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SOMMAIRE

R.S^TREET,Polynomials as Spans 113

A. CAMPBELL &E.LANARI, On truncated quasi-categories 154 P.GAUCHER,Flows revisited: The Model Category Structure

and its left determinedness 208

POLYNOMIALS AS SPANS

Ross STREET

Résumé.L’article définit les polynômes dans une bicatégorieM. Les poly- nômes dans les bicatégories SpnC des spans dans une catégorie finiment complèteC coïncident avec les polynômes dansC comme définis par Nicola Gambino et Joachim Kock, et par Mark Weber. LorsqueM estcalibré, nous obtenons une autre bicatégoriePolyM. Nous démontrons que les polynômes dansM ont des représentations comme pseudofoncteursM^op Ñ Cat. En utilisant destabulations, nous produisons des calibrations pour la bicatégorie des relations dans une catégorie régulière et pour la bicatégorie des distribu- teurs entre catégories, en fournissant ainsi de nouveaux exemples de bicaté- gories de “polynômes”.

Abstract. The paper defines polynomials in a bicategory M. Polynomi- als in bicategories SpnC of spans in a finitely complete category C agree with polynomials inC as defined by Nicola Gambino and Joachim Kock, and by Mark Weber. WhenM iscalibrated, we obtain another bicategory PolyM. We see that polynomials inM have representations as pseudofunc- torsM^op Ñ Cat. Usingtabulations, we produce calibrations for the bicat- egory of relations in a regular category and for the bicategory of two-sided modules (distributors) between categories thereby providing new examples of bicategories of “polynomials”.

Keywords. span; partial map; powerful morphism; polynomial functor; ex- ponentiable morphism; calibrated bicategory; right lifting.

Mathematics Subject Classification (2010).18C15, 18D05, 18C20, 18F20.

∗The author gratefully acknowledges the support of Australian Research Council Dis- covery Grants DP160101519 and DP190102432.

1. Introduction

Polynomials in an internally complete (= “locally cartesian closed”) cate- goryE were shown by Gambino-Kock [9] to be the morphisms of a bicate- gory. Weber [30] defined polynomials in any categoryC with pullbacks and proved they formed a bicategory. While both these papers are quite beau- tiful and accomplish further advances, I felt the need to better understand the composition of polynomials. Perhaps what I have produced is merely a treatment of polynomials for bicategory theorists.

The starting point was to view polynomials as spans of spans so that composition could be viewed as the more familiar composition of spans us- ing pullbacks; see Bénabou [3]. A polynomial fromX toY in a categoryC is a diagram of the shapeX Ð^mÝݲ E Ý^mÝѹ SÑÝ^p Y withm2a powerful (= expo- nentiable) morphisms inC. Such diagrams can be thought of as generalizing spans: a span X Ý^pmÝÝÝÝ^2,S,pqÑ Y amounts to the case whereE “ S andm1 is the identity. Our simple idea was to make the diagram more complicated by including an identity thus:

X Ð^mÝݲ E Ý^mÝѹ SÐÝ^1S SÝÑ^p Y ,

resulting in a span

X ÐÝÝÝÝÝÝ^pm1,E,m2q S ÝÝÝÝÑ^p1S,S,pq Y

of spans fromX toY.

Of course, the bicategory of spans does not have all bicategorical pull- backs. Fortunately, polynomials are not general spans and sufficient pull- backs can be constructed. Indeed, that is what Weber’s distributivity pull- backs around a pair of composable morphisms in C construct. That con- struction requires the use of powerful morphisms in C. Here we define a morphism in a bicategory to be a right lifterwhen every morphism into its codomain has a right lifting through it. For spans inC to be right lifters, one leg must be powerful.

We introduce the termcalibrationfor a class of morphisms, calledneat, in a bicategory; the technical use of this word comes from Bénabou [4] who used it for categories. A bicategory with a distinguished calibration is called calibrated. Polynomials in a calibrated bicategory M are spans with one

leg a right lifter and the other leg neat. This suffices for the construction of a tricategory [10] of polynomials in M in which all the 3-morphisms are invertible. However, for two reasons, we decided to centre attention here on the bicategory PolyM obtained by taking isomorphism classes of 2-morphisms. One reason is that it covers our present examples, the other is the possibility of iterating the construction without moving to higher level categories.

Apolynomic bicategory M is one in which the neat morphisms are all the groupoid fibrations (see Section 3) inM. We prove that SpnC is poly- nomic for any finitely complete C. In this case the polynomials are the polynomials inC in the sense of Weber [30].

The bicategory RelE of relations in a regular category E is calibrated by morphisms which are isomorphic to graphs of monomorphisms inE. In Example 10.3 for E a topos, we give a reinterpretation of the bicategory of polynomials inRelE as a Kleisli construction.

By providing a calibration for the bicategoryModof two-sided modules between categories, we obtain another example. Again, in Example 10.6, we give a reinterpretation of the bicategory of polynomials in Modas a Kleisli construction.

It must be pointed out that the meaning of polynomial in a bicategory is different from the meaning in Section 4 of Weber [30] which is about polynomials in 2-categories. Weber is dealing with the 2-category as aCat- enriched category, taking the polynomials to be diagrams of the same shape as in the case of ordinary categories, and accommodating the presence of 2-cells. In particular, if a category is regarded as a 2-category with only identity 2-cells, then his polynomials in the 2-category are just polynomials in the category. To define a polynomial, in the sense of this paper, in such a 2-category would require the specification of a calibration on the category and then a polynomial would reduce to a single morphism (called “neat”) in that calibration.

I am grateful to the Australian Category Seminar, especially Yuki Mae- hara, Richard Garner, Michael Batanin and Charles Walker, for comments during and following my talks on this topic. I am also particularly grateful to the diligent and insightful referee for suggesting important improvements, mainly that I should add the detail to the previously vaguely expressed Ex- amples 10.3 and 10.6; there are some facts involved that may be unfamiliar.

2. Bipullbacks and cotensors

Recall that the pseudopullback (also called iso-comma category) of two functors C ÝÑ^F E Ð^PÝ D is the category F{psP whose objects pC, α, Dq consist of objectsC P C andDP D withα :F C ÝÑ^– P D, and whose mor- phismspu, vq: pC, α, Dq Ñ pC¹, α¹, D¹qconsist of morphismsu : C Ñ C¹ inC andv :DÑD¹inD such thatpP vqα “α¹pF uq. We have a universal square of functors

F{psP

cod

dom //D

F

ks ^ξ

–

C _P ^//E

(2.1)

containing an invertible natural transformationξ.

A square

P

c

d //A

n

ks ^θ

–

B _p ^//C

(2.2)

in a bicategory A is abipullbackof the cospanA ÝÑⁿ C ÝÑ^p B when, for all objectsKofA, the induced functor

ApK, Pq ÝÑApK, nq{_psApK, pq, u ÞÑ pdu, θu, cuq, is an equivalence of categories.

In a bicategoryA, we writeA²for the (bicategorical) cotensor (or power) ofAwith the ordinal2; this means that the categoryApK, A²qis equivalent to the arrow category ofApK, Aq, pseudonaturally inK PA. The identity morphism inApA², A²qcorresponds to a morphism (arrow)

A²

c

77

d

''ó^λ A

inApA², Aq.

Example 2.1. ForA “ V -Catin the sense of [14], theV-category A² is the usual arrowV-category.

Example 2.2. Recall (from [5] for instance) the definition of the bicate- goryV-ModofV-categories and their modules for a nice symmetric closed monoidal base category V. The objects are V-categories. The homcate- gories are defined to be theV-functor categories

V-ModpA, Bq “ rB^opbA,Vs

whose objects m : B^op bA Ñ V are called modules from A toB. Com- position is defined by the coendspn˝mqpc, aq “ şb

mpb, aq bnpc, bq. Let I_˚2 denote the free V-category on the category2. The cotensor of the V- category Awith the ordinal 2 in the bicategory V-Mod is theV-category pI_˚2q^opbA. This is because of the calculation

V-ModpK,pI_˚2q^opbAq “ rI_˚2bA^opbK,Vs – rA^opbK,V²s – rA^opbK,Vs² –V-ModpK, Aq².

Let B0 : 1 Ñ 2 be the functor 0 ÞÑ 1; it is right adjoint to ! : 2 Ñ 1. It follows that c : A² ÑA inV-ModisppI_˚!q^op bAq_˚ : pI_˚2q^op bA Ñ A.

In particular, when V “ Set, c is the modulepr_2˚ induced by the second projection functor2^opˆAÑA.

Remark 2.3. The phenomenon described in Example 2.2 has to do with the fact that the pseudofunctorp´q˚ :V-CatÑV-Mod, taking eachV-functor f :A Ñ B to the modulef_˚ :A Ñ B withf_˚pb, aq “ Bpb, f aq, preserves bicolimits andV-Modis self dual.

3. Groupoid fibrations

Letp:E ÑBbe a functor. A morphismχ:e¹ ÑeinEis calledcartesian¹ forpwhen the square (3.3) is a pullback for allk PE.

Epk, e¹q ^{Epk,χq //}

p

Epk, eq

p

Bppk, pe¹q

Bppk,pχq //Bppk, peq

(3.3)

1Classically called “strong cartesian”

Note that all invertible morphisms in E are cartesian. If pis fully faithful then all morphisms ofEare cartesian.

We call the functorp:E ÑB agroupoid fibrationwhen

(i) for all objectse P E and morphisms β : b Ñ pe inB, there exist a morphismχ:e¹ ÑeinEand isomorphismb –pe¹ whose composite withpχisβ, and

(ii) every morphism ofE is cartesian forp.

From the pullback (3.3), it follows that groupoid fibrations are conservative (that is, reflect invertibility).

We call the functorp : E Ñ B anequivalence relation fibration orer- fibration when it is a groupoid fibration and the only endomorphisms ξ : x ÑxinE which map to identities underpare identities. It follows (using condition (ii)) thatpis faithful. Note that ifpis an equivalence then it is an er-fibration.

WriteGFibB for the 2-category whose objects are groupoid fibrations p: E ÑB, and whose hom categories are given by the following pseudop- ullbacks.

GFibBpp, qq

//rE, Fs

rE,qs

ks ^–

1 rps

//rE, Bs

(3.4)

So objects of GFibBpp, qq are pairs pf, φqwhere f : E Ñ F is a functor and φ : qf ñ pis an invertible natural transformation. If φ is an identity thenpf, φqis calledstrict.

LetGpd be the 2-category of groupoids, functors and natural transfor- mations. Write HompB^op,Gpdq for the 2-category of pseudofunctors (=

homomorphisms of bicategories [3])T :B^op ÑGpd, pseudo-natural trans- formations, and modifications [16].

Recall that the Grothendieck construction pr : oT Ñ B on a pseud- ofunctor T : B^op Ñ Gpd is the projection functor from the category oT whose objects are pairs pt, bq with b P B and t P T b, and whose mor- phisms pτ, βq : pt, bq Ñ pt¹, b¹q consist of morphisms β : b Ñ b¹ in B andτ : t Ñ pT βqt¹ in T b. This construction is the object assignment for a

2-functor

o: HompB^op,Gpdq ÝÑGFibB (3.5) which actually lands in the sub-2-category of strict morphisms. Note that the pullback

T b ^//

oT

pr

1 rbs //B

is also a bipullback (see [12]); this suggests that we can reconstruct a pseud- ofunctor T from a groupoid fibrationp : E Ñ B by defining T b to be the pseudopullback ofrbs :1 ÑB andp.

Proposition 3.1. The 2-functor(3.5)is a biequivalence.

A category which is both a groupoid and a preorder is the same as an equivalence relation; that is, a set of objects equipped with an equivalence relation thereon. LetERbe the 2-category of equivalence relations, functors and natural transformations. Note that the 2-functorSet ÑERtaking each set to the identity relation is a biequivalence. Write ERFibB for the full sub-2-category ofGFibB with objects the er-fibrations.

Proposition 3.2. The biequivalence(3.5)restricts to a biequivalence o: HompB^op,ERqÝÑ^„ ERFibB ,

and so further restricts to a biequivalence rB^op,SetsÝÑ^„ ERFibB .

LetE andB be bicategories. Baklovi´c [2] and Buckley [6] say that a morphism x : Z Ñ X inE is cartesianfor a pseudofunctorP : E Ñ B when the following square is a bipullback inCatfor all objectsK ofE.

EpK, Zq

P

EpK,xq //EpK, Xq

P

ks ^–

BpP K, P Zq _{BpP K,P xq} ^//BpP K, P Xq

(3.6)

A 2-cell σ : x¹ ñ x : Z Ñ X in E is called cartesian for P when it is cartesian (as a morphism of EpZ, Xq) for the functor P : EpZ, Xq Ñ BpP Z, P Xq. Note that all equivalences are cartesian morphisms and all invertible 2-cells are cartesian.

Definition 3.3. A pseudofunctorP :E ÑBis agroupoid fibrationwhen (i) for all X P E andf : B Ñ P X inB, there exist a morphism x :

Z ÑX inE and an equivalenceB »P Zwhose composite withP x is isomorphic tof,

(ii) every morphism ofE is cartesian forP, and (iii) every 2-cell ofE is cartesian forP.

A morphism p : E Ñ B in a tricategory T is called agroupoid fibration when, for all objects K of T, the pseudofunctor T pK, pq : T pK, Eq Ñ T pK, Bqis a groupoid fibration between bicategories.

Definition 3.4. A pseudofunctorF :A ÑB is calledconservativewhen (a) if F f is an equivalence in B for a morphism f in A then f is an

equivalence;

(b) ifF αis an isomorphism inB for a 2-cellαinA thenαis an isomor- phism.

A morphism f : A Ñ B in a tricategory T is conservative when, for all objects K of T , the pseudofunctor TpK, fq : T pK, Aq Ñ TpK, Bq is conservative.

Proposition 3.5. Groupoid fibrations are conservative.

Proof. If P xis an equivalence, we see from the bipullback (3.6) that each functorEpK, xqis too. Since these equivalences can be chosen to be adjoint equivalences, they become pseudonatural in K and so, by the bicategorical Yoneda Lemma [23], are represented by an inverse equivalence for x. This proves (a) in the Definition of conservative. Similarly, for (b), look at the pullback (3.3) for the functorp“ pEpZ, XqÝÑ^P BpP Z, P Xqq.

For a pseudofunctorP : E Ñ B between bicategories, write B{P for the bicategory whose objects are pairspB ÝÑ^f P E, Eq, whereEis an object ofE andf : B Ñ P E is a morphism ofB, and whose homcategories are defined by pseudopullbacks

B{Pppf, Eq,pf¹, E¹qq

d

c //EpE, E¹q

P

BpP E, P E¹q

–

Bpf,1q

BpB, B¹q

Bp1,f¹q

//BpB, P E¹q

(3.7)

WriteE{E forB{P in the caseP is the identity pseudofunctor ofE. There is a canonical pseudofunctor J_P : E{E Ñ B{P taking the object pX ÝÑ^u E, EqtopP X Ý^{P u}ÝÑP E, Eq.

Proposition 3.6. The pseudofunctorP :E ÑB between bicategories sat- isfies condition (i) in the Definition 3.3 of groupoid fibration if and only if J_P :E{E ÑB{P is surjective on objects up to equivalence. Also,P :E Ñ B satisfies condition (ii) if and only if the effect ofJP : E{E Ñ B{P on homcategories is an equivalence. Condition (iii) is automatic if all 2-cells in E are invertible.

Example 3.7. Each biequivalence of bicategories is a groupoid fibration.

Example 3.8. LetHbe an object of the bicategoryB. WriteB{H for the bicategory B{P whereP is the constant pseudofunctor1 Ñ B atH. The

“take the domain” pseudofunctor

dom :B{H ÑB (3.8)

is a groupoid fibration. For, it is straightforward to see that the canonical pseudo-functorpB{Hq{pB{Hq Ñ B{domis a biequivalence, so it remains to prove each 2-cell

σ :pg, ψq ñ pf, φq:pAÝÑ^a Hq Ñ pB ÑÝ^b Hq

inB{H is cartesian for (3.9). The condition for a 2-cell ispbσqψ “φ. Take another 2-cellτ : ph, θq ñ pf, φqinB{H (so thatpbτqθ “ φ) and a 2-cell υ : h ñ g inB such that συ “ τ. Then we havepbσqpbυqθ “ pbσυqθ “ pbτqθ “ φ “ pbσqψ with bσ invertible. Sopbυqθ “ ψ. We conclude that υ :ph, θq ñ pg, ψqis a 2-cell inB{H, as required.

Note that (3.8) is not a local groupoid fibration in general; that is, the functor dom_a,b : B{HpA ÑÝ^a H, B ÝÑ^b Hq Ñ BpA, Bqis generally not a groupoid fibration.

Example 3.9. Apparently more generally, letf :HÑK be a morphism in the bicategoryB. Write

f_˚ :B{H ÑB{K (3.9)

for the pseudofunctor which composes with f. On applying Example 3.8 with B and H replaced by B{K and H ÝÑ^f K, up to biequivalence we obtain this example.

Proposition 3.10. Up to biequivalence, pseudofunctors of the form (3.8) are precisely the groupoid fibrations P : E Ñ B for which the domain bicategory has a terminal object. Moreover, if such aP has a left biadjoint, it is a biequivalence.

Proof. Let T be a terminal object of E. Make a choice of morphism!E : E ÑT for each objectE ofE. Then the pseudofunctor

JˆP :E ÝÑB{P T , E ÞÑ pP E ÝÝÑ^P!E P Tq

is a biequivalence overB; it is a tripullback of the biequivalence J_P along the canonicalB{P T ÑB{P. SoP is biequivalent to (3.8) withH “P T.

For the second sentence, if we suppose thedomof (3.8) has a left biad- joint then it preserves terminal objects. The bicategoryB{Hhas the termi- nal object1H :HÑH. SoH “dompH ÝÑ^1H Hqis terminal inB. Sodom is a biequivalence.

4. Spans in a bicategory

Spans in a bicategory A with bipullbacks (= iso-comma objects) will be recalled; compare [23] Section 3.

AspanfromXtoY in the bicategoryA is a diagramX ÐÝ^u S ÝÑ^p Y; we writepu, S, pq:X ÑY. The composite ofX ÐÝ^u SÑÝ^p Y andY ÐÝ^v T ÑÝ^q Z is obtained from the diagram

P

pr₁



pr₂

S ^–_ðù

u



p

T

v



q

X Y Z

(4.10)

(where P is the bipullback of pand v) as the spanX ÐÝÝ^upr1 P ÝÝÑ^qpr2 Z. A morphismpλ, h, ρq:pu, S, pq Ñ pu¹, S¹, p¹qof spansis a morphismh:S Ñ S¹inM equipped with invertible 2-cells as shown in the two triangles below.

S

h

u

xx

p

&&

λ–ð ^ρ–ð

X S¹

u¹

oo

p¹

//Y

(4.11)

A 2-cell σ : h ñ k : pu, S, pq Ñ pu¹, S¹, p¹qbetween such morphisms is a 2-cellσ:h ñk :S ÑS¹inM which is compatible with the 2-cells in the triangles in the sense thatλ“λ¹.u¹σandρ¹ “p¹σ.ρ. We writeSpnApX, Yq for the bicategory of spans fromX toY.

Composition pseudofunctors

SpnApY, Zq ˆSpnApX, Yq ÝÑSpnApX, Zq

are defined on objects by composition of spans (4.10) and on morphisms by using the universal properties of bipullback.

In this way, we obtain a tricategorySpnA. The associator equivalences are obtained using the horizontal and vertical stacking properties of pseudop- ullbacks. The identity span on X has the form p1_X, X,1_Xqand the unitor equivalences are obtained using the fact that a pseudopullback of the cospan X ÑÝ^f Y ÐÝ^1Y Y is given by the span X ÐÝ^1X X ÝÑ^f Y equipped with the canonical isomorphism1_Yf –f –f1_X inA.

For e : X Ñ Y in A, let e_˚ “ p1_X, X, fq : X Ñ Y. Notice that e^˚ “ pe, X,1Xq : Y Ñ X is a right biadjoint for e_˚ in the tricategory SpnA.

Proposition 4.1. Let A be a finitely complete bicategory. The following conditions on a spanpu, S, pqfromXtoY inA are equivalent:

(i) the morphismpu, S, pq : X Ñ Y has a right biadjoint in the tricate- gorySpnA;

(ii) the morphismu:SÑXis an equivalence inA;

(iii) the morphismpu, S, pq:X ÑY is equivalent inSpnA tof_˚for some morphismf inA;

(iv) the morphismpu, S, pq:X ÑY is a groupoid fibration in the tricate- gorySpnA.

Proof. The equivalence of (i), (ii) and (iii) is essentially as in the case where A is a category; see [7]. We will prove the equivalence of (ii) and (iv). Put S“SpnA to save space. Using Proposition 3.10 and the fact that the span K Ð^prÝݹ KˆX Ý^prÝѲ Xis a terminal object in the bicategorySpK, Xq, we see that the pseudofunctor PK :“ SpK, Xq ^SpK,p˚u

˚q

ÝÝÝÝÝÝÑ SpK, Yqis a groupoid fibration if and only if the canonical pseudofunctorJ_PK in the diagram (4.12) is a biequivalence.

SpK, Xq

SpK,u^˚q

xx

J_PK

**» +3

SpK, Sq

J_SpK,p˚q //SpK, Yq{ppr₁, KˆS, ppr₂q

(4.12)

However, we see easily thatJ_PK does factor up to equivalence as shown in (4.12) whereJ_SpK,p˚q is a biequivalence. Sop_˚u^˚ : X Ñ Y is a groupoid fibration if and only if SpK, u^˚qis a biequivalence for all K; that is, if and onlyuis an equivalence inA.

Remark 4.2. (a) In fact (ii) implies (iv) in Proposition 4.1 requires no assumption on the bicategory A. For, it is straightforward to check (compare Example 3.9) that p_˚ : SpnApK, Sq Ñ SpnApK, Yqis a groupoid fibration for all K; this does not even require bipullback in A since we only need the hom bicategories ofSpnA.

(b) Given thatp_˚ is a groupoid fibration, we can prove the converse (iv) implies (ii) by noting that p_˚u^˚ is a groupoid fibration if and only if u^˚ is (compare (i) of Proposition 5.2). So, providedSpnApK, Yqhas a terminal object (as guaranteed by the finite bicategorical limits in A), we deduce thatu^˚ is a biequivalence using Proposition 3.10 and u_˚ %u^˚.

Remark 4.3. IfC is a finitely complete category (regarded as a bicategory with only identity 2-cells) then the tricategory SpnC has only identity 3- cells; it is a bicategory. We are interested in spans in such a bicategory A “SpnC. The problem is that bipullbacks do not exist in this kind ofA in general. Hence we must hone our concepts to restricted kinds of spans in A.

5. More on bipullbacks and groupoid fibrations

In Section 4, we defined groupoid fibrations in a tricategory. This applies in a bicategoryA regarded as a tricategory by taking only identity 3-cells. Then the 2-cells in each ApA, Bq are invertible (identities in fact) so condition (iii) of Definition 3 is automatic.

Proposition 5.1. Supposep: E Ñ B is a morphism in a bicategoryA for whichE² andB²exist. Thenp:E ÑB is a groupoid fibration inA if and only if the square

E²

p²

c //E

p

ks ^–

B² _c ^//B

is a bipullback.

Proof. Since all concepts are defined representably, it suffices to check this for the bicategoryA “ Catwhere the bipullback ofcand pis the comma categoryB{p. So the square in the Proposition is a bipullback if and only if the canonical functor j_p : E² ÑB{pis an equivalence. We have the result by looking at Proposition 3.6 in the bicategory case.

Proposition 5.2. SupposeA is a bicategory.

(i) Supposer–pqwithpa groupoid fibration inA. Thenris a groupoid fibration if and only ifqis.

(ii) In the bipullback(2.2)inA, ifpis a groupoid fibration then so isd.

(iii) Suppose(2.2)is a bipullback inA andpis a groupoid fibration. For any square

K

v

u //A

n

ks ^ψ

B _p ^//C

(5.13)

inA withψ not necessarily invertible, there exists a diagram K

h

v

xx

u

&&

ðλ ^ρ–ð

B ^oo _c P

d //A ,

(5.14)

(withλinvertible if and only ifψis) which pastes onto(2.2)to yieldψ.

This defines on objects an inverse equivalence of the functor from the category of suchpλ, h, ρqto the category of diagrams(5.13)obtained by pasting onto(2.2).

Proof. These are essentially standard facts about groupoid fibrations, es- pecially (i) and (ii). For (iii) we use the groupoid fibration property of p to lift the 2-cell ψ : nu ñ pv to a 2-cell χ : w ñ v with an invert- ible 2-cell ν : nu ñ pw such that ψ “ ppχqν. Now use the bipullback property of (2.2) to factor the square ν : nu ñ pw as a span morphism pσ, h, ρq :pw, K, uq Ñ pc, P, dqpasted onto (2.2). Thenλis the composite ofσandχ.

The next result is related to Proposition 5 of [22].

Proposition 5.3. In the bipullback square(2.2)in the bicategoryA, ifpis a groupoid fibration andnhas a right adjointn %uthenchas a right adjoint c%vsuch that the mateθˆ:dvñupofθ :ndñpcis invertible.

Proof. Letε : nu ñ1C be the counit ofn % u. By the groupoid fibration property ofp, there existsχ :w ñ1_Band an invertibleτ :nup ñpw such that ppχqτ “ εp. By the bipullback property of (2.2), there exists a span morphism

B

v

up

xx

w

&&

λ–ð ^ρ–ð

A P

oo d

c //B whose pasting ontoθ isτ. Thenc %v with counitcv ^ρ

´1

ùùñ wùñ^χ 1B and we see thatθˆ“ρis invertible.

Proposition 5.4. SupposeC is a category with pullbacks. Then the pseudo- functorp´q˚ :C ÑSpnC takes pullbacks to bipullbacks.

Proof. Let the spanpp, P, qq:AÑBbe the pullback of the cospanpf, C, gq inC. Consider a square

X

pr,T ,sq

pu,S,vq //A

f˚

ks ^ψ

–

B _g

˚

//C

inSpnC. The isomorphismψ amounts to an isomorphismh: pu, S, f vq Ñ pr, T, gsq of spans. In particular, f v “ gsh, so, by the pullback property, there exists a uniquet:S ÑP such thatpt“v andqt“sh. Then we have a morphism of spans

X

pu,S,tq

pr,T ,sq

xx

pu,S,vq

&&

λ–ð ^ρ–ð

B _q P

˚

oo p˚

//A

in whichλish:pu, S, qtq Ñ pr, T, sqandρis an identity.

6. Lifters

LetM be a bicategory.

We use the notation

Y

n

K

rifpn,uq

88

u &&

$

Z

(6.15)

to depict a right lifting rifpn, uq (see [27]) of u through n. The defining property is that pasting a 2-cell v ùñ rifpn, uq onto the triangle to give a 2-cellnvùñudefines a bijection.

A morphismn : Y ÑZ is called aright lifterwhenrifpn, uqexists for allu:K ÑZ.

Example 6.1. Left adjoint morphisms in any M are right lifters (since the lifting is obtained by composing with the right adjoint).

Example 6.2. Composites of right lifters are right lifters.

Example 6.3. SupposeM “SpnC withC a finitely complete category. If f :AÑB is powerful (in the sense of [25], elsewhere called exponentiable, and meaning that the functor C{B ÑC{A, which pulls back alongf, has a right adjoint Πf) in C then f^˚ : B Ñ Ais a right lifter. The formula is rifpf^˚,pv, T, qqq “ pw, U, rqwhere

pU Ý^pw,rqÝÝÑK ˆBq “Π_1KˆfpT ÝÝÑ^pv,qq KˆAq.

Example 6.4. Suppose m “ pm₁, E, m₂q is a morphism in M “ SpnC withC a finitely complete category. Thenmis a right lifter if and only ifm₁ is powerful. The previous Examples imply “if”. Conversely, we can apply the Dubuc Adjoint Triangle Theorem (see Lemma 2.1 of [25] for example) to see that MpK, m₁^˚qhas a right adjoint for all K becauseMpK, mq – MpK, m2˚qMpK, m1˚

qand the unit ofm2˚ %m2˚is an equalizer. Taking K to be the terminal object, we conclude thatm₁is powerful.

Example 6.5. Let E be a regular category and let RelE be the locally or- dered bicategory of relations inE as characterized in [7]. The objects are the same as forE and the morphismspr₁, R, r₂q:X ÑY are jointly monomor- phic spans X Ð^rݹ R Ý^rѲ Y in E. LetSubX “ RelEp1, Xq be the partially ordered set of subobjects of X P E. Iff : Y Ñ X is a morphism of E then pulling back subobjects ofXalongf defines an order-preserving func- tion f^´1 : SubX Ñ SubY whose right adjoint, if it exists, is denoted by

@f : SubY Ñ SubX. A similar analysis as in the span case yields that pr₁, R, r₂q:X ÑY is a right lifter inRelE if and only if@r1 exists.

Proposition 6.6. Suppose(2.2) is a bipullback inM withp a groupoid fi- bration. Ifn is a lifter then so iscand, for all morphismsb : K Ñ B, the canonical 2-cell

d˝rifpc, bq ùñrifpn, p˝bq is invertible.

Proof. For allK PM, we have a bipullback square MpK, Pq

MpK,cq

MpK,dq //MpK, Aq

MpK,nq

ks ^–

MpK, Bq

MpK,pq //MpK, Cq

in Cat with MpK, pqa groupoid fibration and MpK, nq % rifpn,´q. By

Proposition 5.3,MpK, cqhas a right adjoint, so thatcis a lifter, andMpK, dqrifpc,´q – rifpn,´qMpK, pq. Evaluating this last isomorphism atbwe obtain the iso-

morphism displayed in the present Proposition.

7. Distributivity pullbacks

We now recall Definitions 2.2.1 and 2.2.2 from Weber [30] ofpullback and distributivity pullback around a composable pair pf, gq of morphisms in a categoryC.

X ^{p //}

q

Z ^{g //}A

f

Y _r ^//B

(7.16)

A pullback pp, q, rqaroundZ ÝÑ^g A ÑÝ^f B is a commutative diagram (7.16) in which the spanpq, X, gpqis a pullback of the cospanpr, B, fqinC.

A morphism t : pp¹, q¹, r¹q Ñ pp, q, rq of pullbacks around pf, gq is a morphism t : Y¹ Ñ Y in C such that rt “ r¹. For such a morphism, using the pullback properties, it follows that there is a unique morphism s : X¹ Ñ X inC such that ps “ p¹ andqs “ tq¹.² This gives a category PBpf, gq. It is worth noting, also using the pullback properties, that the commuting squareqs“ tq¹ exhibits the spanps, X¹, q¹qas a pullback of the cospanpq, Y, tq.

The diagram (7.16) is called a distributivity pullback aroundpf, gqwhen it is a terminal object of the categoryPBpf, gq.

Y

p˚q^˚

r˚ //B

f^˚

ks

–

Z _g

˚ //A

(7.17)

Proposition 7.1. LetZ ÑÝ^g AÝÑ^f B be a composable pair of morphisms in a category C with pullbacks. The diagram(7.16)is a pullback aroundpf, gq in the category C if and only if there is a square of the form (7.17) in the bicategory SpnC. The diagram (7.16) is a distributivity pullback around pf, gqinC if and only the diagram(7.17)is a bipullback inSpnC.

Proof. Passage around the top and right sides of (7.17) produces the pullback span of the cospan Y ÝÑ^r B ÐÝ^f A. Passage around the left and bottom sides produces the left and top sides of (7.16). That these passages be isomorphic says (7.16) is a pullback.

Suppose (7.16) is a distributivity pullback. We will show that (7.17) is a bipullback. Take a square of the form

K

u˚v^˚

s˚t^˚ //B

f^˚

ks

–

Z _g

˚

//A

(7.18)

2Rather than the singlet, Weber’s definition takes the pairps, tq as the morphism of pullbacks aroundpf, gqalthough he has a typographical error in the conditionrt“r¹.

inSpnC. This square amounts to a diagram S

a

u //

v

~~

Z ^{g //}A

f

K ^oo _t T _s ^//B

in C in which the right-hand region is a pullback around pf, gq. By the distributivity property, there exists a unique pairph, kqsuch that the diagram

S

h

a //

u

~~

T

s

k

Z ^oo _p X _q ^//Y _r ^//B

commutes; moreover, the square is a pullback. Thus we have a span mor- phism

K

pt,T ,kq

u˚p^˚

xx

s˚t^˚

&&

λ–ð ^ρ–ð

Z Y

p˚q^˚

oo r˚

//B

which pastes onto (7.17) to yield (7.18); in factρis an identity. To prove the bipullback 2-cell property, suppose we have span morphismse:pv¹, S¹, u¹q Ñ pv, S, uq and j : pt¹, T¹, s¹q Ñ pt, T, sq such that composing the first with g_˚ is the composite of the second with f^˚. Then, in obvious notation, j : pu¹, a¹, s¹q Ñ pu, a, sqis a morphism inPBpf, gq. By the terminal property ofpp, q, rq, we havek¹ “kj. This gives the span morphismj :pt¹, T¹, k¹q Ñ pt, T, kqwhich is unique as required.

Conversely, suppose (7.17) is a bipullback. We must see thatpp, q, rqis terminal in PBpf, gq. Take another objectpp¹, q¹, r¹qof PBpf, gq. We have the square

Y¹

p¹_˚q^1˚

r¹_˚

//B

f^˚

ks

–

Z _g

˚

//A

which allows us to use the bipullback property to obtain a span morphism pp¹_˚q^1˚, Y¹, r_˚¹q Ñ pp_˚q^˚, Y, r_˚q

in SpnC which is compatible with the squares. Since the span Y¹ Ñ Y in this morphism composes with r_˚ to be isomorphic to r_˚¹, we see that it has the form k_˚ for some k : Y¹ Ñ Y in C. Thus we have our unique k :pp¹, q¹, r¹q Ñ pp, q, rqinPBpf, gq.

8. Polynomials in calibrated bicategories

Recall from [3] Section 7 that the Poincaré category ΠK of a bicategory K has the same objects as K , however, the homsetΠK pH, Kqis the set π₀pK pH, Kqqof undirected path components of the homcategoryK pH, Kq.

Composition is induced by composition of morphisms in K . Theclassify- ing categoryClK ofK is obtained by taking isomorphism classes of mor- phisms in each categoryK pH, Kq. IfK is locally groupoidal thenΠK is equivalent toClK .

We adapt Bénabou’s notion of “catégorie calibrée” [4] to our present purpose.

Definition 8.1. A class P of morphisms, whose members are called neat (“propres” in French), in a bicategoryM is called acalibration ofM when it satisfies the following conditions

P0. all equivalences are neat and, ifpis neat and there exists an invertible 2-cellp–q, thenqis neat;

P1. for all neatp, the compositep˝qis neat if and only ifqis neat;

P2. every neat morphism is a groupoid fibration;

P3. every cospan of the form

S ÝÑ^p Y ÐÝⁿ T ,

withn a right lifter andpneat, has a bipullback (8.19) inM withp˜ neat.

P

˜ n

˜

p //T

n

ks ^θ

–

S _p ^//Y

(8.19)

A bicategory equipped with a calibration is calledcalibrated.

Notice that the classGFof all groupoid fibrations in any bicategoryM satisfies all the conditions for a calibration except perhaps the bipullback existence part of P3 (automatically p˜will be a groupoid fibration by (ii) of Proposition 5.2).

A bicategoryM is calledpolynomicwhenGFis a calibration ofM. Definition 8.2. LetM “ pM,Pqbe a calibrated bicategory. Apolynomial pm, S, pqfromXtoY inM is a span

X Ð^mÝS ÝÑ^p Y

in M withm a right lifter and pneat. A polynomial morphism pλ, h, ρq : pm, S, pq Ñ pm¹, S¹, p¹qis a diagram

S

h

m

xx

p

&&

ðλ ^ρ–ð

X S¹

m¹

oo

p¹

//Y

(8.20)

in whichρ(but not necessarilyλ) is invertible. By part (i) of Proposition 5.2 we know that h must be a groupoid fibration. (Indeed, by condition P1, h is neat; this is not really needed and is the only use made herein of the

“only if ” in P1.) We call pλ, h, ρq strong when λ is invertible. A 2-cell σ : h ñ k : pm, S, pq Ñ pm¹, S¹, p¹qis a 2-cell σ : h ñ k : S Ñ S¹ in M compatible with λ and ρ. By Proposition 3.5, we know that σ must be invertible. WritePolyMpX, Yqfor the Poincaré category of the bicategory of polynomials fromX toY so obtained.

We write h “ rλ, h, ρs : pm, S, pq Ñ pm¹, S¹, p¹qfor the isomorphism class of the polynomial morphism pλ, h, ρq : pm, S, pq Ñ pm¹, S¹, p¹q. We also writeλ_h andρ_h when several morphisms are involved.

Proposition 8.3. If C is a finitely complete category then the bicategory SpnC is polynomic.

Proof. Take a cospanS ÝÑ^p Y ÐÝⁿ T inSpnC withpa groupoid fibration and n “ pn1, F, n2qa lifter. By Proposition 4.1, we can supposepis actuallyp_˚ for somep:S ÑY inC. From Example 6.4, we know thatn₁is powerful.

Form the pullback spanS ÐÝ^f P ÝÑ^g F of the cospanSÑÝ^p Y Ðⁿݲ F inC. By Proposition 5.4, we have a bipullback

P

f˚

g˚ //F

pn2q˚

ks ^–

S _p

˚ //Y .

Since n1 is powerful, Proposition 2.2.3 of Weber [30] implies we have a distributivity pullback

V ^{a //}

q

P ^{g //}F

n1

W _r ^//T

aroundpn₁, gq. By Proposition 7.1, we have the bipullback W

a˚q^˚

r˚ //T

pn1q^˚

ks

–

P _g

˚

//F

inSpnC. Paste this second bipullback on top of the first to obtain a bipull- back of the cospanS ÝÑ^p˚ Y ^pn2q˚pn1q

˚

ÐÝÝÝÝÝÝT as required.

The class of equivalences in any bicategory is a calibration. In Sec- tion 10, we will provide an example of a calibration strictly between equiv- alences andGP.

In a calibrated bicategoryM, polynomials can be composed as in the diagram (8.21); this is made possible by Example 6.2, Proposition 6.6, con- dition P3 and the “if” part of condition P1. Identity spans are also identity

polynomials.

P

ðù–θ

˜ n



˜ p

S

m

 ^p

T

 n

q

X Y Z

(8.21)

Indeed, this composition of polynomials is the effect on objects of functors

˝: PolyMpY, Zq ˆPolyMpX, Yq ÝÑPolyMpX, Zq. (8.22) The effect on morphisms is defined using part (iii) of Proposition 5.2 as follows. Take morphismsh : pm, S, pq Ñ pm¹, S¹, p¹qandk : pn, T, qq Ñ pn¹, T¹, q¹q. We have a square

P

h˜n

k˜p //T¹

n¹

ks ^ψ

S¹

p¹ //Y in which

ψ “ pn¹kp˜ù^λù^kñ^p˜ np˜ù^θ–ñp˜n ù^ρùù^hn–˜ñp¹h˜nq.

Now we use Proposition 5.2 to obtain, in obvious primed notation, a diagram P

`

h˜n

xx

k˜p

&&

ðσ ^τ–ð

S¹ P¹

˜ n¹

oo

˜ p¹

//T¹

which leads to the polynomial morphism

ppλ_hnqpm˜ ¹σqq, `,pq¹τqpρ_kpq˜ :pm˜n, P, qpq ÝÑ pm˜ ¹n˜¹, P¹, q¹p˜¹q

whose isomorphism class is the desired

k˝h:pn, T, qq ˝ pm, S, pq Ñ pn¹, T¹, q¹q ˝ pm¹, S¹, p¹q.

Proposition 8.4. There is a bicategory PolyM of polynomials in a cali- brated bicategory M. The objects are those of M, the homcategories are thePolyMpX, Yq. Composition is given by the functors(8.22). The vertical and horizontal stacking properties of bipullbacks provide the associativity isomorphisms.

We writePoly_sM for the sub-bicategory ofPolyM obtained by restrict- ing to the strong polynomial morphisms.

Example 8.5. IfC is a finitely complete category then the bicategoryPolySpnC is biequivalent to the bicategory denoted by Poly_C in Gambino-Kock [9]

and by PolypCq in Walker [28]. Moreover, Poly_sSpnC is biequivalent to Walker’s bicategoryPoly_cpCq. Note that the isomorphism classeshof poly- nomial morphisms have canonical representatives of the formf_˚(since each spanpu, S, vq:U ÑV withuinvertible is isomorphic top1_U, U, v u^´1q).

Proposition 8.6. If the bicategoryM is calibrated then, for eachK P M, there is a pseudofunctorH^K : PolyM ÝÑCattaking the polynomialXÐ^mÝ S ÝÑ^p Y to the composite functor

MpK, XqÝ^rifpm,´qÝÝÝÝÑMpK, SqÝÝÝÝÑ^MpK,pq MpK, Yq.

The 2-cellh:pm, S, pq Ñ pn, T, qqinPolyM is taken to the natural trans- formation obtained by the pasting

MpK, Sq

MpK,hq

MpK,pq

))MpK, Xq

rifpm,´q 55

rifpn,´q ))

λˆ

MpK, YqMpK,ρq

MpK, Tq

MpK,qq

55

where ˆλ is the mate under the adjunctions of the natural transformation MpK, λq:MpK, nqMpK, hq ñ MpK, mq.

Proof. We will show that polynomial 2-cells

α:pλ_h, h, ρ_hq ñ pλ¹, h¹, ρ¹q:pm, S, pq Ñ pn, T, gq

are taken to identities. Since MpK, λq “

´MpK, nqMpK, hqùùùùùùùùùñ^{MpK,nqMpK,αq} MpK, nqMpK, h¹q

MpK,λ¹q

ùùùùùñMpK, mq

¯ ,

it follows that λˆ “

´MpK, hqrifpm,´q ^MpK,αqrifpm,´q

ùùùùùùùùùñMpK, h¹qrifpm,´q

ˆλ¹

ùñrifpn,´q

¯ .

Using this and thatρ¹ “ pgαqρ, we have the identity

pgλˆqpρrifpm, uq “ pgλˆ¹qpρ¹ rifpm, uq:f rifpm, uq ùñg rifpm, uq

induced byαas claimed.

ThatHK is a pseudofunctor follows from Proposition 6.6.

We can put somewhat more structure on the image of the pseudofunctor HK. Recall the definition (for example, in [21] Section 3) of the 2-category V-ActofV-actegories for a monoidal categoryV.

Composition in M yields a monoidal structure on the category VK “ MpK, Kq and a right VK-actegory structure on each category HKX “ MpK, Xq:

´ ˝ ´:MpK, Xq ˆMpK, Kq ÝÑK pK, Xq.

We can replace the codomainCatofH^Kin Proposition 8.6 byVK-Act.

To see this, we need aVK-actegory morphism structure on each functor H^Kpm, S, pq “ prifpm,´q :MpK, Xq ÑMpK, Yq.

However, for each a P VK and u P MpK, Xq, we have the canonical mrifpm, uqaùñ^$a uawhich induces a 2-cellrifpm, uqañrifpm, uaq. Whisker- ing this with p : S Ñ Y, we obtain the component at pu, aq of a natural transformation:

MpK, Xq ˆVK prifpm,´qˆ1_VK

´˝´ //MpK, Xq

prifpm,´q

+3

MpK, Yq ˆVK ´˝´ //MpK, Yq.

The axioms for an actegory morphism are satisfied and eachH^Kpm, S, pqis a 2-cell inVK-Act.

In fact, we have a pseudofunctor

H: PolyM ÝÑHompM^op,Actq

whereActis the 2-category of pairspV,Cqconsisting of a monoidal cate- goryV and a categoryC on which it acts.

9. Bipullbacks from tabulations

Tabulations in a bicategory, in the sense intended here, appeared in [7] to characterize bicategories of spans.

For any bicategory M, we write M˚ for the sub-bicategory obtained by restricting to left adjoint morphisms. For each left adjoint morphism f :X ÑY inM, we writef^˚ :Y ÑX for a right adjoint.

Definition 9.1. The bicategory M is said tohave tabulations from the ter- minalwhen the following conditions hold:

(i) the bicategory M˚ has a terminal object 1 with the property that, for all objectsU, the unique-up-to-isomorphism left-adjoint morphism

!_U :U Ñ1is terminal in the categoryMpU,1q;

(ii) for each morphismu: 1ÑX inM, there is a diagram(9.24), called atabulationofu, in whichp: U ÑX is a left adjoint morphism and such that the diagram

MpK, Uq

MpK,pq

//1

ru!Ks

λ +3

MpK, Xq

1_MpK,Xq //MpK, Xq,

(9.23)

where the natural transformationλhas componentpw ÝÝÑ^ρuw u!_UwÝÑ^u!

u!K at w P MpK, Uq, exhibits MpK, Uqas a bicategorical comma object inCat.

U

!U



p

ks ^ρu

1 _u ^//X

(9.24)

Remark 9.2. (a) The bicategorical comma property of the diagram (9.23) impliespis an er-fibration inM.

(b) Notice that condition (ii) in this Definition does agree with combined conditions T1 and T2 in the definition of tabulation in [7] for mor- phisms with domain 1. This is because all left adjoints K Ñ 1 are isomorphic to!_K using condition (i) of our Definition.

(c) Using (b) and Proposition 1(d) of [7], we see that the matep!^˚_U ñ u ofρ_u :p ñu!_U is invertible. Let us denote the unit of the adjunction

!U %!^˚_U byηU : 1U ùñ!^˚_U !U. So we can replaceuup to isomorphism byp!^˚_U andρ_ubypη_U.

(d) IfM has tabulations from the terminal and we have a morphism p : U Ñ X such that (9.23) has the bicategorical comma property with u“p!^˚_U : 1ÑXandρu “pηUthenpis a left adjoint. This is because a tabulation of u : 1 Ñ X does exist in which the right leg is a left adjoint and the comma property implies the right leg is isomorphic to pefor some equivalencee.

(e) Another way to express the comma object condition (9.23) is to say (9.25) is a bipullback forˆλw“λ_w “ ppw ÝÝÑ^ρuw u!_UwÝÑ^u! u!_Kq.

MpK, Uq

ˆλ

! //1

ru!Ks

MpK, Xq² _cod ^//MpK, Xq

(9.25)

LetTabbe the class of morphismsp : U Ñ X inM which occur in a tabulation (9.24).

Theorem 9.3. The class Tabis a calibration for any bicategoryM which has tabulations from the terminal.

Proof. We must prove properties P0–P3 for a calibration. Property P0 is obvious. For P1, take V ÝÑ^q U ÑÝ^p X with p P Tab. By (c) and (d) of Remark 9.2, pcan be assumed to come from the tabulation ofu“p!^˚_U. Ifq is to come from a tabulation it must be of v “q!^˚_V. Ifpq is to come from a tabulation it must be ofw“pq!^˚_V “pv. Contemplate the following diagram in which theλˆcomes fromv.

MpK, Vq

λˆ

! //1

rq!^˚_V!Ks

MpK, Uq² _cod ^//

MpK,pq²

MpK, Uq

MpK,pq

MpK, Xq²

cod //MpK, Xq

By Remark 9.2(a), pis a groupoid fibration; incidentally, this gives P2. So the bottom square is a bipullback (see Proposition 5.1). Therefore, the top square is a bipullback if and only if the pasted square is a bipullback. By Remark 9.2(d) and (e), this says q P Tab if and only if pq P Tab. This proves P1.

It remains to prove P3. We start with a cospanZ ÝÑ^p C Ð^mÝ B withm a right lifter andp P Tab. Putz “ p!^˚_Z : 1Ñ C and tabulatey “ rifpm, zq : 1 ÑBasy “r!^˚_Y forr :Y ÑB inTab. Using the tabulation property of Z, we inducenand invertibleθas in the diagram (9.26) in which the triangle containing$exhibits the right liftingrifpm, zq.

Y

ùñ–θ

!Y

&&

n

r

ùñ

Z ^{!Z //}

p

1

 z

B _m ^//C

gηZùñ

“

Y

rηYùñ

!Y //

r

1

z

y

B _m ^//C

ùñ$

(9.26)

It is the region containing θin (9.26) that we will show is a bipullback. For

allK PM, we must show that the left-hand square in the diagram MpK, Yq

MpK,rq

MpK,nq //MpK, Zq ^//

MpK,pq

– +3

1

rz!Ks

λ +3

MpK, Bq

MpK,mq //MpK, Cq

1_MpK,Cq //MpK, Cq

is a bipullback. However, the right-hand square has the comma property. So the bipullback property of the left-hand square is equivalent to the comma property of the pasted diagram. However, using (9.26), we see that the pasted composite is equal to the pasted composite

MpK, Yq

MpK,rq

//1 ^//

ry!Ks

λ +3

1

rz!Ks

r$!Ks+3

MpK, Bq ₁

MpK,Bq

//MpK, Bq

MpK,mq

//MpK, Cq

Here, the left-hand square has the comma property and y!_K is the value of the right adjoint rifpm,´q to MpK, mq at z!_K. So the pasted composite does have the comma property, as required.

10. Calibrations of Spn C , of Rel E and of Mod

If C is a category with finite limits, its terminal object 1 clearly has the property (i) in the definition of tabulations from the terminal. Then, from Remark 9.2 and [7], we know thatSpnC has tabulations from the terminal.

A span from1toXhas the formp!, U, pq:1ÑXfor somep:U ÑX inC. A tabulation of the span is provided by the diagram

U

!U˚



p˚

!!ks ^ρu

1 p!,U,pq //X .

It follows thatTabconsists of spans of the formp_˚ for some morphismpin C. Using Proposition 4.1, we deduce: