ⊗ A A r / / X ⊗ A X ⊗ A G ( R )   X ⊗ A X p / / X A R   µ / / where the square is the counit for K ⊣ G ; and the triangle, the counit for J ⊣ F , is an Eilen b erg-Mo ore coalgebra fo r the comonad G ( R ). The arrow comp onent of the Eilen b erg - Mo ore coalgebra is necessarily of the form ( u, v ), where u and v are maps, and it also follo ws that w e hav e ( u, v ) ( u, v ) ∗ ∼ = G ( R ). Thus we hav e v u ∗ ∼ = r ( u, v )( p ( u, v )) ∗ ∼ = r ( u, v )( u, v ) ∗ p ∗ ∼ = r G ( R ) p ∗ ∼ = R where the first t w o isomorphisms are trivial, the third arises from the in v ertibilit y of the mate of γ as an Eilen b erg-Mo ore structure, and the fourth is in v ertibilit y of µ , as in Theorem 4.3. 4.8. Theorem. F or a c a rtesia n bic ate gory B with Eilenb er g-Mo or e obje cts f o r Com o n- ads and for which Maps ar e Comonadi c , Map B has pul l b acks satisfying the Be ck c ondition 18 (me aning that for a pul l b ack squar e N A a / / P N p   P M r / / M A b   ≃ / / (5) the mate pr ∗ / / a ∗ b of ap ∼ = br in B , with r esp e ct to the adjunc tion s r ⊣ r ∗ and a ⊣ a ∗ , is invertible). Proo f. Giv en the cospan a : N / / A o o M : b in Map B , let P together with ( r , σ, p ) b e a tabulation for a ∗ b : M / / N . Then pr ∗ / / a ∗ b , the mate of σ : p / / a ∗ br with resp ect to r ⊣ r ∗ , is inv ertible by Theorem 4.7. W e hav e also ap / / br , the mate of σ : p / / a ∗ br with resp ect to a ⊣ a ∗ . Since A is discrete, ap / / br is also in v ertible and is the only 2-cell b etw een the comp osite maps in question. If w e ha v e also u : N o o T / / M : v , for maps u and v with au ∼ = bv , then the mate u / / a ∗ bv ensures that the span u : N o o T / / M : v factors through P b y an es sen tially unique map w : T / / P with pw ∼ = u and r w ∼ = v . 4.9. Proposition. In a c artesian bic ate gory with Eilenb er g-Mo or e ob j e cts for Comon - ads and for which Maps a r e Comonadic, ev e ry sp an of maps x : X o o S / / A : a gives rise to the fol lowi n g tabulation di agr am : S A a ' ' O O O O O O O O O O X S 7 7 x o o o o o o o o o o X A ax ∗   aη x O O Proo f. A general tabulation counit ω R : v R / / Ru R is giv en in terms of the Eilen b erg- Mo ore coalgebra for the comonad ( u, v )( u, v ) ∗ and necessarily ( u, v )( u, v ) ∗ ∼ = G ( R ). It follo ws that for R = ax ∗ , it suffices to sho w that G ( ax ∗ ) ∼ = ( x, a )( x, a ) ∗ . Consider the 19 diagram (with ⊗ suppressed): X X A X AA X S A X X A X xA                X S A X AA X aA   ? ? ? ? ? ? ? ? ? ? ? ? ? S A X S A ( x,S ) A ? ? ? ? ?   ? ? ? ? ? S A X S X S X S A X ( S,a )             S A X S S S A ( S,a )                S X S ( x,S )   ? ? ? ? ? ? ? ? ? ? ? ? ? X A X X A dA   ? ? ? ? ? ? ? ? ? ? ? ? ? S A X A xA                S A X X A X A X AA X d                X S X A X a   ? ? ? ? ? ? ? ? ? ? ? ? ? X S X AA S X A ( x,a )   S X A ( x,a )   The comonoid G ( ax ∗ ) can b e read, f r o m left t o righ t, along the ‘W’ shape of the lo w er edge as G ( ax ∗ ) ∼ = X d ∗ .X aA.X x ∗ A.dA . But eac h of the squares in the diagram is a (pro duct-absolute) pullbac k so that with Prop osition 4.8 at hand w e can contin ue: X d ∗ .X aA.X x ∗ A.dA ∼ = X a.X ( S , a ) ∗ . ( x, S ) A.x ∗ A ∼ = X a. ( x, S ) . ( S, a ) ∗ .x ∗ A ∼ = ( x, a )( x, a ) ∗ as required. 5. Chara cteriza t ion of Bicatego r ies of Spans 5.1. If B is a cartesian bicategory with Map B essen tially lo cally discrete then eac h slice Map B / ( X ⊗ A ) is a lso essen tially lo cally discrete and w e can write Span Map B ( X , A ) for the categories obtained b y ta king the quotients of the equiv alence relations com- prising the hom categories o f the Map B / ( X ⊗ A ). Then w e can construct functors C X,A : Span Map B ( X, A ) / / B ( X , A ), where for an arro w in Span Map B ( X , A ) as sho wn, A N _ _ b ? ? ? ? ? ? ? ? ? ? ? ? ? M A a                M N h   X N ? ? y              M X x   ? ? ? ? ? ? ? ? ? ? ? ? ? M N h   w e define C ( y , N , b ) = by ∗ and C ( h ) : ax ∗ = ( bh )( y h ) ∗ ∼ = bhh ∗ y ∗ bǫ h y ∗ / / by ∗ . If Map B is know n to hav e pullbac ks then the Span Map B ( X, A ) b ecome the hom-categories for a 20 bicategory Span Map B and w e can consider whether the C X,A pro vide the effects on homs for an iden tit y-on-ob jects pseudofunctor C : Span Map B / / B . Consider Y A N Y y                N A b   ? ? ? ? ? ? ? ? ? ? ? ? ? N A   ? ? ? ? ? ? ? ? ? ? ? ? ? N M M A                A X M A a                M X x   ? ? ? ? ? ? ? ? ? ? ? ? ? N M P N p                P M r   ? ? ? ? ? ? ? ? ? ? ? ? ? (6) where the square is a pullbac k. In somewhat abbreviated notatio n, what is needed further are coheren t, in v ertible 2-cells e C : C N .C M / / C ( N M ) = C P , for eac h compo sable pair of spans M , N , a nd coheren t, in v ertible 2- cells C ◦ : 1 A / / C (1 A ), for eac h o b ject A . Since the iden tit y span on A is (1 A , A, 1 A ), and C (1 A ) = 1 A . 1 ∗ A ∼ = 1 A . 1 A ∼ = 1 A w e ta k e the inv erse of this comp osite for C ◦ . T o giv e the e C t ho ugh is to giv e 2-cells y b ∗ ax ∗ / / y pr ∗ x ∗ and since spans of the form (1 N , N , b ) and ( a, M , 1 M ) arise as special cases, it is easy to ve rify that to give the e C it is necessary and sufficien t to give coheren t, in v ertible 2- cells b ∗ a / / pr ∗ for eac h pullback square in Map B . The in v erse of suc h a 2-cell pr ∗ / / b ∗ a is the mate of a 2- cell bp / / aq . But b y discretene ss a 2-cell bp / / aq mu st b e essen tially an iden tit y . Th us, definabilit y of e C is equiv alen t to t he inv ertibility in B of the mate pr ∗ / / b ∗ a of the iden tity bp / / ar , fo r each pullbac k square as displa y ed in (6). In short, if Map B has pullbac ks and these satisfy the Bec k condition as in Prop osition 4.8 then w e ha v e a canonical pseudofunctor C : Span Map B / / B . 5.2. Theorem. F or a b ic ate gory B the fo l lowing ar e e quivalent: i ) T her e is a bie quivalenc e B ≃ Span E , for E a c ate gory with finite limits; ii ) Th e bic ate gory B is c artesian, e ach c omonad has an Eilenb er g-Mo or e o b je ct, and every m ap is c omon a dic. iii ) The bic a te gory Map B is an essential ly lo c al ly discr ete bic a te gory with finite limi ts, satisfying in B the Be ck c ondition for pul lb acks of maps, and the c anonic al C : Span Map B / / B is a bie quivalenc e of bic ate gories. Proo f. That i ) implies ii ) follo ws from our discussion in the In tro duction. That iii ) implies i ) is trivial so we show that ii ) implies iii ). W e hav e already observ ed in Theorem 3.14 that, for B cartesian with ev ery ob ject discrete, Map B is essen tially lo cally discrete and we hav e seen b y Prop ositions 2.3 and 21 3.5 that, in a cartesian bicategory in whic h Maps are Comonadic, ev ery ob ject is discrete. In Theorem 4.8 w e ha v e seen that, for B satisfying the conditions of ii ), Map B has pullbac ks, and hence all finite limits and, in B the Bec k condition holds f o r pullbac ks. Therefore we ha v e the canonical pseudofunctor C : Span Map B / / B dev elop ed in 5.1. T o complete the pro of it suffices to sho w that the C X,A : Span Map B ( X , A ) / / B ( X , A ) are equiv alences of categories. Define functors F X,A : B ( X , A ) / / Span Map B ( X , A ) by F ( R ) = F X,A ( R ) = ( u, τ R, v ) where τ R A v ' ' O O O O O O O O O O X τ R 7 7 u o o o o o o o o o X A R   ω / / is the R -comp onen t of the counit for ι ⊣ τ : G / / Map B . F or a 2-cell α : R / / R ′ w e define F ( α ) to be the essen tially unique map satisfying τ R τ R ′ F ( α ) / / τ R ′ A v ′   ? ? ? ? ? ? ? ? ? ? ? ? ? X τ R ′ ? ? u ′              X A R ′   ω ′ / / τ R X u 7 7 o o o o o o o o o o o o o o o o o o o o o o τ R A v ' ' O O O O O O O O O O O O O O O O O O O O O O O = τ R A v   ? ? ? ? ? ? ? ? ? ? ? ? ? X τ R ? ? u              X A X A R   X A R ′   ω / / α / / (W e r emark that ess en tial uniqueness here means that F ( α ) is determined to within unique in v ertible 2- cell.) Since ω : v / / Ru has mate v u ∗ / / R in v ertible, b ecause ( v , τ R , u ) is a tabula t ion of R , it follow s that w e ha v e a na t ur a l isomor phism C F R / / R . O n the other hand, starting with a span ( x, S, a ) from X to A w e ha v e as a consequence of Theorem 4 .9 that ( x, S, a ) is part of a ta bula t io n of ax ∗ : X / / A . It follo ws that we hav e a na tural isomorphism ( x, S, a ) / / F C ( x, S, a ), whic h completes the demonstration t ha t C X,A and F X,A are in v erse equiv alences. 6. Direct sums in bicatego ries of spa ns In t he previous section w e gav e a c haracterization of those (cartesian) bicategories of the form Span E for a category E with finite limits. In this final se ction w e give a refinemen t, sho wing that Span E has direct sums if and only if the original category E is lextens iv e [CL W]. Direct sums are of course understo o d in the bicategorical sense. A zer o ob je ct in a bicategory is an ob ject whic h is b oth initial and terminal. In a bicategory with finite pro ducts and finite copro ducts in whic h the initial ob ject is also terminal there is a 22 canonical induced arrow X + Y / / X × Y , and w e say that the bicategor y has dir e ct sums when this map is a n equ iv alence . 6.1. Remark. Just as in the case of ordinary categories, the existence of direct sums giv es rise to a calculus of matrices. A morphism X 1 + . . . + X m / / Y 1 + . . . + Y n can b e represen ted by an m × n matrix of mo r phisms b et w een the summands, and comp osition can b e represen t ed b y matrix m ultiplication. 6.2. Theorem. L et E b e a c ate gory w ith finite limits, an d B = Span E . Then the fol lowing ar e e quivalent: i ) B has dir e ct sums; ii ) B has finite c opr o ducts; iii ) B has finite pr o ducts; iv ) E is lextensive. Proo f. [ i ) = ⇒ ii )] is trivial. [ ii ) ⇐ ⇒ iii )] follows from the fact that B op is biequiv alen t to B . [ ii ) = ⇒ iv )] Supp ose that B has finite copro ducts, and write 0 for the initial o b ject and + for the copro ducts. F or eve ry ob ject X there is a unique span 0 o o D / / X . By uniquenes s, any map in to D m ust b e inv ertible, and an y t w o suc h with the same domain m ust b e equal. Th us when we comp ose the span with its opp osite, as in 0 o o D / / X o o D / / 0, the resulting span is just 0 o o D / / 0. Now b y the univ ersal prop ert y o f 0 once again, this mu st just b e 0 o o 0 / / 0, and so D ∼ = 0, and our unique span 0 / / X is a map. Clearly copro ducts of maps are copro ducts, and so the copro duct injections X + 0 / / X + Y and 0 + Y / / X + Y are also maps. Thus the copro ducts in B will restrict to E pro vided that the co diago nal u : X + X o o E / / X : v is a map for all ob jects X . No w the fact that the co diag o nal comp osed with the first injection i : X / / X + X is the iden tit y tells us that w e ha v e a diagram as on the left b elo w X i ′   ? ? ? ? ? ? ?               1   X i ′   ? ? ? ? ? ? ?               i   X i   ? ? ? ? ? ? ?               E u          v   ? ? ? ? ? ? ? X i   ? ? ? ? ? ? ?               E u          u   ? ? ? ? ? ? ? X X + X X X X + X X + X in whic h the square is a pullback; but then the diagra m on t he right shows that the comp osite of u : X + X o o E / / X + X : u with the injection i : X / / X + X is just i . Similarly its comp osite with the other injection j : X / / X + X is j , and so u : X + X o o E / / X + X : u is the iden t ity . This prov es t ha t the co diagonal is indeed a map, and so that E has finite 23 copro ducts; w e ha v e a lr eady assumed that it has finite limits. T o see that E is lextensiv e observ e that w e ha v e equiv alences E / ( X + Y ) ≃ B ( X + Y , 1) ≃ B ( X , 1) × B ( Y , 1) ≃ E /X × E / Y . [ iv ) = ⇒ i )] Supp ose tha t E is lextens iv e. Then in particular, it is distributiv e, so that ( X + Y ) × Z ∼ = X × Z + X × Y , and we ha v e B ( X + Y , Z ) ≃ E /  ( X + Y ) × Z  ≃ E / ( X × Z + Y × Z ) ≃ E / ( X × Z ) × E / ( Y × Z ) ≃ B ( X , Z ) × B ( Y , Z ) whic h shows that X + Y is the copro duct in B ; but a similar argumen t shows that it is also the pro duct. 6.3. Remark. The implication iv ) ⇒ i ) was prov ed in [P&S , Section 3]. 6.4. Remark. The equiv alence ii ) ⇔ iv ) can b e seen as a sp ecial case o f a more general result [H&S] characterizing colimits in E whic h are also (bicategorical) colimits in Span E . 6.5. Remark. There is a corresp onding result in v olving partial maps in lextens iv e categories, although the situation there is more complicated as one do es not ha v e direct sums but only a we ak ened r elat io nship b et w een pro ducts and copro ducts, and a similarly w eak ened calculus of matrices. See [C&L, Section 2 ]. There is also a n ullary v ersion of the theorem. W e simply recall that a n initial ob ject in a category E is said to b e strict , if an y morphism in to it is inv ertible, and t hen leav e the pro of to the reader. Once again the equiv alence ii ) ⇔ iv ) is a special case of [H&S]. 6.6. Theorem. L et E b e a c ate gory w ith finite limits, an d B = Span E . Then the fol lowing ar e e quivalent: i ) B has a zer o obje ct; ii ) B has an initial obje ct; iii ) B has a terminal obje ct; iv ) E has a strict initial o b j e ct. References [CKWW] A. Carb oni, G.M. Kelly , R.F.C. W alters , and R.J. W o o d. Ca rtesian bicategor ies II, The ory Appl. Cate g. 19 (2008 ), 93– 124. [CL W] A. Carb oni, Stephen Lack, and R.F.C. W a lters. Int ro duction to extensive and distributive categ o ries. J. Pur e Appl. Algebr a 84 (1 9 93), 1 45–15 8. [C&W] A. Ca rb oni and R.F.C. W alters. C a rtesian bicategorie s . I. J. Pure Appl. A lgebr a 49 (1987 ), 11– 32. 24 [C&L] J .R.B. Co ck ett and Stephen Lack. Res tr iction categ ories I II : colimits, partial limits, a nd extensivity , Math. Struct. in Comp. Scienc e 17 (2007), 77 5–817 . [LSW] I. F ranco Lop ez, R. Street, and R.J. W o o d, Duals Inv ert, Applie d Cate goric al Struct ur es , to app ear. [H&S] T. Heindel and P . Sob o ci ´ nski, V an Ka mpe n colimits a s bicolimits in Span, L e cture Notes in Computer Scienc e, (CALCO 2 009), 5 728 (2009), 335– 349. [P&S] Elango Panc hadcharam and Ross Street, Mack ey functor s on compact closed catego ries, J. H omotopy and Rela te d Structure s 2 (200 7 ), 261 –293. [ST] R. Street. The formal theor y of monads, J. Pur e Appl. Algebr a 2 (1972), 14 9 –168 . [W&W] R.F.C. W a lters and R.J . W o o d. F ro be nius ob jects in cartesia n bica teg ories, The ory Appl. Cate g. 20 (2008), 25–47 . Scho ol of Computing and Mathematics University of Western Sydney L o cke d Bag 1797 South Penrith DC NSW 1797 Austr alia and Dip artimento di Scienze del le Cultu r a Politiche e del l’Informazione Universit` a del l Insubria, Italy and Dep artment of Mathematics and St atistics Dalhousie University Halifax, NS, B3H 3J5 Canada Email: s.l ack@uw s.edu .au, ro bert. walter s@uninsubria.it, rjwood@ dal.c a