Tesis de Licenciatura 2- ltered bicolimits and nite...

UNIVERSIDAD DE BUENOS AIRES

Facultad de Ciencias Exactas y Naturales

Departamento de Matematica

Tesis de Licenciatura

2-filtered bicolimits and finite weighted bilimitscommute in Cat

Nicolas Abel Canevali

Director: Eduardo J. Dubuc

Fecha de Presentacion: 16 de marzo de 2016

... cercare e saper riconoscerechi e cosa,

in mezzo all’inferno,non e inferno,e farlo durare,

e dargli spazio.

Italo Calvino, Le citta invisibili

2-FILTERED BICOLIMITS AND FINITE WEIGHTED

BILIMITS COMMUTE IN Cat

NICOLAS ABEL CANEVALI

Introduccion

Los lımites y los colımites son importantes construcciones categoricasuniversales que se remontan al origen mismo de la teorıa de categorıas.Por ejemplo, el producto cartesiano de dos conjuntos S y T es el productocategorico en la categorıa Ens de conjuntos (pequenos): un conjunto

S × T con dos funciones proyeccion sobre los factores S × T π1 // S

y S × T π2 // T , universal entre todas las ternas (Z, λ1, λ2) con Z un

conjunto y funciones Zλ1 // S y Z

λ2 // T . En otras categorıas estaconstruccion universal da lugar a distintas nociones: en un poset la mismadescripcion define el ınfimo de dos elementos; en una categorıa de modulossobre un anillo base se corresponde con la suma directa. Un tipo de colımite

es el coegalizador: el coegalizador de dos flechas Af //

g// B es un objeto

C con una flecha Bh // C con hf = hg, universal entre todos los pares

(Z, k) con Bk // Z tales que kf = kg.

La nocion de lımite o colımite en una categorıa C se aplica a cualquier

diagrama, donde por diagrama entendemos un funtor B F // C , con Bla categorıa de indexacion. Los lımites y colımites no necesariamente exis-ten en general, pero cuando existen y tenemos un funtor en dos variables

A×P F // C , podemos tomar lımite o colımite en cualquiera de las dosvariables para obtener un funtor en la otra variable. Por ejemplo, limA←−

F da

un funtor P 7→ limA←−F (A,P ), es decir, que a P le asigna el lımite del funtor

F (−, P ). Luego podemos tomar lımite o colımite en la variable restante. Entotal, hay cuatro combinaciones posibles de lımite o colımite en una u otravariable, y del orden en que se toman los lımites o colımites. El hecho de quelos lımites esten dados por una propiedad de tipo terminal (por ejemplo, losmorfismos canonicos dados por la propiedad universal del producto llegan

al objeto universal: Z∃! // S × T ) hace que tomar lımite en una variable

primero y luego en la otra de lo mismo que hacerlo al reves, porque se definenpor propiedades universales del mismo lado. Es decir,

(0.1) limA←−limP←−

F ∼= limP←−limA←−

F

3

4 NICOLAS ABEL CANEVALI

y de hecho, esto coincide con tomar el lımite de F como un funtor (enuna variable) desde la categorıa producto limA×P←−

F . Mas precisamente, la

ecuacion 0.1 significa que se tienen morfismos canonicos en ambas direcciones,construidos mediante las propiedades universales y los conos universales delos lımites involucrados, que determinan un isomorfismo. En breve, podemosdecir que los lımites conmutan con los lımites.

Los colımites estan dados por una propiedad universal de tipo inicial.Por ejemplo, en el caso del coproducto, los morfismos canonicos salen del

objeto universal: S∐T

∃! // Z . Como es de esperar, al estar dados por

propiedades universales del mismo lado, tambien tenemos que los colımitesconmutan con los colımites

colimA−→colimP−→

F ∼= colimP−→colimA−→

F

(que es lo mismo que colimA×P−→F ).

Esto deja de valer en general cuando se combina un lımite con un colımite:tenemos un morfismo canonico que llamamos “de comparacion” solo en unadireccion

colimA−→limP←−

F♦ // limP←−

colimA−→F

que no podemos asegurar que sea un isomorfismo. Luego, resulta naturalpreguntarse por condiciones sobre A, P , C y F que permitan concluir que ♦sea un isomorfismo.

Un contexto donde esto sucede es bajo las hipotesis de que A sea filtrante,P sea finita, y C sea la categorıa de los conjuntos. Decimos entonces quelos colımites filtrantes y los lımites finitos conmutan en la categorıa de losconjuntos:

(0.2) colimA−→limP←−

F ∼= limP←−colimA−→

F

Esta es una propiedad esencial de la categorıa de los conjuntos, y caracterizaa los colımites filtrantes en Ens: una categorıa A es filtrante precisamente silos colımites indexados por A conmutan con todos los lımites finitos en Ens.La ecuacion 0.2 tambien vale en otras categorıas, como las categorıas dehaces sobre un sitio pequeno, y tiene muchas aplicaciones.

Por ejemplo, podemos definir la nocion de presentacion finita en unacategorıa arbitraria C: un objeto H ∈ C es de presentacion finita si el

funtor C[H,−] // Ens representado por H preserva colımites filtrantes. Puede

verificarse que esta nocion tiene sentido en tanto caracteriza los conjuntosfinitos si C = Ens y coincide con la definicion usual de anillo de presentacionfinita (cociente de un anillo de polinomios en finitas variables por un idealfinitamente generado) si C es la categorıa de anillos. Un requisito importante esque la presentacion finita sea preservada por colımites finitos. La demostracionde este hecho sigue de la siguiente cadena de isomorfismos (donde asumimos

2-FILTERED BICOLIMITS AND FINITE WEIGHTED BILIMITS COMMUTE IN Cat 5

que tenemos un diagrama finito (Hi)i con objetos de presentacion finita, yun diagrama filtrante (Yα)α):

Ens(colimi−→Hi, colimα−→

Yα) ∼= limi←−Ens(Hi, colimα−→

Yα)

∼= limi←−colimα−→

Ens(Hi, Yα)

∼= colimα−→limi←−Ens(Hi, Yα)

∼= colimα−→Ens(colimi−→

Hi, Yα)

Vemos que el hecho de que los colımites filtrantes conmutan con los lımitesfinitos en Ens es esencial en esta demostracion.

Para un segundo ejemplo, si tenemos un funtor

Cp // Ens

se tiene una nocion clara de lo que significa que sea exacto, es decir, quepreserva lımites finitos. Si la categoria C no admite todos los lımites finitos,esta condicion tiene sentido pero es algo vacua. La definicion correcta es la

de funtor playo: el funtor Cp // Ens es playo si su diagrama Γp, tambien

llamado categorıa de elementos, es una categorıa filtrante (un objeto de estacategorıa es un par (x,C) con x ∈ pC y una flecha (x,C) → (y,D) esta

dada por un morfismo Cf // D en C tal que (pf)(x) = y).

Un hecho fundamental es que en el diagrama

C h //

p

��

∼=

EnsCop

p∗

��Ens

el funtor p∗ es exacto precisamente cuando Γp es filtrante, es decir, cuandop es playo. Podemos demostrar una de estas implicaciones utilizando laconmutacion de colımites filtrantes y lımites finitos en Ens: puede verse queel funtor p∗ es de hecho

p∗ = colim−→

(x,C)∈Γp

evC(−)

(donde evC es el funtor EnsCop → Ens de evaluacion en un objeto C ∈ C). Si

Γp es filtrante, entonces p∗ es un colımite filtrante de funtores de evaluacion,que preservan todos los lımites porque los lımites se calculan punto a puntoen las categorıas de funtores. Luego p∗ es un colımite filtrante de funtoresexactos, y es exacto por un argumento analogo al ejemplo anterior, medianteel hecho de que los colımites filtrantes conmutan con los lımites finitos en lacategorıa de los conjuntos.

Puede demostrarse que si C tiene lımites finitos, entonces p es exacto si ysolo si es playo.


La demostracion de la conmutacion de colımites filtrantes y lımites finitosen Ens se hace mediante la construccion del colımite filtrante, que generalizala construccion del anillo de germenes de funciones continuas a valores realesen un punto de un espacio topologico.

Estamos interesados en formular y verificar la ecuacion 0.2, para funtoresa valores en los conjuntos, en una version 2-categorica, para 2-funtores avalores en las categorıas. Este resultado es importante en el desarrollo de lateorıa de 2-funtores 2-playos y en la teorıa de 2-pro-objetos y sus 2-categorıasde modelos de Quillen (ver [8] y [6]). Para esto debemos realizar variasadaptaciones. Naturalmente, las categorıas A y P pasan a ser 2-categorıasy el funtor F pasa a ser un 2-funtor. Los lımites que deben considerarse enla 2-categorıa Cat son no solo los conicos sino los pesados por un segundo

2-funtor P W // Cat . Ası, por ejemplo, podemos fijar A ∈ A y considerar el

lımite de F (A,−) pesado por W , que denotamos wlim←−

W F (A,−). De hecho,

en una 2-categorıa, varias nociones distintas de propiedad universal tienensentido y en el caso que nos concierne utilizamos la de bilımite. La igualdad enla formula 0.2, que entendemos como expresando que cierta funcion canonicaes biyectiva, va a significar que un determinado funtor canonico es unaequivalencia de categorıas. De esta forma, buscamos demostrar la propiedadfundamental de conmutacion

bicolimA−→wbilimW

P←−F ' wbilimW

P←−bicolimA−→

F

cuando A×P F // Cat es un 2-funtor, A es una 2-categorıa 2-filtrante yel lımite pesado por W es finito, en algun sentido relevante para 2-categorıas:va a ser necesario imponer condiciones de finitud no solo sobre la 2-categorıaP, sino tambien sobre el peso W .

Este es un importante resultado que es inedito, y el objetivo de obteneruna demostracion fue el punto de partida de esta tesis.

Asumimos familiaridad con los conceptos y herramientas de la teorıabasica de categorıas para una lectura de este trabajo.

En la seccion 2 introducimos las definiciones de 2-categorıas, 2-funtores,2-transformaciones naturales y modificaciones que aparecen en la teorıade 2-categorıas, junto con ejemplos y variantes. Luego damos nociones defiltrabilidad y finitud aplicables al contexto 2-categorico en la seccion 3 yvarias definiciones utilizadas de 2-lımites en la seccion 4. En la seccion 5trabajamos algunos ejemplos de 2-lımites finitos y pseudocolımites 2-filtrantescon sus construcciones en la 2-categorıa de categorıas, y en la seccion 6construimos 2-lımites y bilımites a partir de lımites mas simples, siguiendo [15],[28] y [10]. La seccion 7 contiene las contribuciones originales de este trabajo:demostramos la ecuacion 0.2, en su version 1-categorica, y seguidamente la


conmutacion de bicolımites 2-filtrantes con biproductos finitos, biegalizadoresy bicotensores con una categorıa finita, por separado, para concluir luego elresultado principal. Finalmente, en la seccion 8, que es independiente conrespecto a nuestro resultado principal, damos definiciones 2-categoricas deends y coends, conceptos que surgieron durante el transcurso del estudio dela teorıa relevante para este trabajo, y las aplicamos para la demostracionde algunas propiedades basicas de los lımites pesados.


Contents

1. Introduction 92. Basic elements of 2-categories 132.1. 2-categories 132.2. 2-functors 212.3. 2-natural transformations 232.4. Modifications 253. Notions of filteredness and finiteness 283.1. Filtered categories 283.2. Finite weights 294. Notions of limits and colimits in a 2-category 315. Examples in Cat 375.1. Product 375.2. 2-product 385.3. Pseudoproduct 405.4. Biproduct 405.5. Equalizer 405.6. 2-equalizer 415.7. Pseudoequalizer 425.8. Biequalizer 465.9. Cotensor 465.10. Pseudocotensor 505.11. Bicotensor 505.12. Inserter and iso-inserter 515.13. Comma-object 535.14. Biequifier and biidentifier 555.15. Descent object 565.16. 2-filtered pseudocolimit of categories 576. Constructions of all limits in terms of simpler limits 616.1. Construction of weighted 2-limits 616.2. Construction of bilimits 647. Commutation of filtered bicolimits

and finite weighted bilimits in Cat 677.1. The 1-dimensional case 677.2. The 2-dimensional case 737.3. 2-product 737.4. Pseudoequalizer 797.5. Cotensor product 877.6. Main result 888. Ends 899. Bibliography 96


1. Introduction

Limits and colimits are important categorical constructions that goback to the very origin of category theory. For example, the cartesianproduct of two sets S and T is the categorical product in the categoryEns of (small) sets: a set S × T together with projection functions

into the factors S × T π1 // S and S × T π2 // T universal among all

triples (Z, λ1, λ2) with Z a set and functions Zλ1 // S and Z

λ2 // T .In other categories, this universal construction gives rise to differentnotions: in a poset the same description defines the infimum of twoelements; in a category of modules over a base ring it corresponds to thedirect sum. A type of colimit is the coequalizer: the coequalizer of two

arrows Af //

g// B is an object C with an arrow B

h // C satisfying

hf = hg, universal among all pairs (Z, k) with Bk // Z such that kf = kg.

The notion of limit or colimit in a category C applies to any diagram,

where by diagram we understand a functor B F // C , with B the indexingcategory. Limits and colimits don’t necessarily exist in general, but when

they exist and we have a functor in two variables A×P F // C , we cantake the limit or the colimit in any of the two variables to obtain a functorin the other variable. For example, limA←−

F is a functor P 7→ limA←−F (A,P )

sending P to the limit of the functor F (−, P ). Then we can take the limitor the colimit in the remaining variable. In total, there are four possiblecombinations of limit and colimit in one or the other variable, and in the orderin which the limits or colimits are taken. The fact that limits are given by auniversal property of terminal type (for example, the canonical morphismsgiven by the universal property of the product go into the universal object:

Z∃! // S × T ) results in that taking limit first in one variable and then in

the other gives the same result as the other way around, because they aredefined by universal properties on the same side. That is

(1.1) limA←−limP←−

F ∼= limP←−limA←−

F

and in fact, this coincides with taking the limit of F as a functor (in onevariable) from the product category limA×P←−

F . More precisely, the equation

1.1 means that we have canonical morphisms in both directions, constructedfrom the universal properties and the universal cones of the limits involved,that determine an isomorphism. In a nutshell, we can say that limitscommute with limits.

Colimits are given by a universal property of initial type. For example, inthe case of the coproduct, canonical morphisms go out of the universal object:

S∐T

∃! // Z . As expected, since they are given by universal properties


on the same side, we also have that colimits commute with colimits

colimA−→colimP−→

F ∼= colimP−→colimA−→

F

(which is the same as colimA×P−→F ).

This is no longer true when we combine a limit with a colimit: we havea canonical morphism that we call a “comparison” morphism only in onedirection

colimA−→limP←−

F♦ // limP←−

colimA−→F

which we can’t guarantee to be an isomorphism. It is then natural to askfor conditions on A, P, C and F that allow us to conclude that ♦ is anisomorphism.

A context where this happens is under the hypotheses of A being filtered,P finite and C the category of sets. We then say that filtered colimits andfinite limits commute in the category of sets:

(1.2) colimA−→limP←−

F ∼= limP←−colimA−→

F

This is an essential property of the category of sets, and characterizes filteredcolimits in Ens: a category A is filtered precisely when colimits indexed byA commute with all finite limits in Ens. The equation 1.2 also holds inother categories, like the categories of sheaves on a small site, and has manyapplications.

For example, we can define the notion of finite presentation in an arbi-trary category C: an object H ∈ C is of finite presentation if the functor

C[H,−] // Ens represented by H preserves filtered colimits. It can be checked

that this notion makes sense inasmuch as it characterizes finite sets if C = Ensand coincides with the usual definition of finitely presented ring (quotientof a polynomial ring in a finite number of variables by a finitely generatedideal) if C is the category of rings. An important requirement is that finitepresentation be preserved by finite colimits. The proof of this fact followsfrom the following chain of isomorphisms (where we assume that we have afinite diagram (Hi)i with objects of finite presentation, and a filtered diagram(Yα)α):

Ens(colimi−→Hi, colimα−→

Yα) ∼= limi←−Ens(Hi, colimα−→

Yα)

∼= limi←−colimα−→

Ens(Hi, Yα)

∼= colimα−→limi←−Ens(Hi, Yα)

∼= colimα−→Ens(colimi−→

Hi, Yα)

We see that the fact that filtered colimits commute with finite limits in Ensis essential in this proof.

As a second example, if we have a functor

Cp // Ens


there is a clear notion of what it means for it to be exact, that is, that itpreserves finite limits. If the category C doesn’t admit all finite limits, thiscondition makes sense but is somewhat vacuous. The correct definition is

that of flat functor: the functor Cp // Ens is flat if its diagram Γp (also

called category of elements) is a filtered category (an object of this categoryis a pair (x,C) with x ∈ pC and an arrow (x,C) → (y,D) is given by a

morphism Cf // D in C such that (pf)(x) = y).

A fundamental fact is that in the diagram

C h //

p

��

∼=

EnsCop

p∗

��Ens

the functor p∗ is exact precisely when Γp is filtered, that is, when p is flat.We can prove one of this implications using the commutation of filteredcolimits and finite limits in Ens: it can be seen that the functor p∗ is in fact

p∗ = colim−→

(x,C)∈Γp

evC(−)

(where evC is the functor EnsCop → Ens by evaluation at an object C ∈ C).

If Γp is filtered, then p∗ is a filtered colimit of evaluation functors, whichpreserve all limits because in functor categories limits are computedpointwise. Then p∗ is a filtered colimit of exact functors, and is exact byan argument analogous to the example above, using the fact that filteredcolimits commute with finite limits in the category of sets.

It can be shown that if C has finite limits, then p is exact if and only if itis flat.

The proof of the commutation of filtered colimits and finite limits in Ensis done by means of the construction of the filtered colimit, which generalizesthe construction of the ring of germs of continuous real-valued functions in apoint of a topological space.

We are interested in formulating and verifying the equation 1.2, forset-valued functors, into a 2-categorical version, for category-valued2-functors. This result is important in the development of the theory of2-flat 2-functors and in the theory of 2-pro-objects and their Quillen model2-categories (see [8] and [6]). In order to do this, we must perform a num-ber of adaptations. Naturally, the categories A and P will be 2-categories,and the functor F will be a 2-functor. The limits that must be consideredin the 2-category Cat are no longer just the conical ones, but also limits

weighted by a second 2-functor P W // F . In this way, for example, wecan fix A ∈ A and we can consider the limit of F (A,−) weighted by W ,which we denote wlim

←−W F (A,−). In fact, in a 2-category, many different


notions of universal property make sense and in the case that we consider,we utilize that of bilimit. The equality in formula 1.2, that we understandas expressing that a certain canonical function is bijective, will mean that acertain canonical functor is an equivalence of categories. Then, we wish toprove the fundamental commutation property


P←−F ' wbilimW

P←−bicolimA−→

F

when A×P F // Cat is a 2-functor, A is a 2-filtered 2-category and thelimit weighted by W is finite, in some sense that is relevant for 2-categories:it will be necessary to impose finiteness conditions not only on the 2-categoryP, but also on the weight W .

This is an important result which is original, and the objetive of obtaininga proof for this result was the starting point for this thesis.

We assume familiarity with the concepts and tools in the basic theory ofcategories for reading this work.

In section 2, we introduce the definitions of 2-categories, 2-functors,2-natural transformations and modifications that appear in 2-category theory,together with examples and variants. We then give notions of filterednessand finiteness applicable to the 2-categorical context in section 3 and variousutilized definitions of 2-limits in section 4. In section 5 we work throughsome examples of finite 2-limits and 2-filtered pseudocolimits with theirconstructions in the 2-category of categories, and in section 6 we construct2-limits and bilimits from simpler limits, following [15], [28] and [10]. Section7 contains the original contributions of this work: we prove the equation 1.2in its 1-categorical version, and following that we prove the commutation offiltered bicolimits and finite biproducts, biequalizers and bicotensors witha finite category, separately, to later conclude the main result. Finally, insection 8, which is independent with respect to our main result, we give2-categorical definitions of ends and coends, concepts that came up duringthe course of the study of the relevant theory for this work, and we applythem in proofs of certain basic properties of weighted 2-limits.


2. Basic elements of 2-categories

A 2-category is a Cat-enriched category, that is, a category in which thehom-sets are themselves categories. This definition would be technicallysufficient to give the concepts of 2-categories, 2-functors, and 2-naturaltransformations simply as Cat-enriched versions of categories, functors, andnatural transformations. However, it is useful to have a more elementarydefinition of these concepts, in terms of objects, morphisms and 2-cells. Thedefinitions we give below are those of strict 2-categories and 2-functors, withon-the-nose equalities, but more lax versions are also useful. An introductionto these topics can be found in [4, chapter 7], [7], [17], or [19].

2.1. 2-categories.

Definition 2.1. A 2-category A is given by

• a collection of objects or 0-cells

A,B,C, . . .

• for each pair of objects A, B, a collection of morphisms (or arrows,or 1-cells) between them, each one indicated as follows

Af // B

• for each pair of parallel morphisms, Af //

g// B , a collection of 2-cells

(or 2-morphisms, or 2-arrows) each one indicated as follows

A

f))

g55⇓α B

• a distinguished identity morphism for each object idA : A→ A• a distinguished identity 2-cell for each morphism idf : f ⇒ f• a way of (horizontally) composing two morphisms with compatible

domain and codomain

Af // B , B

g // C 7→ Agf // C

• a way of vertically composing two 2-cells with compatible domainand codomain, between the same objects

A

f))

g55⇓α B

A

g))

h

55⇓β B

7→ A

f))

h

55⇓β◦α B


• a way of horizontally composing two 2-cells with compatible domainand codomain

A

f))

g55⇓α B , B

h))

k

55⇓β C 7→ A

hf))

kg

55⇓β∗α C

(this composite 2-cell is often denoted βα by simple juxtaposition)

subject to certain axioms:

• compositions are associative (for morphisms f , g, h, and 2-cells α, β,γ such that the compositions make sense in each case)

f ◦ (g ◦ h) = f ◦ (g ◦ h)

α ◦ (β ◦ γ) = α ◦ (β ◦ γ)

α ∗ (β ∗ γ) = α ∗ (β ∗ γ)

• identities are neutral for compositions

Af // B

idB // B = Af // B

AidA // A

f // B = Af // B

A

f

!!

g

??⇓α⇓idgg

// B = A

f))

g55⇓α B

A

f

!!

g

??⇓idf⇓αf

// B = A

f))

g55⇓α B

A

f))

g55⇓α B

idB // B = A

f))

g55⇓α B

AidA // A

f))

g55⇓α B = A

f))

g55⇓α B

• the interchange law:

(δ ◦ γ) ∗ (β ◦ α) = (δ ∗ β) ◦ (γ ∗ α) in A

f##

h

==⇓α⇓β

g // B

f ′

##

h′

==⇓γ⇓δ

g′ // C

Observation 2.2. We draw

A

f))

g55⇓α B

h // C


and

Ah // B

f))

g55⇓α C

to denote the horizontal composition of 2-cells idh ∗α or α ∗ idh, respectively.This operation of composing horizontally with an identity 2-cell is calledwhiskering.

Example 2.3. Any category C can be made into a 2-category by only addingidentity 2-cells for each morphism in C.

Example 2.4. Just as Ens is the paradigmatic category, Cat can be consid-ered as a 2-category, with natural transformations as the 2-cells.Given categories, functors and natural transformations as in

A

F

H

??

⇓α

⇓βG // B

the vertical composition β ◦ α : F ⇒ H is given by components as

(β ◦ α)X = βX ◦ αXwhere X is an object in A.Given categories, functors and natural transformations as in

AF""

G

==⇓α BH""

K

==⇓β C

we can define the horizontal composition β ∗ α : HF ⇒ KG in two ways:either

βGX ◦H(αX)

orK(αX) ◦ βFX

These compositions coincide because

βGX ◦H(αX) = K(αX) ◦ βFX

holds by naturality of Hβ +3 K with respect to the morphism

FXαX // GX . We observe that the first definition corresponds to

β ∗ α = (β ∗ idG) ◦ (idH ∗ α)

and the second one to

β ∗ α = (idK ∗ α) ◦ (β ∗ idF )

Cat can also be considered as an ordinary category, consisting of categoriesand functors. In general, any 2-category becomes an ordinary category bydiscarding the data for the 2-cells. To make this distinction explicit, wewill sometimes say “1-category” to refer to an ordinary category (no 2-cellsinvolved).


Example 2.5. The category Top of topological spaces and continuous fun-cions can be regarded as a sort of higher dimensional category, with homo-topies between continuous functions as 2-cells, homotopies between thosehomotopies as 3-cells, and so on. We can discard the information abovethe second level to obtain a 2-category: it has topological spaces as objects,continuous functions as morphisms, and homotopies (up to homotopy, tomake composition associative) as 2-cells. Furthermore, since every 2-cellis invertible, it is called a (2,1)-category (a (n,s)-category has k-cells fork = 0, 1, . . . , n, where every k-cell for k > s is invertible).

Example 2.6. If C is a 2-category, we denote Cop the 2-category obtainedby reversing only the 1-cells. There are other variants that involve reversingthe 2-cells.

We can organize this information in the external definition of a categoryenriched over Cat (here we consider Cat as a 1-category). This gives thefollowing equivalent description.

Definition 2.7. A 2-category A is given by

• a collection of objects• for any pair of objects A, B, a category A(A,B)• for any three objects A, B, C, a functor

A(A,B)×A(B,C)cA,B,C // A(A,C)

• for each object A, a functor

{•} uA // A(A,A)

(here {•} is the terminal category consisting of one object and theidentity morphism)

satisfying

• associativity: given any four objects A, B, C, D the following diagramcommutes

(2.8)

A(A,B)×A(B,C)×A(C,D)1×cB,C,D //

cA,B,C×1

��

≡

A(A,B)×A(B,D)

cA,B,D

��A(A,C)×A(C,D) cA,C,D

// A(A,D)


• identity: given objects A and B, the following diagrams commute(2.9)

A(A,B)

≡

{•} × A(A,B)∼=oo

uA×1

��

A(A,B)

≡

A(A,B)× {•}∼=oo

1×uB

��A(A,A)×A(A,B)

cA,A,B

ee

A(A,B)×A(B,B)

cA,B,B

ee

These notions correspond in the following way:

• the functors cA,B,C give horizontal composition of morphisms and2-cells• the identity morphism on an object and identity 2-cell on this mor-

phism are given by the functors uA• the associativity diagram translates into associativity of horizontal

composition of morphisms and 2-cells• the identity diagrams say that identity morphisms and identity 2-cells

on identity morphims are neutral elements for horizontal compositionof morphisms and 2-cells, respectively• associativity and identity axioms for the vertical composition of

2-cells follow from the fact that for any two objects A and B we haveA(A,B) a “vertical” category• the interchange law is equivalent to functoriality of cA,B,C

We will often write the hom-category A(A,B) as [A,B].

In many cases it is useful to consider a weakening of the axioms of2-categories. Namely, we could want the associativity and identity diagrams2.8 and 2.9 to commute up to specified natural isomorphisms for each choiceof objects in a coherent way, which can be made precise. This gives riseto the notion of bicategory. We refer to [4, chapter 7], [20] or [1] for anintroduction to bicategories.

Example 2.10. (this is in [2, example 2.6], with more details in [25]) Givena category C with pullbacks, we can define a bicategory of spans with objectsthose of C and morphisms A→ B given by spans

(f, g) =

Xf

~~

g

A B


A 2-cell (f, g) ⇒ (f ′, g′) in this bicategory is given by a morphism h in Cmaking the two triangles commute

Xg

f

~~h

��

A B

X ′g′

>>

f ′

``

These compose vertically in the obvious way. Now, to compose two spans

X1

f1

~~

g1

X2

f2

~~

g2

A B B C

we take a pullback to obtain a new span:

Yp

~~

q

X1

f1

~~

g1

X2

f2

~~

g2

A B C

The choice of a pullback for each pair of objects X1, X2 must be given inadvance so that composition of morphisms is well-defined in this bicategory.However, if we have a third span

X3

f3

~~

g3

A B

we can compose the three morphisms in two different ways:

Z1

r1

~~ s1

Y1

p1

~~

q1

X1

f1

~~

g1

!!

X2

f2

}}

g2

X3

f3

~~

g3

A B C D


or

Z2

r1

~~

s2

Y2

p2

~~

q2

X1

f1

~~

g1

X2

f2

~~

g2

!!

X3

f3

}}

g3

A B C D

corresponding to the compositions

(f3, g3) ◦ ((f2, g2) ◦ (f1, g1))

and

((f3, g3) ◦ (f2, g2)) ◦ (f1, g1)

respectively. Now we see that to have strict associativity we must havechosen the pullbacks in some way such these two compositions are equal:in particular, Z1 and Z2 must coincide. (For example, in Ens we have anisomorphism A×X (B×Y C) ∼= (A×X)×Y C but not an equality.) Instead, wecan ask for associativity up to a certain natural isomorphism: the universalproperties of the spans involved in these two diagrams allow us to constructan invertible 2-cell

(f3, g3) ◦ ((f2, g2) ◦ (f1, g1))⇒ ((f3, g3) ◦ (f2, g2)) ◦ (f1, g1)

This consists of an isomorphism Z1 → Z2 in C that we can obtain by firstusing the universal property of the pullback Y2 and the morphisms q1 ◦ r1

and s1, giving us a morphism h : Z1 → Y2. Then from the universal propertyof Z2 with the morphisms h and p1 ◦ r1 we reach the desired morphism. Theinverse of this morphism can be obtained arguing symmetrically.

Definition 2.11. Two objects A and B in a 2-category are isomorphic,denoted

A ∼= B

if there are morphisms

Af //

Bgoo

that compose to the identity morphisms in both directions:

gf = idA

fg = idB

Observation 2.12. This definition also works in a 1-category since 2-cellsare not involved.

When the 2-category is Cat , we obtain the following


Definition 2.13. Two categories A and B are isomorphic, denoted

A ∼= Bif there exist functors

AF // BGoo

that compose to the identity functors in both directions:

GF = IdA

FG = IdB

In a 2-category there is a second notion of similarity between objects:

Definition 2.14. Two objects A and B in a 2-category are equivalent,denoted

A ' Bif there are morphisms

Af //

Bgoo

that compose to morphisms that are isomorphic to the identity morphismsin both directions: there exist invertible 2-cells α and β such that

A

gf##

idA

==⇓α∼= A B

fg##

idB

==⇓β∼= B

In particular, we have

Definition 2.15. Two categories A and B are equivalent, denoted

A ' Bif there exist functors

AF // BGoo

such that the compositions in both directions are naturally isomorphic tothe identity functors: there exist natural isomorphisms α, β

GFα∼= +3 IdA FG

β∼= +3 IdB

In practice, it is usually the case that one of the functors is canonicallydefined, and the existence of the other follows from the following observation.

Observation 2.16. In [22, theorem 4.1], there is a proof of the fact that

given a functor A F // B , it is sufficient for the existence of G such that thepair (F,G) defines an equivalence of categories that F satisfy the followingconditions

• F is full:A(A,B)→ B(FA,FB)

is surjective for any A,B ∈ A


• F is faithful:

A(A,B)→ B(FA,FB)

is injective for any A,B ∈ A

• F is essentially surjective: given an object B ∈ B, there exists anA ∈ A such that B is isomorphic to FA

Observation 2.17. In the light of observation 2.16 we give the following

Definition 2.18. A functor A F // B is an equivalence if there exists a

functor B G // A such that the pair (F,G) is an equivalence. The functorG is called a quasi-inverse for F .

We remark that this definition has a direction (it is not symmetric on Fand G). The functor G is not determined by F : there can be different choicesfor G that result in the pair (F,G) defining an equivalence of categories,although any two such choices are naturally isomorphic.

2.2. 2-functors.

A 2-functor between 2-categories F : A −→ B is an assignment of objects,morphisms and 2-cells in A, to objects, morphisms and 2-cells in B, respec-tively, that preserves all identities and compositions. In particular, it is afunctor between the underlying categories.

Definition 2.19. Given 2-categories A and B, a 2-functor A F // B gives

• for any object A of A, an object FA of B

• for any morphism Af // B in A, a morphism FA

Ff // FB in B

• for any 2-cell A

f))

g55⇓α B in A, a 2-cell FA

Ff++

Fg

33⇓Fα FB in A

such that

F (g ◦ f) = Fg ◦ FfF (β ◦ α) = Fβ ◦ FαF (δ ∗ γ) = Fδ ∗ Fγ

whenever g and f are morphisms, and α, β, γ, δ are 2-cells such that thespecified compositions make sense in each case, and

F (idA) = idFA

F (idf ) = idFf

with idA the identity morphism on an object A and idf the identity 2-cellon a morphism f .


Observation 2.20. Since we will be working mostly with 2-functors withthe 2-category Cat as codomain, we work out precisely how such a functor

A H // Cat acts.

Given a 2-cell A

f))

g55⇓α B in A, we obtain a corresponding 2-cell Hα in Cat ,

that is, a natural transformation between functors

HA

Hf++

Hg

33⇓Hα HB

Naturality of this HfHα +3 Hg means that for each X

u // Y in HA the

following diagram commutes

(Hf)(X)(Hf)(u) //

(Hα)X

��

≡

(Hf)(Y )

(Hα)Y

��(Hg)(X)

(Hg)(u)// (Hg)(Y )

Observation 2.21. It can be easily checked that the composition of two

2-functors A F // B G // C is a 2-functor A GF // C .

The external definition of a 2-functor is as follows.

Definition 2.22. Given 2-categories A and B, a 2-functor A F // B is anassignment of an object FA of B for each object A of A, and a collection offunctors

A(A,B)FA,B // B(FA,FB)

preserving horizontal compositions:

A(A,B)×A(B,C)FA,B×FB,C //

cA,B,C

��

≡

B(FA,FB)× B(FB,FC)

cFA,FB,FC

��A(A,C)

FA,C

// B(FA,FC)

for all objects A, B, C; and identities:

{•}

≡

uA //

uFA

A(A,A)

FA,A

��B(FA,FA)


for any object A of A.

Again, we can weaken this definition: if we ask that the diagramsexpressing preservation of compositions and identities commute only up to aspecified natural isomorphism in a coherent way we obtain the notion of apseudofunctor.

The precise definition will not be necessary, but we make the observationthat any 2-functor is a pseudofunctor, with identities as the chosen naturalisomorphisms.

Example 2.23. We have representable 2-functors: for A an object in A,

the 2-functor A[A,−] // Cat is defined on objects and arrows as for ordi-

nary representable functors: if B is an object in A, we have [A,B] thehom-category between A and B, and if f : B −→ C in A, the functorf∗ = [A, f ] : [A,B] −→ [A,C] is defined by postcomposition

h 7→ fh

α : h⇒ h′ 7→ idf ∗ α

For B

f))

g55⇓α C in A, [A,α] : [A, f ] =⇒ [A, g] is a natural transformation

defined on components by

[A,α]h = α ∗ idh

Similarly, we have contravariant 2-functors Aop[−,A] // Cat .

2.3. 2-natural transformations.

Definition 2.24. Given parallel 2-functors between 2-categories AF //

G// B ,

a 2-natural transformation η : F =⇒ G is a family of morphismsηA : FA −→ GA in B (called component of η at A) for each object Aof A, compatible with morphisms and 2-cells:

• for all Af // B in A the following diagram commutes

(2.25)

FA

≡

ηA //

Ff

��

GA

Gf

��FB ηB

// GB


• for all A

f))

g55⇓α B in A we have

FA

Ff++

Fg

33⇓Fα FBηB // GB = FA

ηA // GA

Gf++

Gg

33⇓Gα GB

Observation 2.26. 2-natural transformations can be composed both verti-cally and horizontally, with the same definitions as for natural transforma-tions, given in example 2.4.

Example 2.27. Continuing example 2.23, if we have Af // A′ in A,

there is a 2-natural transformation [f,−] : [A,−] ⇒ [A′,−] : A −→ Catby precomposition. Its component at the object B is a functor[f,−]B : [A′, B] −→ [A,B] defined as

g 7→ gf

α : g ⇒ h 7→ α ∗ idf

There is a useful relaxation of the notion of 2-natural transformation.Since the diagram 2.25 is of objects and arrows in B, a 2-category, we couldask that it be commutative only up to specified invertible 2-cells in a coherentway.

Definition 2.28. Given parallel 2-functors between 2-categories AF //

G// B ,

a pseudonatural transformation η : F =⇒ G is a family of morphismsηA : FA −→ GA in B for each object A of A, together with invertible 2-cells

ηf for each Af // B in A

FAηA //

Ff

��⇐=ηf

GA

Gf

��FB ηB

// GB

given in a coherent way:

• ηidA = idηA

FAηA //

≡

GA

FA ηA// GA


• ηgf = (ηg ∗ idFf ) ◦ (idGg ∗ ηf )

FAηA //

Ff

��⇐=ηf

GA

Gf

��FB

ηB //

Fg

��⇐=ηg

GB

Gg

��FC ηC

// GC

=

FAηA //

F (gf)

��⇐=ηgf

GA

G(gf)

��FC ηC

// GC

and compatible with 2-cells: for all A

f))


FAηA //

Ff

Fα⇐Fg

��⇐=ηf

GA

Gf

��FB ηB

// GB

=

FAηA //

Fg

��⇐=ηg

GA

Gf

Gα⇐Gg

��FB ηB

// GB

(idηB ∗ Fα) ◦ ηf = ηg ◦ (Gα ∗ idηA)

We observe that if the invertible 2-cells are the identities we get back thedefinition of 2-natural transformation.

Observation 2.29. Pseudonatural transformations can also be composedboth vertically and horizontally. This requires an elaborate but straightfor-ward verification.

2.4. Modifications.

We give the definition of modification between two pseudonatural trans-formations, which, as we have seen, subsumes the case of 2-natural transfor-mations.

Definition 2.30. Given 2-functors and pseudonatural transformations

AF))

G

55⇓η ⇓σ B , a modification between them ζ : η σ is a collection of 2-cells

ηA ⇒ σA in B for each object A in A, such that for any Af // B in A we

have


FAηA //

Ff

��

⇐=ηf

GA

Gf

��

=

FA

ηA++

⇓ζAσA

33

Ff

��⇐=σf

GA

Gf

��FB

ηB++

⇓ζBσB

33 GB FB σB// GB

(ζB ∗ idFf ) ◦ ηf = σf ◦ (idGf ∗ ζA)

Observation 2.31. Given any 2-cell A

f))


(ζB ∗ Fα) ◦ ηf = ((idσB ◦ ζB) ∗ (Fα ◦ idFf )) ◦ ηf identities

= (idσB ∗ Fα) ◦ (ζB ∗ iFf) ◦ ηf interchange law

= (idσB ∗ Fα) ◦ σf ◦ (idGf ∗ ζA) ζ modification

= σg ◦ (Gα ∗ idσA) ◦ (idGf ∗ ζA) σ pseudonatural

= σg ◦ ((Gα ◦ idGf ) ∗ (idσA ◦ ζA)) interchange law

= σg ◦ (Gα ∗ ζA) identities

So in fact ζ is compatible with all 2-cells α in A, not just identity 2-cells onmorphisms:

(ζB ∗ Fα) ◦ ηf = σg ◦ (Gα ∗ ζA)

Observation 2.32. Given modifications between 2-natural transformations

αζ β

ρ γ

with α, β, γ : F ⇒ G : A −→ B, we can compose them vertically: ρ ◦ ζ isgiven by components as

(ρ ◦ ζ)X = ρX ◦ ζXif X ∈ A. Horizontal composition is similarly defined: given 2-categories,2-functors, 2-natural transformations and modifications as in

A

F

H

??

⇓α ζ ⇓β

⇓γ ρ ⇓δG // B

we can define the horizontal composite ρ ∗ ζ by components as

(ρ ∗ ζ)X = ρX ∗ ζXif X ∈ A.

It can be checked that these compositions, together with the obviousidentities, define a 2-category [A,B] of 2-functors, 2-natural transformations


and modifications (see observation 2.26).

Example 2.33. Continuing the examples of 2-functors and 2-naturaltransformations [A,−] and [f,−] given by objects and morphisms in a2-category A (examples 2.23 and 2.27), we can consider the modification

[α,−] associated to a 2-cell A

f))

g55⇓α B .

For allX inA, the component [α,−]X : [f,−]X =⇒ [g,−]X : [B,X] −→ [A,X]

is a natural transformation (a 2-cell in Cat), and for Bs // X , we have

[α,−]X,s = ids ∗ αAt this point we can speak of the Yoneda embedding in the 2-categoricalsetting: the assignment A 7→ [A,−] is a 2-functor h : A −→ [A, Cat ]op (see[7]).

Theorem 2.34. Given a 2-functor A F // Cat , we have an isomorphismof categories

2Nat([A,−], F ) ∼= FA

where the right side is the category of 2-natural transformations [A,−]⇒ Fand modifications between them. �

This theorem can be seen as a particular case of the enriched Yonedalemma when the enriching category is Cat , or it can be proved by elementarymeans. Setting F = [A′,−] we conclude that h : Aop −→ [A, Cat ] is fullyfaithful as an ordinary functor and locally fully faithful. This will allow usto see A as a (full and locally full) subcategory of [A, Cat ]op, useful in theconstruction of the 2-filtered pseudocolimit (see section 5.16).

Example 2.35. We can instead consider [A,B]p: 2-functors, pseudonaturaltransformations and modifications also organize themselves into a 2-category.The identity on objects and the full inclusions

[A,B](F,G) �� i // [A,B]p(F,G)

determine a locally full (but not full) 2-functor

[A,B]i // [A,B]p

For the 2-functor A // [A, Cat ]opp there is an adequate version of the

Yoneda lemma (see [7, proposition 1.1.18]).


3. Notions of filteredness and finiteness

3.1. Filtered categories.

We first recall the definition of filteredness in an ordinary category.

Definition 3.1. A category A is filtered if

• it is non-empty• for any two objects there is a third object further ahead: given objectsA, B there exists an object C and arrows as in

(3.2)

A

%%C

B

99

• any two parallel arrows are equalized further ahead: given f and gthere exists C and an arrow h such that hf = hg in

(3.3) Af //

g// B

h // C

Observation 3.4. It is easy to prove that this is equivalent to the conditionthat every finite diagram in A admits a cone. This is done in [23, lemma 6.1].

Example 3.5. Every directed poset is filtered when seen as a category.

Example 3.6. A category with a terminal object is filtered.

Example 3.7. A category with binary products and coequalizers is filtered.

In [10] we find the following

Definition 3.8. A 2-category A is 2-filtered if

• it is non-empty• (F0) for any two objects A, B there is a third object further ahead

A

%%C

B

99

• (FF1) given E1

f1 ??

g1 ��

A

B

, E2

f2 ??

g2 ��

A

B

there exist invertible 2-cells

E1

f1 ??

g1 ��

⇓γ1∼=

Au��

Bv

??C , E2

f2 ??

g2 ��

⇓γ2∼=

Au��

Bv

??C


• (F2) given (not necessarily invertible) 2-cells E

f ??

g ��

⇓γ1

Au1��

Bv1

??C1 ,

E

f ??

g ��

⇓γ2

Au2��

Bv2

??C2 there exist invertible 2-cells α, β such that

E

f??

g ��

⇓γ1

Au1

��

Bv1

??

v2 ��

⇓α∼=

C1w1

��

C2

w2

??C

= E

f??

g ��

⇓γ2

Au2

��

u1??

⇓β∼=

C1w1

��

Bv2

??C2

w2

??C

This 2-categorical analogue of filteredness can be formulated in differentways. In particular, the notion of bifiltered category appears in [18] (alsosee [10, definition 2.6]). In [5, section 5], these definitions are shown to beequivalent to the condition that every finite diagram admits a pseudocone.

Definition 3.8 can be weakened to give the concepts of pre-2-filterednessand pseudo-2-filteredness, which can be found in [10]. For the results that wewant to prove, however, we need to use the whole strength of the definitionof 2-filtered 2-category.

We observe that an ordinary 1-category is filtered if and only if it is2-filtered when seen as a 2-category with identity 2-cells, so that 2-filterednessis a generalization of 1-filteredness.

3.2. Finite weights.

In the 1-dimensional case, a limit of a functor P F // C is calledfinite when the category P is finite, and we know that finite limits canbe constructed from binary products and equalizers, as we will showlater on, and that these limits commute with filtered colimits. In the

2-dimensional case, we also have a weight (a 2-functor) P W // Catand the finite condition just on P is not sufficient to guarantee thatthe 2-limit wlim

←−W F can be constructed from the corresponding three

types of basic finite 2-limits, nor that such a limit will commute with


2-filtered 2-colimits: the weight functor must also satisfy finiteness conditions.

In [15] and [16], Kelly gives a definition of a finite 2-limit to be one withthe indexing 2-category P having a finite collection of objects and finitelypresentable hom-categories, and such that for all P ∈ P also WP is afinitely presentable category. This is precisely the type of 2-limit that can beconstructed from binary 2-products, 2-equalizers and cotensor products witha finite category (in fact, cotensor products with the category 2 = {0→ 1}suffice). However, we find convenient to use a simplified definition, which isenough for our purposes.

Definition 3.9. A 2-functor P W // Cat is a finite weight wheneverthe indexing 2-category P as well as every category WP for P ∈ P isfinite (this means a finite collection of objects and morphisms in the caseof a category, and also a finite collection of 2-cells in the case of a 2-category).

A weighted 2-limit wlim←−

W F is called finite when its weight P W // Cat is

a finite weight. The same terminology applies for bilimits and pseudolimits.


4. Notions of limits and colimits in a 2-category

We speak mostly about limits, as the discussion can be dualized to referto colimits.

In an ordinary category C, the definition of a limit can be given in several

equivalent ways. The most elementary way of saying that a functor B F // Chas a limit L, with morphisms L

πB // FB for each object B of B, is thatthis data defines a cone for F , i.e. the following diagram commutes for all

Bf // B′

L

πB

��

πB′

��

≡

FBFf

// FB′

and every other such cone (Z, (µB)B) factorizes uniquely through (L, (πB)B)

Z

∃!

��≡ ≡

µB

��

µB′

��

L

πB

��

πB′

��

≡

FBFf

// FB′

Definition 4.1. Given categories B, C and an object Z ∈ C, there is afunctor

B ∆Z // Csending every object B 7→ Z and every morphism f 7→ idZ . This is theconstant functor at Z. In fact, we can see ∆ as a functor

C ∆ // CB

with object function Z 7→ ∆Z in a straightforward way.

Using this definition, (L, (πB)B) is a limit for F when ∆Lπ +3 F is a

natural transformation that induces by postcomposition a (natural) bijectionfor all Z in B

C(Z,L)π∗∼= // Nat(∆Z , F )


The set Nat(F,G) of natural transformations between two functorsF,G : B −→ C is in fact the hom-set in the functor category CB, so wehave a natural bijection

C(Z,L)π∗∼= // CB(∆Z , F )

Again, we can rephrase this in terms of another ubiquitous notion incategory theory: (L, π) is a limit of F when it is a representation of thefunctor CB(∆−, F ) : B −→ Ens, which maps

Z 7→ CB(∆Z , F )

If the limit exists for all diagrams F of shape B in C, we can say thatlim←−

: CB −→ C is a functor, which is right adjoint to the diagonal ∆ : C −→ CB

C(Z, lim←−

F ) ∼= CB(∆Z , F )

In the general V-enriched case, these definitions of limit don’t make sense,because we cannot speak of morphisms Z → FB, and ∆ : C −→ CB mightnot exist (see [13]). We observe that we have

CB(∆Z , F ) ∼= EnsB(∆1, C(Z,F−))

because a natural transformation ∆1 ⇒ C(Z,F−) amounts to a func-tion from the singleton set {∗} −→ C(Z,FB) for each B in B satisfyingnaturality conditions, and a function from the singleton set is just an element.

The functor ∆1 might still not make sense in the enriched setting, butwe can now substitute it with any other (enriched) functor W that we callthe weight. This gives the definition of a weighted (or indexed) limit forcategories enriched on an arbitrary monoidal category V , which can be foundin [30].

Definition 4.2. Given V-functors B W // V and B F // C , a W -weightedlimit of F , or limit of F weighted by W is a representation of the functor

Z 7→ [B,V](W, C(Z,F−))

where [B,V ] is the enriched functor category, and C(Z,F−) is just C(Z,−) ◦ F .This representation consists of an object wlim

←−W F in C and the unit: a

V-natural transformation π : W =⇒ C(wlim←−

W F, F−). We also denote the

limit object by {W,F}.

An introduction to weighted limits and colimits can be found in [3], [27],[26, chapter 7], or [12].

From now on, we will restrict ourselves to the case V = Cat , which hasmany special features. In particular, we do have the diagonal 2-functor ∆.This gives the following definition, that can be found, for example, in [15]:


Definition 4.3. (weighted 2-limit) Given 2-functors B W // Cat and

B F // C , a W -weighted 2-limit of F , or 2-limit of F weighted by W is a2-representation of the functor

Z 7→ [B, Cat ](W, C(Z,F−))

sending an object Z in C to the category of 2-natural transformationsand modifications between the 2-functors W and C(Z,F−). That is, anisomorphism of categories

C(Z,wlim←−

W F ) ∼= [B, Cat ](W, C(Z,F−))

2-natural in Z.This representation consists of an object wlim

←−W F in C and a 2-natural

transformation π : W =⇒ C(wlim←−

W F, F−). The object wlim←−

W F is also

denoted {W,F}.

We call this a 2-limit, to emphasize that the universal property of therepresenting object is an isomorphism of categories, and not just a bijectionof sets as in the ordinary 1-dimensional definition of a limit.

Observation 4.4. In [15, section 3], there is an observation that when a2-category C admits a specific kind of weighted 2-limit (all cotensors with thecategory 2 = {0→ 1}), the 2-dimensional property of any limit follows fromthe 1-dimensional universal property. This observation can be proved from[14, theorem 4.85]. This means that in Cat (a 2-category that admits cotensorswith 2 = {0 → 1}), some verifications could be omitted when we want toprove that a given construction is the solution of the 2-dimensional universalproperty. In practice, however, an explicit description of the unique 2-cellsdefined by 2-dimensional universal properties may be needed. Furthermore,it is instructive to give the explicit formulation involving the 2-dimensionalaspect, and we choose to do so in many cases. Cotensors are defined in 5.10.

We can dualize this definition to obtain one of weighted 2-colimits.

Definition 4.5. (weighted 2-colimit) Given 2-functors Bop W // Cat

and B F // C , a W -weighted 2-colimit of F , or 2-colimit of F weighted byW is a 2-representation of the functor

Z 7→ [Bop, Cat ](W, C(F−, Z))

sending an object Z in C to the category of 2-natural transformationsand modifications between the 2-functors W and C(F−, Z). That is, anisomorphism of categories

C(wcolim−→

W F,Z) ∼= [Bop, Cat ](W, C(F−, Z))

2-natural in Z.This representation consists of an object wcolim

−→W F in C (also denoted W ∗F )

and a 2-natural transformation λ : W =⇒ C(F−,wcolim−→

W F ).


With the definitions we’ve given so far, there is a possible variation to thenotion of a 2-limit or 2-colimit. In the formula

[B, Cat ](W, C(Z,F−))

we can replace the 2-functor 2-category [B, Cat ] of 2-functors, 2-naturaltransformations and modifications by the 2-category [B, Cat ]p of 2-functors,pseudonatural transfomations, and modifications.

Definition 4.6. (weighted pseudolimit) Given 2-functors B W // Cat

and B F // C , a W -weighted pseudolimit of F , or pseudolimit of F weightedby W is a 2-representation of the functor

Z 7→ [B, Cat ]p(W, C(Z,F−))

and is denoted wpslim←−

W F .

The weighted pseudocolimit wpscolim−→

W F is analogously defined.

We are interested in the representation being given by an equivalenceinstead of an isomorphism. We have, for example, the following

Definition 4.7. (weighted bilimit) Given 2-functors B W // Cat and

B F // C , a W -weighted bilimit of F , or bilimit of F weighted by W is abirepresentation of the functor

Z 7→ [B, Cat ]p(W, C(Z,F−))

That is,C(Z,wbilim

←−W F ) ' [B, Cat ]p(W, C(Z,F−))

is an equivalence of categories.

Observation 4.8. From the definitions above it follows that any two so-lutions of the universal problem will be equivalent objects. Thus, whenpseudolimits exist, they will be equivalent to any choice of bilimits on thesame data, but not isomorphic.

Whenever the weight of a weighted 2-limit or 2-colimit is the functor ∆1,we call such 2-limits or 2-colimits conical, as they can be defined in termsof universal appropriate cones. When a 2-limit is conical, we omit the letterw in the notation, as conical limits coincide with the classical notions of2-limits and 2-colimits.

Example 4.9. We have

(4.10) [B, Cat ]p(∆1, C(Z,F−)) ∼= [B, C]p(∆Z , F )

as in the unenriched, 1-categorical case. A conical pseudolimit of a functor

A F // C can be defined as a representation of the functor

Z 7→ [B, C]p(∆Z , F )


It is given by a pseudocone with vertex pslim←−

F for the functor F : a pseudo-

natural transformation

∆pslim←−

Fπ +3 F

or just written

pslim←−

Fπ +3 F

universal in the sense that such that every other pseudocone with vertex Zan object of C

Zµ +3 F

factorizes through π in a unique way

Z∃! //

µ�&

≡

pslim←−

F

π

��F

This is not sufficient to specify what a pseudolimit is, since it implies thatpostcomposition with the universal pseudocone π

C(Z,pslim←−

F )π∗ // [B, C]p(∆Z , F )

is a bijection for all Z, and not an isomorphism of categories. The additionalrequirement is that any modification between two pseudocones with the samevertex

Z

µ%-

ρ19

ζ F

must also factorize uniquely through π: if we have

Z∃!u //

µ�&

≡

pslim←−

F

π

��F

Z∃!v //

ρ�&

≡

pslim←−

F

π

��F

then for all modifications

Z

µ%-

ρ19

ζ F

there exists a unique 2-cell in C

Z

u ++

v33

⇓α pslim←−

F

such that

Z

u ++

v33

⇓α pslim←−

Fπ +3 F ≡ Z

µ%-

ρ19

ζ F


Conical 2-limits can be described in these terms making the obvioussubstitutions: instead of pseudonatural transformations ∆Z =⇒ F whichare pseudocones, we consider 2-natural transformations ∆Z =⇒ F which wemight call 2-cones. Of course, the case of colimits is identical once we revertall the arrows.

We will say 2-filtered bicolimit to refer to a conical bicolimit where theindexing category (the shape of the diagram) is 2-filtered.

It can be instructive to spell out the details and draw the pseudocones aswe usually draw cones in the 1-categorical case (see [7]). We take up thisproposal for particular examples of limits and colimits in the next section.


5. Examples in Cat

5.1. Product.

We will consider binary products, but all the discussion in this subsectioncan be generalized to arbitary products in a straightforward way.

If C is an ordinary category (a 1-category) and A and B are two objectsin C, we say that a third object denoted A×B with projections π1, π2

A

A×B

π1;;

π2 ##B

is a product of A and B if it is universal in the sense that every other suchdiagram factorizes uniquely through it: for any C, µ1 and µ2, there exists aunique morphism that we can call (µ1, µ2) such that the triangles commute

(5.1)

A

C∃!(µ1,µ2) //

µ111

µ2 --

≡

≡A×B

π199

π2%%B

This is just the definition of a limit in a 1-category, as we discussed insection 4, where the functor is a diagram of shape {• •} with image {A B}.We can say that

C(Z,A×B) ∼= C(Z,A)× C(Z,B)

is a natural bijection by postcomposition. We observe that

C(Z,A)× C(Z,B) ∼= (C × C)((Z,Z), (A,B)) ∼= C{• •}(∆Z , F )

so that this is indeed the usual representability formula

(5.2) C(Z,A×B) ∼= C{• •}(∆Z , F )

This limit exists in Cat : the product of two categories is given bythe usual construction with objects and morphisms of A × B given bypairs of objects or morphisms (the first in A and the second in B) respectively.

We now consider 2-categorical analogues of this limit. This will consistin giving elementary definitions of the different variations on the notion oflimit, applied to the case of a binary product.


5.2. 2-product.

The first way of turning the product into a 2-categorical limit is by replacingthe natural bijection of sets in equation 5.2 by a 2-natural isomorphism ofcategories

C(Z,A×B) ∼= C{• •}(∆Z , F )

where now C is a 2-category. If this representing object A×B exists, it isthe 2-product of A and B.Analogous to the isomorphism 4.10, we have

[B, Cat ](∆1, C(Z,F−)) ∼= [B, C](∆Z , F )

in general. Thus A×B is defined by the formula

C(Z,A×B) ∼= [{• •}, Cat ](∆1, C(Z,F−))

and it is then an example of a conical limit.In addition to the 1-dimensional universal property of the usual productgiven by diagram 5.1, if we have two cones

A A

C

µ1>>

µ2

C

ρ1>>

ρ2 B B

with corresponding morphisms into A×B given by the universal property

C(µ1,µ2) // A×B C

(ρ1,ρ2) // A×B

then 2-cells

C

(µ1,µ2)**

(ρ1,ρ2)

44⇓ξ A×B

must correspond bijectively to modifications µ ρ by postcompositionwith π. Since {• •} is discrete, a modification µ ρ amounts to a pair of2-cells µ1 ⇒ ρ2, µ1 ⇒ ρ2 satisfying no compatibility equations.

The universal property of the 2-product then includes the 1-dimensionalone of the ordinary product. That means that if the 2-product exists, thenit is an ordinary product as well. If we have a construction of the productA×B of two objects in a category C, the 2-product of the same objects, if itexists, must be isomorphic to the given construction.

In Cat , the 2-product exists, and is given by the same usual constructionof the ordinary product.


Proposition 5.3. Given categories A and B, there exists a category A× Bwith functors

A

A× B

π1;;

π2 ##B

such that postcomposition with (π1, π2) gives an isomorphism of categories

[Z,A× B]π∗∼= // Cone(Z)

between the hom-category [Z,A×B] and the category of cones with vertex Zfor the diagram {A B} and modifications between them, for all categoriesZ.

Proof. To verify this, we construct the category A× B and the projectionsπ1 and π2 as in the 1-categorical construction of the product, and onlycheck the 2-dimensional universal property since we already know that the1-dimensional property holds.Given a category Z and functors F : Z −→ A, G : Z −→ B, F ′ : Z −→ Aand G′ : Z −→ B, and natural transformations ζ1 : F =⇒ F ′, ζ2 : G =⇒ G′,we want to see that there exists a unique 2-cell (a natural transformation)γ : (F,G)⇒ (F ′, G′) such that

ζ1 = idπ1 ∗ γζ2 = idπ2 ∗ γ

this gives only one option for the components of γ

ζ1,Z = π1(γZ)

ζ2,Z = π2(γZ)

if Z is an object of Z, so that it must be

γZ = (ζ1,Z , ζ2,Z)

which gives us the uniqueness. In order to prove existence we need to check

that γ defined in this way is a natural transformation: for all Zf // Z ′ in Z

(FZ,GZ)(ζ1,Z ,ζ2,Z)

//

≡(Ff,Gf)

��

(F ′Z,G′Z)

(F ′f,G′f)

��(FZ ′, GZ ′)

(ζ1,Z′ ,ζ2,Z′ )// (F ′Z ′, G′Z ′)

but this is just, component-wise, the naturality of ζ1 and ζ2. �

We remark that this verification of the 2-dimensional universal propertyis unnecessary, because of observation 4.4.


5.3. Pseudoproduct.

We can define a less strict version of the 2-categorical product by us-ing pseudocones and pseudonatural transformations instead of 2-cones and2-natural transformations:

C(Z,A×B) ∼= [{• •}, Cat ]p(∆1, C(Z,F−))

However, if B is a discrete 2-category (just a set) a pseudonatural transforma-

tion between 2-functors BF //

G// C is the same as a 2-natural transformation,

and in fact, it is just a collection of morphisms FB → GB in C, one for eachobject B, satisfying no compatibility conditions. In this case, we have

[{• •}, Cat ]p(∆1, C(Z,F−)) ∼= [{• •}, Cat ](∆1, C(Z,F−))

so the pseudoproduct is the same as the 2-product.

5.4. Biproduct.

Definition 5.4. (biproduct) The biproduct of two objects A and B in Cis a pair of morphisms Z

h // A and Zk // B inducing an equivalence

of categories between C(C,Z) and C(C,A)× C(C,B), for all objects C ∈ C.

Observation 5.5. This definition can be given in elementary terms, resultingin a sentence involving the 1-dimensional and the 2-dimensional universalproperties. This is done by carefully unfolding the definition of equivalence(definition 2.15) or its characterization (in observation 2.16). We will beconsidering biproducts in the 2-category Cat , which admits pseudoproducts(see subsection 5.3), so that we can work with these because of observation4.8.

5.5. Equalizer. In an ordinary category C, an equalizer of

Af //

g// B

is an object E with morphism Ei // A such that

Ei // A

f //

g// B

commutes, that is, fi = gi, universal in the sense that for every other

Af //

g// B

Z

c

??


with fc = gc there is a unique dashed morphism making the triangle commutein

E≡

i // Af //

g// B

Z

∃!c′OO

c

??

This is the same as saying that if F is a diagram of shape {• ⇒ •} with

image Af //

g// B in C, postcomposition with i gives a bijection of sets

C(Z,E) ∼= [{•⇒ •}, C](∆Z , F )

The equalizer of two functors AF //

G// B exists in Cat : it is given by the

subcategory of A consisting of objects and morphisms equalized by F andG.

5.6. 2-equalizer.

We now consider

C(Z,E) ∼= [{•⇒ •}, C](∆Z , F )

a natural isomorphism of categories instead of just a bijection of sets. Since

[{•⇒ •}, C](∆Z , F ) ∼= [{•⇒ •}, Cat ](∆1, C(Z,F−))

this is a conical 2-limit: the weight is ∆1.

We spell out the 2-dimensional universal property of the 2-equalizer: if wehave fd = gd in

E≡

i // Af //

g// B

Z

∃!d′OO

d

??

and a 2-cell

Z

c))

d

55⇓α A

such thatidf ∗ α = idg ∗ α

then there exists a unique 2-cell Z

c′))

d′55⇓ϕ E such that

Z

c′))

d′55⇓ϕ E

i // A = Z

c))

d

55⇓α A


that is, idi ∗ ϕ = α.

Observation 5.6. Also in this case the 2-equalizer is given in Cat by theusual construction of the equalizer. We suppose given categories and functors

AF //

G// B

and define E as the subcategory of A with morphisms and objects those

equalized by F and G, with E I // A the inclusion. If a functor Z C // A

equalizes F and G, the functor Z C′ // E induced by the universal propertyis given by C ′(A) = C(A) on objects and by C ′(f) = C(f) on morphisms.

Given a second functor Z D // A and a natural transformation

ZC))

D

55⇓α A

such that idF ∗ α = idG ∗ α, the corresponding natural transformation

ZC′((

D′66⇓ϕ E

must satisfy

idI ∗ ϕ = α

If Z is an object in Z, the equation I(ϕz) = αz must hold. This gives onlyone possible definition ϕz = αz, and it is well-defined (i.e. it is a morphismin E) because

F (αZ) = (idF ∗ α)Z

= (idG ∗ α)Z

= G(αZ)

Naturality of ϕ is equivalent to naturality of α.

As before in proposition 5.3, this verification is unnecessary.

5.7. Pseudoequalizer.

We define the pseudoequalizer E as the conical pseudolimit of the same

diagram Af //

g// B :

C(Z,E) ∼= [{•⇒ •}, Cat ]p(∆1, C(Z,F−))

Since the indexing category {•⇒ •} has non-identity morphisms, we can’texpect this new definition to be equivalent to that of the 2-equalizer, because


the pseudocone condition is not vacuous in this case. In fact, a pseudoe-qualizer is given by a pseudocone, that is morphisms i and p and invertible2-cells λ1 and λ2 as in

⇓λ1∼=

A

f

��E

i

??

p //

i��

B

⇑λ2∼=

A

g

??

These 2-cells have no compatibility conditions since there are no non-trivialcompositions in {•⇒ •}. The 1-dimensional universal property states thatfor every other such pseudocone

⇓ε1∼=

A

f

��Z

h

??

q //

h��

B

⇑ε2∼=

A

g

??

there is a unique morphism Zk // E such that the composition of k and the

universal pseudocone (i, p, λ1, λ2) gives this other pseudocone (h, q, ε1, ε2):

ik = h

pk = q

λ1 ∗ idk = ε1

λ2 ∗ idk = ε2

The 2-dimensional universal property says that given a third pseudocone

⇓ε′1∼=

A

f

��Z

h′??

q′ //

h′ ��

B

⇑ε′2∼=

A

g

??

and the corresponding morphism Zk′ // E , modifications

(h, q, ε1, ε2) (h′, q′, ε′1, ε′2)


correspond bijectively to 2-cells k ⇒ k′. That is, given 2-cells ψA : h⇒ h′

and ψB : q ⇒ q′ such that

ε′1 ◦ (idf ∗ ψA) = ψB ◦ ε1

ε′2 ◦ (idg ∗ ψA) = ψB ◦ ε2

there is a unique 2-cell

Z

k))

k′55⇓ϕ E

such that

idi ∗ ϕ = ψA

idp ∗ ϕ = ψB

Pseudoequalizers also exist in Cat , but are no longer given by the ordinary1-categorical construction of the equalizer. We did not find a reference forthe construction of pseudoequalizers in Cat .

Proposition 5.7. Given functors and categories AF //

G// B , there exists a

category E, together with functors E I // A , E P // B and natural isomor-phisms λ1 : FI =⇒ P , λ2 : GI =⇒ P (that is, a pseudocone for the diagram

AF //

G// B ) that induces a natural isomorphism of categories between functors

into E and pseudocones of this diagram by postcomposition

I∗ : [Z, E ]∼= // PsCon(Z)

Proof. We propose a category E as a candidate for the pseudoequalizer, andprove that it satisfies the desired property.An object of E is given by a tuple (A,B, γ, δ), with A ∈ A, B ∈ B, and

γ : FA∼= // B and δ : GA

∼= // B isomorphisms in B. (In particular, wehave FA ∼= GA.)

An arrow (A,B, γ, δ)(α,β) // (A′, B′, γ′, δ′) is given by morphisms

α : A −→ A′, β : B −→ B′ commuting with γ, γ′, δ, δ′:

FA

≡Fα��

γ // B

β��

FA′γ′// B′

and

GA

≡Gα��

δ // B

β��

GA′δ′// B′

The functor I maps (A,B, γ, δ) 7→ A and (α, β) 7→ α. The functor P maps(A,B, γ, δ) 7→ B and (α, β) 7→ β. We define the natural transformations λ1

and λ2 by components: λ1,(A,B,γ,δ) = γ, and λ2,(A,B,γ,δ) = δ.


It is straightforward to verify that these data define a category, functorsand natural isomorphisms.

The functor [Z, E ]I∗ // PsCon(Z) is defined as

Z U // E 7→⇓λ1∗idU∼=

AF

""Z

IU

<<

PU //

IU""

B⇑λ2∗idU∼=

AG

<<

ZU""

V

==⇓µ E 7→ ZIU

$$

IV

::⇓idI∗µ A and ZPU

$$

PV

::⇓idP ∗µ B

To see that this functor I∗ is an isomorphism of categories, we propose an

inverse functor PsCon(Z)T // [Z, E ] .

Given a pseudocone with vertex Z given by

⇓ε1∼=

AF

��Z

H

??

Q //

H��

B⇑ε2∼=

AG

??

we prescribe its image via T to be the functor Z // E that sends

Z 7→ (HZ,QZ, ε1,Z , ε2,Z)

Zm // Z ′ 7→ ( HZ

Hm // HZ ′ , QZQm // QZ ′ )

Given a second pseudocone on Z by the data H ′, Q′, ε′1, ε′2 and a modificationψ between them, i.e. natural transformations ψ1 : H ⇒ H ′ and ψ2 : Q⇒ Q′

such that

ε′1 ◦ (idF ∗ ψ1) = ψ2 ◦ ε1

ε′2 ◦ (idG ∗ ψ1) = ψ2 ◦ ε2

we define T (ψ) to be the natural transformation with components

T (ψ)Z : T (H,Q, ε1, ε2) =⇒ T (H ′, Q′, ε′1, ε′2) : Z −→ E

T (ψ)Z =

(HZ

ψ1,Z // H ′Z , QZψ2,Z // Q′Z

)We leave to the reader the routine verifications that all the data for T iswell-defined and gives an honest functor, and that T and I∗ compose to theidentity functors in both directions. �


5.8. Biequalizer.

Definition 5.8. (biequalizer) The biequalizer of two parallel morphisms

Af //

g// B

in C is a pseudocone with vertex E for this diagram that induces an equiva-lence of categories by postcomposition

C(Z,E) ' PsCon(Z)

between the hom-category C(Z,E) and the category of pseudocones withvertex Z, for all Z ∈ C (see subsection 5.7 where these pseudocones aredescribed).

Observation 5.9. The same observations in 5.5 apply in this case, becauseCat admits pseudoequalizers (proposition 5.7).

5.9. Cotensor.

Recall from definition 4.3 that the weighted 2-limit of a 2-functor

B F // C weighted by B W // Cat is given by the formula

C(Z,wlim←−

W F ) ∼= [B, Cat ](W, C(Z,F−))

When B is the terminal 2-category {•} (with one object, the identitymorphism and the identity 2-cell), we can identify the 2-functors F and Wwith their image:

F = F (•) = C

W = W (•) = AIn this way, F is an object of C and W is a category, and the representingformula is simplified:

C(Z,wlim←−

W F ) ∼= [B, Cat ](W, C(Z,F−))

∼= [{•}, Cat ](W, C(Z,F−))

∼= Cat(W, C(Z,F ))

∼= Cat(A, C(Z,C))

We call this type of limit a cotensor product, and denote it {A, C}.

Definition 5.10. Given an object C in a 2-category C and a category A,the cotensor product of A and C is defined as the 2-representation of thefunctor

Z 7→ Cat(A, C(Z,C))

The cotensor product is also called power. We had denoted the cotensorproduct as {A, C}. In the literature the notations A t C or CA can also befound. We then have

C(Z, {A, C}) ∼= Cat(A, C(Z,C))


We can unfold the definition to obtain an explicit description of thecotensor product: the cotensor product of a category A and an object C ofa 2-category C is given by morphisms and 2-cells

{A, C}

ξA

��

ξA′

��

ξf=⇒

C

for each Af // A′ in A, in a functorial way: whenever we have morphisms

Af // A′

g // A′′ in A, we have

{A, C}

ξA

##

ξA′′

{{

ξ′A

��

ξg=⇒

ξf=⇒

C

=

{A, C}

ξA

��

ξA′′

��

ξgf=⇒

C

i.e. a functor

Aξ // C({A, C}, C)

such that

• (1-dimensional universal property) any other such functor

Aη // C(Z,C)

factorizes uniquely through ξ: there exists a unique Zh // {A, C}

such that for all a in A

Z≡

h //

ηA��

{A, C}

ξA{{C

and that for all Af // A′ in A

Z

ηA′

::

ηA$$

⇓ηf C = Zh // {A, C}

ξA′

99

ξA

%%⇓ξf C

• (2-dimensional universal property) given two functors

Aη //

θ// C(Z,C)


that factorize through ξ as

h∗ ◦ ξ = η

k∗ ◦ ξ = θ

and a collection of 2-cells ξA ∗ idhβA +3 ξA ∗ idk indexed by the

objects of A, which is natural in A, i.e. for all Af // A′ in A

(ξf ∗ idk) ◦ βA = βA′ ◦ (ξf ∗ idk)this β corresponds to a unique 2-cell

hα +3 k

in the sense that for all objects A of A we have

idξA ∗ α = βA

We observe that these diagrams no longer have the shape of a cone: theremay be many morphisms from the vertex to a single object of the diagram

(in this case, the many {A, C}ξA // C ), and 2-cells between these. This is

unlike the limits we considered before (products and equalizers).

The definition 5.10 also works for C a V-category. In that case, A is anyobject of V and the representation is given by a V-natural transformation.In particular, we can consider the 1-dimensional case with V = Ens.

Example 5.11. Given a set S and an object C of a category C we have

C(Z,∏S

C) ∼=∏S

C(Z,C)

∼=∏S

Ens({•}, C(Z,C)

∼= Ens(∐S

{•}, C(Z,C))

∼= Ens(S, C(Z,C))

whenever the product ∏S

C

exists. Thus, the cotensor product in this case is the iterated product (aproduct with all factors the same object), which justifies the notation CS .When C is Ens, for example, CS is the set of functions S −→ C.

Example 5.12. The 2-category Cat also has cotensor products: given cate-gories C and D, their cotensor product is given by the exponential

{C,D} = Cat(C,D) = [C,D] = DC

This can be seen as a particular case of the general formula for 2-limits inCat :

wlim←−

W F = [B, Cat ](W,F )


which we prove in proposition 6.1. When B = {•}, we identify functors Wand F with their images C and D respectively. We have

{C,D} ∼= [{•}, Cat ](C,D) ∼= Cat(C,D) = DC

We remark that this cotensor product is not the same as∏|D|

C (where |D|

denotes the set of objects of D), unless D is discrete.

Dualizing these definitions we obtain the tensor product:

Definition 5.13. Given a category A and an object C in a 2-category C, thetensor product of A and C is defined as the 2-representation of the functor

Z 7→ Cat(A, C(C,Z))

The tensor product is also called copower, and we denote it by A ⊗ C orA ∗ C, and then

A(A⊗ C,Z) ∼= Cat(A, C(C,Z))

Example 5.14. Making the necessary modifications, we get the definitionof the tensor product of a set S and an object C of a 1-category C. Withthe same reasoning as above, we have

C(∐S

C,Z) ∼=∏S

C(C,Z) ∼= Ens(S, C(C,Z))

if the coproduct ∐S

C

exists. In this case, the tensor product is given by

S ⊗ C =∐S

C

In Ens, we also have

S ⊗ C =∐S

C ∼= S × C

Example 5.15. It can be easily checked that the tensor product of twocategories C and D (that is, the tensor product in the 2-category Cat) isgiven by the usual product of categories

C ⊗ D = C × D

Example 5.16. The most familiar example of tensor product (and wherethe name comes from) is given by rings R, S, a right R-module M , an(R,S)-bimodule N and a right S-module P : we have an isomorphism ofabelian groups

HomS(M ⊗R N,P ) ∼= HomR(M,HomS(N,P ))

We observe that this is the same formula that defines the tensor product ingeneral. Choosing R = Z, we get

ModS(M ⊗Z N,P ) ∼= Ab(M,ModS(N,P ))


Since ModS (the category of S-modules) is an Ab-enriched category (whereAb is the category of abelian groups), and M is a Z-module (an abeliangroup), this coincides with the definition of a tensor product in a V-category,for V = Ab.

5.10. Pseudocotensor.

Recall that the cotensor {A, C} is given by the formula (see subsection5.9)

C(Z, {A, C}) ∼= [{•}, Cat ](W, C(Z,F−))

where W and F are the 2-functors sending the object of the terminalcategory {•} to A and C, respectively.

If we want to obtain a pseudolimit version of this particular weighted2-limit, we make the usual adaptations, and define the pseudocotensor{A, C}p by the formula

C(Z, {A, C}p) ∼= [{•}, Cat ]p(W, C(Z,F−))

with the same W and F . However, since {•} has no non-trivial morphismsor 2-cells, a 2-natural transformation W ⇒ C(Z,F−) consists of a functorA → C(Z,C) satisfying no naturality conditions, and this is the samesituation in the case of a pseudonatural transformation. This gives

[{•}, Cat ](W, C(Z,F−)) = [{•}, Cat ]p(W, C(Z,F−))

Observation 5.17. This means that the cotensor product and the pseudo-cotensor product are identical notions. This situation is analogous to that ofthe 2-product and the pseudoproduct, which is described in subsection 5.3.

5.11. Bicotensor.

Definition 5.18. Given a category A and an object C in a 2-category C,the bicotensor {A, C}b is given by a birepresentation

C(Z, {A, C}b) ' Cat(A, C(Z,C))

(compare with definition 5.10).

Observation 5.19. The same observations in 5.5 apply in this case, becauseCat admits cotensors (these are the same as pseudocotensors, by observation5.17). The fact that Cat admits cotensor products is given in 5.12.


5.12. Inserter and iso-inserter.

Definition 5.20. When B is the category {• ⇒ •}, the weight

{•⇒ •} W // Cat has image

10 //

1// 2

(here 1 = {•}, 2 = {0 → 1}, and the functors 0 and 1 are the inclu-sions sending • in 1 to the objects 0 and 1, respectively) and the functor

{•⇒ •} F // C has image

Af //

g// B

the 2-limit wlim←−

W F is called the inserter of f and g.

Unraveling the definitions and skipping redundant information, the inserteris an object wlim

←−W F together with a morphism p and a 2-cell λ

A

f

��wlim←−

W F

p

;;

p##

⇓λ B

A

g

@@

and it is the universal diagram of this form:

• (1-dimensional universal property) for any other diagram

Af

��Z

q??

q ��

⇓µ B

A

g

??

there exists a unique h such that

A

Zh //

q11

q--

≡

≡

wlim←−

W F

p

;;

p##A

and λ ∗ idh = µ.


• (2-dimensional universal property) given a second diagram

Af

��Z

q′??

q′ ��

⇓µ′ B

A

g

??

factorizing through wlim←−

W F via A∃!k // wlim

←−W F , and a 2-cell

qβ +3 q′ such that

µ′ ◦ (idf ∗ β) = (idg ∗ β) ◦ µ

there exists a unique 2-cell hα +3 k such that

β = idp ∗ α

We can replace the category 2 in definition 5.20 by the category I, whichhas two objects 0, 1 and an isomorphism between them, to obtain the notionof iso-inserter.

Definition 5.21. When B is the category {• ⇒ •}, the weight

{•⇒ •} W // Cat has image

10 //

1// I

(here 1 = {•} and the functors 0 and 1 are defined as before; see definition

5.20) and the functor {•⇒ •} F // C has image

Af //

g// B

the 2-limit wlim←−

W F is called the iso-inserter of f and g.

The elementary description of this 2-limit is very similar to that of theinserter of f and g (see discussion following definition 5.20). The 2-cell λ isinvertible, the 1-dimensional property holds for invertible 2-cells µ, and the2-dimensional property remains as stated.

Observation 5.22. Inserters and iso-inserters exist in the 2-category Cat .The iso-inserter of functors and categories

AF //

G// B

is given by a category E with objects pairs (A, ρ) with FAρ∼= // GA and

morphisms (A, ρ)→ (A′, ρ′) given by morphisms Aα // A′ such that the


obvious diagram commutes

FAρ∼= //

Fα

��≡

GA

Gα

��FA′

ρ′∼=// GA′

The universal functor E P // A is given by projection on the first variable

and the universal invertible natural transformation FPλ∼= +3 FQ is given

on components by projection on the second variable. The relevant universalproperties are easy to verify.

The inserter of the diagram

AF //

G// B

can be constructed in the same way, with objects given by pairs (A, ρ) with

FAρ // GA , where ρ is not necessarily invertible.

Observation 5.23. In the description of the pseudoequalizer of functors

and categories AF //

G// B given in 5.7, we might be tempted to compose

isomorphisms FAγ // B and GA

δ // B to obtain an isomorphism

FAδ−1γ // GA . An easy verification shows that this extends to a functor

from the pseudoequalizer into the iso-inserter of F and G, which is in factan equivalence, but not an isomorphism. (The quasi-inverse maps objects(A, ρ) 7→ (A,GA, ρ, idGA) and morphisms α 7→ (α,Gα).) Since limits aredefined representably (see 6.3), this implies that in any 2-category, if boththe iso-inserter and the pseudoequalizer of a diagram

Af //

g// B

exist, then they are equivalent.

For example, the pseudocoequalizer of geometric morphisms of toposesis given in [24, lemma 1.10] using the construction of the iso-inserter indefinition 5.21.

5.13. Comma-object.

Definition 5.24. When B is the category {• → • ← •}, the weight

{• → • ← •} W // Cat has image

10 // 2 1

1oo


(here 1 = {•}, 2 = {0 → 1}, and the functors 0 and 1 are the inclu-sions sending • in 1 to the objects 0 and 1, respectively) and the functor

{• → • ← •} F // C has image

Bf // D C

goo

the limit wlim←−

W F is called the comma-object of f and g.

This is given by an object wlim←−

W F with morphisms u and v and a 2-cell

λ as in

wlim←−

W Fv //

u

��=⇒λ

C

g

��B

f// D

such that

• (1-dimensional universal property) for any other diagram

Zy //

x

��=⇒µ

C

g

��B

f// D

there exists a unique Z∃!h // wlim

←−W F such that

uh = x

vh = y

λ ∗ idh = µ

• (2-dimensional universal property) given a second diagram

Zy′ //

x′

��=⇒µ′

C

g

��B

f// D

factorizing through the limit via Z∃!k // wlim

←−W F , and 2-cells

xβ +3 x′ , y

γ +3 y′ such that

µ′ ◦ (idf ∗ α) = (β ∗ idg) ◦ µ


there exists a unique 2-cell hα +3 k such that

idu ∗ α = β

idv ∗ α = γ

Example 5.25. The 2-category Cat has comma-objects, and these are indeedthe usual comma categories: if we have a span in Cat (categories and functors)

A F // C BGoo

the comma category F ↓ G is defined as follows:

• an object is given by a triple (A,B, f) with A an object in A, B an

object in B and FAf // GB a morphism in C

• a morphism (A,B, f) → (C,D, g) is given by a pair (p, q) of mor-

phisms Ap // C in A and B

q // D in B making the obviousdiagram commute:

FAf //

Fp��

≡

GB

Gq��

FC g// GD

The comma-object diagram in this case is

F ↓ G π2 //

π1

��=⇒λ

B

G

��A

F// C

where π1 and π2 are the projections on the first and second component, andλ is defined by λ(A,B,f) = f .

A very useful particular case of the comma category is the slice categoryC/X over an object X of C, when F is the identity functor on A (thus A = C),B is the terminal category {•} and G(•) = X.

5.14. Biequifier and biidentifier.

The following definitions can be found in [28]. 2-limit versions of thesecan be found in [15].

Definition 5.26. (biequifier) Given 2-cells

A

f**

g44⇓α ⇓β B


in a 2-category C, their biequifier is an arrow

Eh // A

inducing an equivalence of categories between C(Z,E) and the full subcate-

gory of C(Z,A) with objects those Zz // A such that α ∗ idz = β ∗ idz,

for all Z ∈ C. This is the bilimit of F weighted by W where the indexingcategory is

•))55⇓ ⇓ •

and the images of W and F are respectively

1

0))

1

55⇓ ⇓ 2 A

f**

g44⇓α ⇓β B

Definition 5.27. (biidentifier) The biidentifier of an endo-2-cell

A

f))

f

55⇓α B

in C is the biequifier of α and idf .

5.15. Descent object.

The definitions of a truncated bicosimplicial diagram in a 2-category, andof its descent object can be found in [28].

Definition 5.28. A truncated bicosimplicial diagram X in a 2-category Cis a diagram

X0

δ0 //

δ1// X1ιoo

δ0 //

δ2//δ1 // X2

with invertible 2-cells

δiδj−1

σi,j∼= +3 δjδi

for i < j, and for all i

idX0

µi∼= +3 ιδi

Definition 5.29. The descent object of a truncated bicosimplicial diagramX in Cat is a category Desc(X ) with objects pairs (X, θ) with X an object

of X0 and δ0(X)θ∼= // δ1(X) an isomorphism in X1 such that

ι(θ) = µ1,Xµ−10,X and σ1,2,X ◦ δ1(θ) ◦ σ0,1,X = δ2(θ) ◦ σ0,2,X ◦ δ0(θ)

and arrows (X, θ)χ // (X ′, θ′) are morphisms X

χ // X ′ in X0 such that

θ′ ◦ δ0(χ) = δ1(χ) ◦ θ


Notice that there is a functor Desc(X )P // X0 given by projection on the

first variable. The category Desc is a kind of iso-inserter or pseudoequalizer(see observations 5.22 and 5.23).

In a general 2-category C, the descent object of a truncated bicosimpli-

cial diagram X as before is an arrow Dh // X0 and an invertible 2-cell

δ0 ∗ idhω∼= +3 δ1 ∗ idh such that for all objects Z ∈ C these induce an equiv-

alence of categories between C(Z,D) and the descent object Desc(C(Z,X))(where C(Z,X) is a truncated bicosimplicial diagram in Cat): the descent ob-ject of a truncated bicosimplicial diagram is defined representably (comparewith equation 6.3)

C(Z,Desc(X)) ' Desc(C(Z,X))

Observation 5.30. Notice that the descent object of a truncated bicosim-plicial diagram is a weighted bilimit (see [28]).

5.16. 2-filtered pseudocolimit of categories.

In this subsection, we recall the construction of a pseudocolimit in Catwhen the diagram is 2-filtered, given in [10].

Definition 5.31. Given a 2-functor A F // Cat with A 2-filtered, thecategory L(F ) has objects and morphisms given as follows

• an object is a pair (x,A) with x an object of FA

• a premorphism (x,A)(u,ξ,v) // (y,B) is given by arrows A

u // C ,

Bv // C and (Fu)(x)

ξ // (Fv)(y) in FC

• two premorphisms (x,A)(u1,ξ1,v1) // (y,B) and (x,A)

(u2,ξ2,v2) // (y,B)

(between the same objects) are equivalent if there is an homotopybetween them, i.e. invertible 2-cells α, β such that the followingdiagram commutes in FC

F (w1u1)(x)(Fβ)x //

(Fw1)(ξ1)

��

≡

F (w2uw)(x)

(Fw2)(ξ2)

��F (w1v1)(y)

(Fα)y

// F (w2v2)(y)

• a morphism (x,A) // (y,B) is an equivalence class of premor-

phisms (x,A) // (y,B)

We will speak of morphisms and premorphisms interchangeably, as it willbe clear from context whether we are talking about an arrow in L(F ) or a


premorphism that represents it.

Using the 2-Yoneda embedding h : A −→ [A, Cat ]op (theorem 2.34), wehave

x ∈ FA

[A,−]x // F in [A, Cat ]

Fx // [A,−] in [A, Cat ]op

Finally we can make the abuse of notation of writing A for the representable

functor [A,−], and then an object of L(F ) is just Fx // A .

Analogously, we can write

F

x??

y ��

⇓ξ

Au

��

Bv

??C

for a premorphism, and the equivalence relation between two premorphismsis given by the LL equation

F

x??

y ��

⇓ξ1

Au1

��

Bv1

??

v2 ��

⇓α∼=

C1w1

��

C2

w2

??C

= F

x??

y ��

⇓ξ2

Au2

��

u1??

⇓β∼=

C1w1

��

Bv2

??C2

w2

??C

This category is a useful construction of the pseudocolimit of a 2-filtereddiagram in Cat . The proof of the following theorem can be found in [10].

Theorem 5.32. Given a 2-functor A F // Cat with A 2-filtered, we have

a pseudocone Fλ // L(F ) given by the formulas

λA(x) = Fx // A , λA(ξ) = F

x ??

y ��

⇓ξ

Aid��

Aid

??A , λu(x) = F

ux ??

x ��

⇓id

Aid��

Au

??A


for Au // B in A and x

ξ // y in FA.

This pseudocone is universal in the sense that it induces an isomorphism ofcategories by precomposition

[L(F ),Z]λ∗∼= // PsCon(Z)

between the category of functors and natural transformations L(F ) −→ Zand the category of pseudocones and modifications for the diagram F withvertex Z, for all categories Z. �

Observation 5.33. We remark that, like every pseudocolimit, thisconstruction is also a bicolimit (see observation 4.8).

We state a lemma from [10] which is useful to prove the main result.

Lemma 5.34. Given a finite family of 2-cells E

f ??

g ��

A

⇓γi

ui��

Bvi

??Ci with

i = 1, . . . , n, there exist morphisms Au // C , B

v // C , Ciwi // C

and invertible 2-cells αi, βi, for i = 1, . . . , n such that the 2-cells

A u

""⇓βi∼=ui

⇓γiE

f??

g ��

Ciwi // C

B

vi>>

⇓αi∼=

v

<<

are all equal for i = 1, . . . , n. Given a second family of 2-cells H

h ??

l ��

A

⇓δi

ui��

Bvi

??Ci ,

we can assume the same u, v, wi, αi and βi also equalize the 2-cells of thesecond family. �

Observation 5.35. A 2-natural transformation ε : F =⇒ G induces afunctor ε : L(F ) −→ L(G) by the universal property of L(F ) applied to the

pseudocone Fε +3 G

ρG +3 L(G) (where ρG is the universal pseudocone

for G). This functor is defined on objects and arrows of L(F ) as follows


Fa // A 7→ G

εA(a) // A

F

a??

a′ ��

⇓ξ

Au

��

A′v

??C 7→ G

εA(a)??

εA′ (a′) ��

⇓εC(ξ)

Au

��

A′v

??C


6. Constructions of all limits in terms of simpler limits

6.1. Construction of weighted 2-limits.

In Cat , all (small) weighted 2-limits exist:

Proposition 6.1. Given 2-functors BW //

F// Cat , we have

wlim←−

W F = [B, Cat ](W,F )

That is, the W -weighted 2-limit of F is the category of 2-natural transforma-tions and modifications between W and F .

Proof. We want to prove that (see definition 4.3)

Cat(Z, [B, Cat ](W,F )) ∼= [B, Cat ](W, Cat(Z, F−))

for Z ∈ Cat , with unit the obvious 2-natural transformation

Wξ +3 Cat([B, Cat ](W,F ), F−)

Given a 2-natural transformation Fρ +3 Cat(Z, F−) , we have to see that

there exists a unique functor Z H // [B, Cat ](W,F ) such that ρ = H∗ ◦ ξ.This condition implies that we must define h as

H(Z)B(X) = ρB(X)(A)

H(Z)B(f) = (ρB(f))A

H(g)B,X = ρB(X)(g)

if Zg // Z ′ in Z, B ∈ B and f : X → X ′ is an arrow in WB. This

gives uniqueness, and it’s easy to verify functoriality. For the 2-dimensional

universal property, we take a second Fρ′ +3 Cat(Z, F−) with corresponding

ZH′ // [B, Cat ](W,F ) and a modification θ : ρ ρ′. We have to show

that there exists a unique natural transformation Hα +3 H ′ such that

θ = α∗ ◦ ξ. We are forced to define α by components as:

αZ,B,X = θB,X,Z

where B ∈ B, Z ∈ Z and X ∈WB. It’s straightforward to verify that thisα is well-defined and natural. �

Observation 6.2. This allows us to rewrite the definition of a weighted2-limit in an arbitrary 2-category by the formula

(6.3) C(Z,wlim←−

W F ) ∼= wlim←−

W C(Z,F−)

Thus, the notion of 2-limit is given by reducing to the case of 2-limits offunctors into Cat . This representable definition will prove useful later on.This means that weighted limits are preserved by representable 2-functors.


To construct limits in the ordinary 1-dimensional setting, we know that it

is enough to take products and equalizers: the limit of a functor B F // Ccan be constructed as the equalizer of

(6.4)

∏A

FAs //

t//∏

f :A→BFB

where s and t are the unique morphisms defined by

π(f :A→B)s = πB

π(f :A→B)t = (Ff)πA

and the universal cone is given by composing with the projections of∏A

FA:

if Ee //∏A

FA is the equalizer, the components of the cone are πAe.

It’s easy to prove that this equalizer has the same universal property as

the limit: given a cone ∆Zλ +3 F , we can obtain Z

(λA)A //∏A

FA by

the universal property of the product. Naturality of λ coincides with thismorphism (λA)A equalizing s and t:

π(f :A→B)t(λA)A = π(f :A→B)s(λA)A

⇐⇒ (Ff)λA = λB

We then get an induced morphism Zh // E into the equalizer, and this

morphism satisfies

πAeh = λA

for all A ∈ A. �

In the 2-dimensional setting, one more type of 2-limit is needed: cotensorproducts. In fact, compositions of 2-products and 2-equalizers only giveconical 2-limits. The following proposition can be found without proof in[15].

Proposition 6.5. (construction of weighted 2-limits) Given 2-functors

B W // Cat and B F // C , if C has 2-equalizers, 2-products and cotensor

products, then wlim←−

W F can be constructed as the 2-equalizer of

∏B

{WB,FB}s //

t//∏B,C

{B(B,C)×WB,FC}

where s and t are the unique morphisms arising from B(B,C) −→ C(FB,FC)and B(B,C) −→ Cat(WB,WC).


Proof. We will first verify it for the case C = Cat , and then reduce the generalcase to this.

We will prove that wlim←−

W F ∼= [B, Cat ](W,F ) is the 2-equalizer (see

proposition 6.1):

[B, Cat ](W,F )U //

∏B

[WB,FB]S //

T//∏B,C

[B(B,C)×WB,FC]

The functor U is the operation of taking components on 2-natural transfor-mations and modifications. The composite functor πB,CS is given by theformulas

(πB,CS)(α)(f,X) = (αC ◦Wf)(X)

(πB,CS)(α)(ε, u) = αC((Wε)Y ◦ (Wf)(u))

(πB,CS)(η)f,X = (ηC ◦Wf)(X)

if α = ( WBαB // FB )B∈B and β = ( WB

βB // FB )B∈B are families offunctors, η = (ηB)B∈B is a family of natural transformations between them,

B

f))

g55⇓ε C in B and X

u // Y in WB.

Likewise, the composite functor πB,CT is given by the formulas

(πB,CT )(α)(f,X) = (Ff ◦ αB)(X)

(πB,CT )(α)(ε, u) = (Fε)αB(Y ) ◦ Ff(αB(u))

(πB,CT )(η)f,X = Ff(ηB,X)

It’s easy to see from these definitions that πB,CSU = πB,CTU , and then Uequalizes S and T . Given a functor

Zϕ //

∏B

[WB,FB]

also equalizing S and T , we have to define

Z H // [B, Cat ](F,G)

so that HZ is the 2-natural transformation with components those of ϕ(Z),

and H( Zp // Z ′ ) is the modification with components those of ϕ(p),

if we want UH = ϕ. The resulting HZ and Hp are indeed a 2-naturaltransformation and a modification, respectively: this follows from Sϕ = Tϕ.This is the 1-dimensional universal property of the 2-equalizer.

For the 2-dimensional universal property, we take a second

Zψ //

∏B

[WB,FB]

with corresponding

Z K // [B, Cat ](F,G)


and a natural transformation ϕσ +3 ψ such that idS ∗ σ = idT ∗ σ. We

want a natural transformation Hτ +3 K such that idU ∗ τ = σ. This

forces us to define τZ,B = πB(σZ), and this is well-defined and gives anatural transformation.

Now, if C is an arbitrary 2-category, we have the following chain ofisomorphisms, 2-natural in A

C(A, eq(∏B

{WB,FB} ////∏B,C

{B(B,C)×WB), FC} ))

∼= eq(C(A,∏B

{WB,FB} //// C(A,

∏B,C

{B(B,C)×WB), FC} ))

∼= eq(∏B

C(A, {WB,FB}) ////∏B,C

C(A, {B(B,C)×WB), FC} )

∼= eq(∏B

{WB, C(A,FB)} ////∏B,C

{B(B,C)×WB), C(A,FC)} )

∼= wlim←−

W C(A,F−) ∼= C(A,wlim←−

W F )

where we use the representable definition of limits in equation 6.3 to extractthe equalizer, the products and the cotensor products out of the secondargument of C(A,−), in that order, and the fact that the construction worksin Cat .By the 2-Yoneda lemma (theorem 2.34),

wlim←−

W F ∼= eq(∏B

{WB,FB} ////∏B,C

{B(B,C)×WB), FC} )

�

Observation 6.6. When a 2-category C has certain classes of 2-limits, wesay that a certain type of 2-limit can be constructed from these, instead ofjust saying that C admits it. Any 2-functor preserving those 2-limits alsopreserves the new type of 2-limit. This is suggested in [3].

6.2. Construction of bilimits.

In this subsection we reproduce some definitions of bilimits, results andproofs in [28] that are essential in section 7.

Observation 6.7. In [29] and [28], Street considers bilimits in a bicategory,where both the weight W and the diagram F are permitted to be pseudofunc-tors (also called homomorphisms of bicategories). The bilimit wbilim

←−W F is

given by a birepresentation of

Z 7→ Hom[B, Cat ](W, C(Z,F−))


where Hom[B, Cat ] is the bicategory of pseudofunctors, pseudonatural trans-formations and modifications.We will always assume W and F to be (strict) 2-functors, so that this notioncoincides with the definition of bilimit we give in 4.7:

Hom[B, Cat ](W, C(Z,F−)) = [B, Cat ]p(W, C(Z,F−))

Proposition 6.8. The biidentifier of an auto-2-cell

A

f))

g55⇓α B

in a 2-category C can be constructed using biequalizers and biproducts.

Proof. Let α be the arrow A→ {Aut, B} corresponding to the automorphismα in C(A,B), where Aut is the group Z of integers (free group on a one-element

set) seen as a category. Likewise, let f be the arrow A → {Aut, B} corre-

sponding to the automorphism idf in C(A,B). We take Hh // A with

αhσ∼= +3 fh the biequalizer of α and f , and K

k // A with fhτ∼= +3 fh

the biequalizer of f and f . The unique functor {•} −→ Aut induces a

morphism {Aut, B} e // B . We can take Hl // K and kl ∼= h with

eσ isomorphic to τ l by the universal property of K, and Ad // K and

kd ∼= idk with idf isomorphic to τd. The biequalizer of dπ1 and lπ2 (whereπ1 and π2 are the projections from the biproduct A×H) is the biidentifierof α (see definition 5.27). �

Proposition 6.9. If

A

f**

g44⇓α ⇓β B

are 2-cells in a 2-category C with β invertible, then the biequifier of α and βis the biidentifier of β−1α.

Proof. This is straightforward from the following equivalences:

α ∗ idz = β ∗ idz⇐⇒ (β−1 ∗ idz) ◦ (α ∗ idz) = (β−1 ∗ idz) ◦ (β ∗ idz)

⇐⇒ β−1α ∗ idz = idf ∗ idzso that the full subcategory of C(Z,A) with objects those z such thatα ∗ idz = β ∗ idz is the full subcategory of C(Z,A) with objects those zsuch that β−1α ∗ idz = idz (see definitions 5.26 and 5.27). �

Proposition 6.10. The descent object of a truncated bicosimplicial diagram(see definitions 5.28 and 5.29)

X0

δ0 //

δ1// X1ιoo

δ0 //

δ2//δ1 // X2


with invertible 2-cells

δiδj−1

σi,j∼= +3 δjδi

for i < j, and for all i

idX0

µi∼= +3 ιδi

can be constructed using biequalizers and biidentifiers of auto-2-cells.

Proof. First, we take the biequalizer Hh // X0 with

idδ0 ∗ hθ∼= +3 idδ1 ∗ h . Then, we take the biequifier K

k // H of

the two invertible 2-cells idι ∗ θ and (µ1 ∗ idh) ◦ (µ−10 ∗ idh) (since these are

invertible, their biequifier is a biidentifier of the auto-2-cell resulting from

their composition). Finally, we take the biequifier Mm // H of the two

invertible 2-cells

(σ1,2 ∗ idhk) ◦ (idδ1 ∗ θ ∗ idk) ◦ (σ0,1 ∗ idhk)and

(idδ2 ∗ θ ∗ idk) ◦ (σ0,2 ∗ idhk) ◦ (idδ0 ∗ θ ∗ idk)The descent object is given by the object L, the arrow hkm and the invertible2-cell θ ∗ idkm. �

Proposition 6.11. The bilimit wbilim←−

W F of a functor B F // C weighted

by B W // Cat can be constructed as the descent object for the truncatedbicosimplicial diagram

∏A

{WA,FA}bδ0 //

δ1//

∏A,B

{A(A,B)×WA,FB}bιoo

δ0 //

δ2//δ1 //∏A,B,C

{A(B,C)×A(A,B)×WA,FC}b

with arrows given in a similar way to those in proposition 6.5. �

Corollary 6.12. An arbitrary bilimit wbilim←−

W F can be constructed by

means of biproducts, biequalizers and bicotensor products. Moreover, if the

weight B W // Cat is finite, the bilimit can be constructed by means offinite biproducts (or binary products and the terminal object, by induction),biequalizers and bicotensor products with finite categories.

Corollary 6.13. Any 2-functor that preserves biproducts, biequalizers andbicotensor products also preserves all weighted bilimits. If the 2-functorpreserves finite biproducts, biequalizers and bicotensor products with a finitecategory, then it preserves all finite weighted bilimits.


7. Commutation of filtered bicolimitsand finite weighted bilimits in Cat

7.1. The 1-dimensional case.

In ordinary category theory we have the following result.

Theorem 7.1. Given a functor A× J F // Ens , for each A ∈ A andJ ∈ J we have functors

AF (−,J) // Ens and J

F (A,−) // Ens

by fixing one variable. Taking limits and colimits (which are computedpointwise) in the functor categories EnsA and EnsJ , respectively, we obtainfunctors

AlimJ←−

F (−,J)

// Ens and JcolimA−→

F (A,−)// Ens

These have limits and colimits of their own, and there’s a canonical functionof sets

colimA−→limJ←−

F♦ // limJ←−

colimA−→F

which is a bijection provided that A is filtered and J is finite (finite collectionof objects and finite hom-sets).

Proof. This can be proven either directly, or by decomposing the limit. Wewill follow this second method as we will also use it for the proof of thecorresponding result in the 2-dimensional setting.Using the construction of limits stated in diagram 6.4, it’s enough to provethis commutation in the cases that J = {• ⇒ •} (for the equalizer) or Jdiscrete and finite (for the product). We will assume J = {• •} in thissecond case, since any finite product can be constructed by iterated binaryproducts and the case of the terminal product is trivial.

We will use the familiar constructions of limits and colimits in Ens.

Observation 7.2. The limit of a functor B G // Ens is the subset of the

product∏A

GA consisting of those elements (xA)A such that (Gf)(xA) = xB

for all Af // B , with the cone defined by the projections to the factors

of the product (this is just the combination of the usual constructions ofequalizer and product applied to the formula in diagram 6.4).

Observation 7.3. The colimit of a functor B G // Ens with B filtered willbe given in terms of the “germs” construction: the quotient of the disjoint

union∐A

GA by the equivalence relation (x,A) ∼ (y,B) if there exists a C


and arrows f , g as in

Af

%%C

Bg

99

such that (Gf)(x) = (Gg)(y), with the cone defined by the inclusionsinto the disjoint union followed by the quotient map. We write [x,A] for

equivalence class represented by an element (x,A) ∈∐A

GA. When there

is more than one set-valued functor G of which we construct the colimit,we can make the functor G explicit by speaking of G-equivalence classes,denoted [x,A]G.

(1) Case J = {• •}We have a functor A × {• •} −→ Ens, which is the same as two

functors AF //

G// Ens . We define

A F×G // Ens

to be the functor taking products pointwise:

(F ×G)(A) = FA×GA(F ×G)(f) = Ff ×Gf

if A is an object and f is a morphism in A.The function

colim−→

F ×G ♦ // colim−→

F × colim−→

G

maps equivalence classes of pairs to pairs of equivalence classes

[(x, y), A] 7→ ([x,A], [y,A])

To see that it is surjective, we pick an arbitrary

([x,A], [y,B]) ∈ colim−→

F × colim−→

G

By axiom 3.2 there exist an object C and morphisms u, v as in

Au

%%C

Bv

99

Then ♦([((Fu)(x), (Fv)(y)), C]) = ([(Fu)(x), C], [(Fv)(y), C]), andin fact

[x,A] = [(Fu)(x), C]

since


Au

%%C

CidC

99

and

(FC)(Fu)(x) = (Fu)(x)

Analogously,

[y,B] = [(Fv)(y), C]

Then ♦([((Fu)(x), (Fv)(y)), C]) = ([x,A], [y,B]).

To verify injectivity, we pick two elements

[(x1, x2), A], [(y1, y2), B] ∈ colim−→

F ×G

that map to the same element in colim−→

F × colim−→

G via ♦:

([x1, A], [x2, A]) = ([y1, B], [y2, B])

This is the same as the two equations

[x1, A] = [y1, B]

[x2, A] = [y2, B]

which mean that there are objects and morphisms as in the diagrams

Au1

%%

Au2

%%C1 C2

Bv1

99

Bv2

99

with

(Fu1)(x1) = (Fv1)(y1)

(Gu2)(x2) = (Gv2)(y2)

We can apply axiom 3.2 to complete a square

C1

w′1

A

u1??

u2 ��

C ′3

C2

w′2

>>

but it doesn’t have to be commutative. By axiom 3.3 applied to thecompositions


Aw′1u1 //

w′2u2

// C ′3

we can replace C ′3 by another object C3, and the morphisms w′1and w′2 by another pair of morphisms w1, w2 such that the followingdiagram commutes

C1

w1

A ≡

u1??

u2

C3

C2

w2

>>

We now apply axiom 3.3 to

A′w1v1 //

w2v2// C3

to obtain

A′w1v1 //

w2v2// C3

e // C4

with

ew1v1 = ew2v2

Putting all this together, we have

Aew1u1

%%D

Bew2v2

99

and

F (ew1u1)(x1) = F (ew1v1)(y1) = F (ew2v2)(y1)

G(ew1u1)(x2) = G(ew2u2)(x2) = G(ew2v2)(y2)

This means that ((x1, x2), A) ∼ ((y1, y2), B) because

(F ×G)(ew1u1)(x1, x2) = (F ×G)(ew2v2)(y1, y2)

Then

[(x1, x2), A] = [(y1, y2), B]

(2) Case J = {•⇒ •}We have a functor A× {•⇒ •} −→ Ens, which is the same as two


functors AF //

G// Ens with natural transformations ε, η : F ⇒ G.

We define

A E // Ens

to be the functor taking equalizers pointwise (in other words, theequalizer of ε and η in the functor category [A, Ens], which is com-puted pointwise):

EAiA // FA

εA //

ηA// GA

(here we take EA as the subset of FA consisting of the elementsequalized by εA and ηA), and on a morphism f

(7.4) (Ef)(x) = (Ff)(x)

It can be checked that this is well-defined. We then consider thediagram

colim−→

Fε //

η// colim−→

G

where ε and η are the morphisms induced in the colimit. We takethe equalizer of this diagram and obtain

E′i // colim

−→F

ε //

η// colim−→

G

(again, we consider E′ ⊆ colim−→

F ) There is a comparison function

given by the universal property of E′

colim−→

E♦ // E′

that maps an E-equivalence class to an F -equivalence class (seeobservation 7.3)

[x,A]E 7→ [x,A]F

This function is surjective: given [x,A]F ∈ E′, we have

ε([x,A]F ) = η([x,A]F )

because [x,A] belongs to E′. From the definition of ε and η, this is

[εA(x), A]G = [ηA(x), A]G

This means there is a C and morphisms u, v as in

Au

%%C

Av

99

with

(7.5) (Gu)εA(x) = (Gv)ηA(x)


By axiom 3.3, we obtain a D and a morphism w as in

Au //

v// C

w // D

with wu = wv. Then, by equation 7.5 and naturality of ε and η, wehave

εDF (wu)(x) = ηDF (wv)(x) = ηDF (wu)(x)

This equation says that F (wu)(x) is a well-defined element of EDbecause it is equalized by εD and ηD as in

EDiD // FD

εD //

ηD// GD

We can then consider the element [F (wu)(x), D]E ∈ colim−→

E, and in

fact, ♦([F (wu)(x), D]E) = [x,A]F , since we have a diagram

Awu

%%D

DidD

99

withF (wu)(x) = idD(F (wu)(x))

This shows surjectivity of ♦. To see that it is injective, we pick[x,A]E , [y,B]E ∈ colim

−→E with [x,A]F = [y,B]F , that is, there is a

C and morphisms u and v

Af

%%C

Bg

99

with(Fu)(x) = (Fv)(y)

Since on morphisms the functor E is defined by the formula 7.4, weimmediately have

(Eu)(x) = (Ev)(y)

which means [x,A]E = [y,B]E .

�

In the rest of this section we will prove that 2-filtered bicolimits and finitebilimits commute in Cat , which is a 2-categorical version of this classicalresult, by means of a similar general argument: decomposing a finite bilimitinto certain kinds of finite limits which commute with the 2-filtered bicolimit.We assume throughout the rest of this section that A is 2-filtered.


7.2. The 2-dimensional case.

An adaptation of theorem 7.1 to the 2-categorical setting can be found in[11], which considers pseudocolimits and pseudolimits of diagrams in Catindexed by filtered and finite categories (these are conical pseudolimits). Weextend this result to finite weighted bilimits and 2-filtered bicolimits in Cat ,both indexed by 2-categories.

In the following sections, we consider particular classes of finite weightedpseudolimits and show that they commute with 2-filtered pseudocolimits.Finally, we combine these facts in the proof of the main result.

7.3. 2-product.

Definition 7.6. Given 2-functors AF //

G// Cat , the 2-functor A F×G // Cat

is defined by the following assignments (see observation 2.20)

(F ×G)(A) = F (A)×G(A)

(F ×G)(f)(X,Y ) = ((Ff)(X), (Gf)(Y ))

(F ×G)(f)(u, v) = ((Ff)(u), (Gf)(v))

(F ×G)(α)(P,Q) = ((Fα)P , (Gα)Q)

for A

f&&

g

88⇓α B in A, Xu // X ′ in FA, Y

v // Y ′ in GA, and (P,Q) an

object in FA×GA.

It is straightforward to check that this does indeed define a 2-functor.

Observation 7.7. The 2-functor F ×G is the 2-product in the 2-category[A, Cat ] (see [14, section 3.3].)

We consider the bicolimit of F ×G using of the construction in [10] (recallthat this is in fact a pseudocolimit; see definition 5.31 and theorem 5.32).Objects of the category L(F ×G) are given by

F ×G(a,b) // A

(a, b) ∈ FA×GA

Notice that in fact F ×G is a (formal) coproduct since we are in the dualcategory [A, Cat ]op. Premorphisms of L(F ×G) are given by


F ×G

(a,b)??

(a′,b′) ��

⇓(ξ,σ)

Au

��

A′v

??C

(Fu)(a)ξ // (Fv)(a′) in FC

(Gu)(b)σ // (Gv)(b′) in GC

An homotopy between two premorphisms (ξ1, σ1) and (ξ2, σ2) is given byinvertible 2-cells α, β satisfying the LL equation.

(7.8) F ×G

(a,b)??

(a′,b′) ��

⇓(ξ1,σ1)

Au1

��

A′v1

??

v2 ��

⇓α∼=

C1u

��

C2

v

??D

= F ×G

(a,b)??

(a′,b′) ��

⇓(ξ2,σ2)

Au2

��

u1??

⇓β∼=

C1u

��

A′v2

??C2

v

??D

On the other side, each one of AF //

G// Cat is a 2-functor from a

2-filtered category to Cat , so it makes sense to consider the categoriesL(F ) and L(G) and take their product. We have a comparison functor♦ : L(F × G) −→ L(F ) × L(G) given by the universal property of thepseudocolimit L(F ×G), and which is defined as follows.

♦ : L(F ×G) −→ L(F )× L(G)

F ×G

(a,b)??

(a′,b′) ��

⇓(ξ,σ)

Au

��

A′v

??C 7→

F

a??

a′ ��

⇓ξ

Au

��

A′v

??C , G

b??

b′ ��

⇓σ

Au

��

A′v

??C

We observe that equation 7.8 is equivalent to the two LL equations for

the homotopies (α, β) : ξ1 ⇒ ξ2 and (α, β) : σ1 ⇒ σ2, reflecting the fact thatthe functor ♦ is well-defined.


Observation 7.9. The proof of the following theorem is similar to the proofof [10, theorem 2.4] when for P we take the discrete category {• •}. Forexample, objects of L(F ×G) are given by elements (a, b) ∈ FA×GA, whileobjects of L(FP) are given by elements (a, b) ∈ FA× FA.We include the proof also because in this case (of two factors) it is muchclearer.

Theorem 7.10. Given 2-functors AF //

G// Cat with A 2-filtered, the com-

parison functor ♦ : L(F×G) −→ L(F )×L(G) is an equivalence of categories.

Proof. We proceed by showing that it is essentially surjective, full and faithful.This is enough to prove that ♦ is an equivalence by observation 2.16.

(1) Essential surjectivity

We pick an arbitrary object ( Fa // A , G

b // B ) in L(F )× L(G).Using 2-filteredness of A, we take C, u and v as in the followingdiagram.

Au

C

Bv

>>

The invertible morphism

F

ua??

a ��

Cid

��

A

⇓id

u

??C

shows that Fa // A is isomorphic to F

ua // C . Likewise, we have

Gb // B ∼= G

vb // C .

Then ( Fa // A , G

b // B ) ∼= ♦( F ×G(ua,vb) // C ).

(2) Fullness

Given a premorphism ♦( F ×G(a,b) // A ) −→ ♦( F ×G

(a′,b′) // A′ )by the data

F

a??

a′ ��

⇓ξ

Au1

��

A′v1

??C1 , G

b??

b′ ��

⇓σ

Au2

��

A′v2

??C2

we want to find premorphisms δ and ε equivalent to ξ and σ,respectively, that will allow us to pass from C1, C2, u1, u2, v1 and


v2 to a single C, a single u and a single v, respectively. We use2-filteredness of A, axiom FF1 in definition 3.8, to obtain invertible2-cells

A

u1??

u2 ��

⇓γu∼=

C1p

��

C2

q

??C , A′

v1??

v2 ��

⇓γv∼=

C1p

��

C2

q

??C

We can then define u = pu1, v = qv2, and δ and ε as in

F

a??

a′ ��

⇓δ

Au

��

A′v

??C = F

a??

a′ ��

⇓ξ

Au1

��

A′v1

??

v2 ��

⇓γu

C1p

��

C2

q

??C

G

b??

b′ ��

⇓ε

Au

��

A′v

??C = G

b??

b′ ��

⇓σ

Au2

��

u1??

⇓γv

C1p

��

A′v2

??C2

q

??C

Then ξ ∼ δ and σ ∼ ε, so (ξ, σ) = ♦

F ×G

(a,b)??

(a′,b′) ��

⇓(δ,ε)

Au

��

A′v

??C

.

We observe that we essentially follow the same steps as to provefullness of the comparison functor ♦ in theorem 7.23. This is part c)in the proof of [10, theorem 2.4]. See also the proof of [10, lemma 2.3].


(3) FaithfulnessIf we have

F ×G

(a,b)??

(a′,b′) ��

⇓(ξ1,σ1)

Au1

��

A′v1

??C1 , F ×G

(a,b)??

(a′,b′) ��

⇓(ξ2,σ2)

Au2

��

A′v2

??C2

such that ♦(ξ1, σ1) = ♦(ξ2, σ2), that is, ξ1 ∼ ξ2 and σ1 ∼ σ2: thereare homotopies given by invertible 2-cells α1, β1, α2, β2 satisfyingthe LL equations

(7.11) F

a??

a′ ��

⇓ξ1

Au1

��

A′v1

??

v2 ��

⇓α1∼=

C1w1

��

C2

w2

??C

= F

a??

a′ ��

⇓ξ2

Au2

��

u1??

⇓β1∼=

C1w1

��

A′v2

??C2

w2

??C

(7.12) G

b??

b′ ��

⇓σ1

Au1

��

A′v1

??

v2 ��

⇓α2∼=

C1w′1

��

C2

w′2

??C′

= G

b??

b′ ��

⇓σ2

Au2

��

u1??

⇓β2∼=

C1w′1

��

A′v2

??C2

w′2

??C′

Intuitively, we would want ξ1 ∼ ξ2 and σ1 ∼ σ2 via the same pair(α, β) of invertible 2-cells witnessing the homotopy. We can applylemma 5.34 to the families of 2-cells {α1, α2} and {β1, β2} to obtain(invertible!) 2-cells α, β by pasting invertible 2-cells λ1, µ1, λ2, µ2

so that


A′

v1??

v2 ��

⇓α∼=

C1u

��

C2

v

??D =

C1 u

!!⇓λ1∼=w1

⇓α1∼=A′

v1>>

v2

Ch1 // D

C2

w2>>⇓µ1∼=

v

==

=

C1 u

""⇓λ2∼=

w′1 ⇓α2∼=A′

v1>>

v2

C ′h2 // D

C2

w′2>>⇓µ2∼=

v

==

and

A′

u1??

u2 ��

⇓β∼=

C1u

��

C2

v

??D =

C1 u

!!⇓λ1∼=w1

⇓β1∼=A′

u1>>

u2

Ch1 // D

C2

w2>>⇓µ1∼=

v

==

=

C1 u

""⇓λ2∼=

w′1 ⇓β2∼=A′

u1>>

u2

C ′h2 // D

C2

w′2>>⇓µ2∼=

v

==

A simple calculation verifies that if we replace α1 and β1 by α andβ in the LL equation 7.11 it remains true, by pasting appropriatelyλ1 and µ1:

(7.13) F

a??

a′ ��

⇓ξ1

Au1

��

A′v1

??

v2 ��

⇓α∼=

C1u

��

C2

v

??D

= F

a??

a′ ��

⇓ξ2

Au2

��

u1??

⇓β∼=

C1u

��

A′v2

??C2

v

??D

We can do the same in the LL equation 7.12, replacing α and βfor α2 and β2:


(7.14) G

b??

b′ ��

⇓σ1

Au1

��

A′v1

??

v2 ��

⇓α∼=

C1u

��

C2

v

??D

= G

b??

b′ ��

⇓σ2

Au2

��

u1??

⇓β∼=

C1u

��

A′v2

??C2

v

??D

These two resulting equations 7.13 and 7.14 are together equivalentto

F ×G

(a,b)??

(a′,b′) ��

⇓(ξ1,σ1)

Au1

��

A′v1

??

v2 ��

⇓α∼=

C1u

��

C2

v

??D

= F ×G

(a,b)??

(a′,b′) ��

⇓(ξ2,σ2)

Au2

��

u1??

⇓β∼=

C1u

��

A′v2

??C2

v

??D

which expresses that (ξ1, σ1) ∼ (ξ2, σ2) via (α, β), i.e. they define thesame morphism in L(F ×G).

�

Observation 7.15. As usual, by induction, the comparison functor

♦ : L

(∏i

Fi

)−→

∏i

L(Fi)

is an equivalence of categories for any finite collection of 2-functors

A Fi // Cat . This is true for the empty product as well, since in thiscase both the domain and the codomain of ♦ reduce to the terminal category{•}.

7.4. Pseudoequalizer.

We examine the construction of the pseudoequalizer pointwise to determinea 2-functor and then verify that it commutes with a 2-filtered pseudocolimit.


Definition 7.16. Given 2-functors and 2-natural transformations

AF))

G

55⇓ε ⇓η Cat , the pseudoequalizer functor E : A −→ Cat is given on objects

A by the construction in proposition 5.7 applied to FAεA //

ηA// GA , and on

morphisms Af // B and 2-cells A

f&&

g

88⇓α B as follows

Ef : EA→ EB

(a, b, γ, δ) 7→ ((Ff)(a), (Gf)(b), (Gf)(γ), (Gf)(δ))

(ξ, σ) 7→ ((Ff)(ξ), (Gf)(σ))

Eα : Ef =⇒ Eg

(Eα)(a,b,γ,δ) = ((Fα)a, (Gα)b))

Observation 7.17. It can be checked that these assignments yield awell-defined 2-functor E, which, in fact, is the pseudoequalizer in the2-category [A, Cat ] (see [7, proposition 1.2.4]).

We now consider the pseudocolimit of E, via the construction in [10]. Thecategory L(E) has as objects

E(a,b,γ,δ) // A

a ∈ FA, b ∈ GA, εA(a)γ∼= // b ∈ GA, ηA(a)

δ∼= // b ∈ GA

(see definition 5.31) and as premorphisms

E

(a,b,γ,δ)??

(a′,b′,γ′,δ′) ��

⇓(ξ,σ)

Au

��

A′v

??C

(Fu)(a)ξ // (Fv)(a′) , (Gu)(b)

σ // (Gv)(b′)

such that σ ◦Gu(γ) = Gv(δ′) ◦ εC(ξ)

and σ ◦Gu(γ) = Gv(δ′) ◦ ηC(ξ)

Two premorphisms (ξ1, σ1) and (ξ2, σ2) are equivalent and define the samemorphism in L(E) if there is an homotopy between them: α and β invertible


2-cells in A such that the LL equation holds:

(7.18) E

(a,b,γ,δ)??

(a′,b′,γ′,δ′) ��

⇓(ξ1,σ1)

Au1

��

A′v1

??

v2 ��

⇓α∼=

C1u

��

C2

v

??D

= E

(a,b,γ,δ)??

(a′,b′,γ′,δ′) ��

⇓(ξ2,σ2)

Au2

��

u1??

⇓β∼=

C1u

��

A′v2

??C2

v

??D

Following the definitions of E, this is the same as the two equations

(Fα)a′ ◦ (Fw1)(ξ1) = (Fw2)(ξ2) ◦ (Fβ)a

(Gα)b′ ◦ (Gw1)(σ1) = (Gw2)(σ2) ◦ (Gβ)b

which establish the homotopies (α, β) : ξ1 ⇒ ξ2 and (α, β) : σ1 ⇒ σ2.

Given 2-functors and 2-natural transformations AF))

G

55⇓ε ⇓η Cat as in defini-

tion 7.16, we obtain a diagram in Cat by observation 5.35

L(F )ε //

η// L(G)

and we consider its pseudoequalizer

⇓ϕ1∼=

L(F )

ε

##E

I

>>

P //

I

L(G)

⇑ϕ2∼=

L(F )

η

;;

This category E has as objects (see proposition 5.7) Fa // A , G

b // B , G

εA(a) ??

b ��

⇓γ∼=

Au1��

Bv1

??C1 , G

ηA(a) ??

b ��

⇓δ∼=

Au2��

Bv2

??C2

Given a second object in E


Fa′ // A′ , G

b′ // B′ , G

εA′ (a′) ??

b′ ��

⇓γ′∼=

A′u′1��

B′v′1

??C ′1 , G

ηA′ (a′) ??

b′ ��

⇓δ′∼=

Au′2��

B′v′2

??C ′2

a morphism between them is represented by a pair of premorphisms in

L(F ) and L(G) F

a??

a′��

⇓ξ

Ar��

A′r′

??D1 , G

b??

b′��

⇓σ

As��

A′s′

??D2

satisfying σ ◦ γ = γ′ ◦ εD1(ξ) and σ ◦ δ = δ′ ◦ εD2(ξ) in L(G).

There is a comparison functor ♦ : L(E) −→ E given by the universal

property of L(E). Given an object E(a,b,γ,δ) // A , we define ♦(a, b, γ, δ) to

be Fa // A , G

b // A , G

εA(a) ??

b ��

⇓γ∼=

Aid��

Aid

??A , G

ηA(a) ??

b ��

⇓δ∼=

Aid��

Aid

??A

If we have a morphism E

(a,b,γ,δ) ??

(a′,b′,γ′,δ′)��

⇓(ξ,σ)

Au��

A′v

??C in L(E), its image via ♦ is

♦(ξ, σ) =

F

a??

a′��

⇓ξ

Au��

A′v

??C , G

b??

b′��

⇓σ

Au��

A′v

??C

This comparison functor ♦ is in fact an equivalence of categories:

Theorem 7.19. If A is a 2-filtered 2-category, given 2-functors and 2-natural

transformations AF))

G

55⇓ε ⇓η Cat , we call E is the pseudoequalizer of the induced


functors L(F )ε //

η// L(G) and E the pseudoequalizer functor A −→ Cat.

Then, the comparison functor L(E)♦ // E is an equivalence of categories.

That is, we have

L( E // Fε //

η// G ) ' E // L(F )

ε //

η// L(G)

Proof. As before, we show that it is essentially surjective, full and faithful.This is enough to prove that ♦ is an equivalence by observation 2.16.

(1) Essential surjectivity

Given an object

Fa // A , G

b // B , G

εA(a) ??

b��⇓γ∼=

A u1��

Bv1

??C1 , G

ηA(a) ??

b��⇓δ∼=

A u2��

Bv2

??C2

in E , we can form the composition γ−1 ◦ δ in L(G). This compositionof morphisms in L(G) is given along an invertible 2-cell τ that existsby 2-filteredness of A (specifically, axiom FF1 in definition 3.8):γ−1 ◦ δ = γ−1 ◦τ δ. We obtain

γ−1 ◦ δ =

Au2��

⇓δ∼= C1

��⇓τ∼= C3Bv1��

v2 ??

⇓γ−1∼= C2

??G

ηA(a)

??

b //

εA(a)′

��A

u1

??

We refer to [10] for a full explanation of this composition.The element (a, b, γ, δ) is isomorphic to ♦(a, εA(a), id, γ−1 ◦ δ), whichis

F

a // A , GεA(a)// A , G

εA(a) ??

εA(a) ��

⇓id∼=

Aid��

Aid

??A ,

Au2��

⇓δ∼= C1

��⇓τ∼= C3Bv1��

v2 ??

⇓γ−1∼= C2

??G

ηA(a)

??

b //

εA(a)′

��A

u1

??


via the isomorphism

(id, γ−1) =

F ⇓id

a??

a ��

Aid��

Aid

??A , G ⇓γ−1∼=

b??

εA(a) ��

Bv1��

Au1

??C1

with inverse

(id, γ) =

F ⇓id

a??

a ��

Aid��

Aid

??A , G ⇓γ∼=

εA(a) ??

b ��

Au1��

Bv1

??C1

(2) Fullness

Suppose given objects in the image of ♦ and a morphism betweenthem: F

a??

a′��

⇓ξ

Ar��

A′r′

??D1 , G

b??

b′��

⇓σ

As��

A′s′

??D2

: ♦(a, b, γ, δ) −→ ♦(a′, b′, γ′, δ′)

First, we want to obtain equivalent representatives for the morphismsξ and σ, with D1 = D2, r = s and r′ = s′. For this goal, we observethat by application of axiom FF1 of 2-filteredness of A (in definition3.8), as in the proof of fullness in the case of the product (theorem

7.10), we obtain F

a ??

a′��⇓ξ′

Au��

A′ u′

??D , G

b ??

b′��⇓σ′

Au��

A′ u′

??D with ξ ∼ ξ′ in L(F ) and

σ ∼ σ′ in L(G). The equations

σ ◦ γ = γ′ ◦ εD1(ξ)(7.20)

σ ◦ δ = δ′ ◦ εD2(ξ)(7.21)

hold in L(G) because (ξ, σ) is a morphism in E . Since equivalenceof premorphisms is preserved by both composition and the functorsε and η (this only means that composition and the functors arewell-defined with respect to equivalence classes given by homotopiesin L(F )), we can replace ξ by ξ′ and σ by σ′ in the equations 7.20and 7.21 to obtain

σ′ ◦ γ = γ′ ◦ εD(ξ′)

σ′ ◦ δ = δ′ ◦ εD(ξ′)

Then (ξ′, σ′) is also a morphism in E , and it is equal to (ξ, σ). Thisallows us to assume that we start in the simpler case of a morphism


represented by F

a??

a′��

⇓ξ

Au��

A′u′

??D , G

b??

b′��

⇓σ

Au��

A′u′

??D

: ♦(a, b, γ, δ) −→ ♦(a′, b′, γ′, δ′)

The equations σ ◦ γ = γ′ ◦ εD(ξ), σ ◦ δ = δ′ ◦ εD(ξ) hold in L(G),and thus really are equivalences of premorphisms given, each one, bya homotopy. As in the proof of the faithfulness of the comparisonfunctor in the case of the product (theorem 7.10), we can apply lemma

5.34 to assume that it is the same pair

A′u′ ??

u′��⇓ω∼=

Dh��

D k

??R , A′u ??

u��⇓τ∼=

Dh��

D k

??R

of invertible 2-cells in A that witnesses these homotopies. The LLequation for the first homotopy σ ◦ γ ∼ γ′ ◦ εD(ξ) is

Aid��

⇓γ∼= Au��

⇓id D

⇓ω∼=

h��R

Au��

id??

⇓σ D

id??G

εA(a)

??

b //

b′

��A′

u′ ��

u′??

D

k

??

=

D

h

��⇓τ∼=

R

A

u??

u��

⇓εD(ξ) Did��

⇓id Dk

??

A′id��

u′??

⇓γ′∼= A′u′

??G

εA(a)

??

εA′ (a′)//

b′

��A′

id

??

We define α = (Fk)(ξ) and β = (Gω)b′ ◦ (Gh)(σ) ◦ (Gτ−1)b.Precomposing with τ−1∗idεA(a) on both members of this LL equation,we obtain

Aid��

⇓γ Aku��

⇓id RAku��

id??

⇓β Rid

??G

εA(a)

??

b //

b′

��A′

ku′

??

=

Au��

⇓εD(ξ) Dk��

⇓id RA′id��

u′??

⇓γ′∼= A′ku′

??G

εA(a)

??

εA′ (a′)//

b′

��A′

id

??

This says that

(7.22) β ◦G(ku)(γ) = G(ku′)(γ′) ◦G(k)(εD(ξ))


Since

G(k)(εD(ξ)) = (Gk ◦ εC)(ξ)

= (εR ◦ F (k))(ξ)

= εR(α)

because ε is a 2-natural transformation F ⇒ G, the equation 7.22 isequivalent to

β ◦G(ku)(γ) = G(ku′)(γ′) ◦ εR(α)

In the same manner, we can obtain

β ◦G(ku)(δ) = G(ku′)(δ′) ◦ ηR(α)

These two equations are precisely the condition that (α, β) defines amorphism in E :

E

(a,b,γ,δ)??

(a′,b′,γ′,δ′) ��

⇓(α,β)

Aku

��

A′ku′

??R

If we apply ♦ to this morphism we get

♦(α, β) =

F

a??

a′��

⇓α

Aku��

A′ku′

??R , G

b??

b′��

⇓β

Aku��

A′ku′

??R

Finally, we observe that α ∼ ξ and β ∼ σ, since α and β wereobtained from ξ and σ respectively by pasting of invertible 2-cells.Then in E we have ♦(α, β) = (ξ, σ).

(3) Faithfulness

Given morphisms E

(a,b,γ,δ) ??

(a′,b′,γ′,δ′)��

⇓(ξ1,σ1)

Au1��

A′v1

??C1 and E

(a,b,γ,δ) ??

(a′,b′,γ′,δ′)��

⇓(ξ2,σ2)

Au2��

A′v2

??C2 in

L(E) with ♦(ξ1, σ1) = ♦(ξ2, σ2), this is ξ1 ∼ ξ2 in L(F ) and σ1 ∼ σ2

in L(G): there exist invertible 2-cells α1, β1, α2, β2 satisfying the LL


equations

F

a??

a′ ��

⇓ξ1

Au1

��

A′v1

??

v2 ��

⇓α1∼=

C1w1

��

C2

w2

??C

= F

a??

a′ ��

⇓ξ2

Au2

��

u1??

⇓β1∼=

C1w1

��

A′v2

??C2

w2

??C

and

G

b??

b′ ��

⇓σ1

Au1

��

A′v1

??

v2 ��

⇓α2∼=

C1w′1

��

C2

w′2

??C′

= G

b??

b′ ��

⇓σ2

Au2

��

u1??

⇓β2∼=

C1w′1

��

A′v2

??C2

w′2

??C′

We proceed as in the proof of the fullness of ♦ for the product(theorem 7.10). Thus we can assume α1 = α2 and β1 = β2. Thesetwo equations are then precisely the equations for homotopy between(ξ1, σ1) and (ξ2, σ2) in L(E) given in diagram 7.18. We conclude thatthese premorphisms define the same morphism in L(E).

�

7.5. Cotensor product. The following is proved in [10].

Theorem 7.23. Given a 2-functor A F // Cat with A 2-filtered and a

finite category P, we can consider the 2-functor FP : A −→ Cat definedby FP(A) = (FA)P and similarly for morphisms and 2-cells. Then, thecomparison functor ♦ : L(FP) −→ L(F )P (given by the universal propertyof L(FP) as in theorem 5.32) is an equivalence of categories.


7.6. Main result. We are now in conditions to state and prove the mainresult of this thesis, which is the following

Theorem 7.24. 2-filtered bicolimits and finite weighted bilimits commute

in Cat. More precisely, given a 2-functor A×P F // Cat with A 2-filtered

and P W // Cat a finite weight, the comparison functor


P←−F

♦ // wbilimWP←−

bicolimA−→F

is an equivalence of categories.

Proof. We apply the construction of the bilimit given in 6.2 to decompose theweighted bilimit wbilim

←−W

Pas a composition of biequalizers, finite biproducts

and bicotensors with a finite category. It this way, it suffices to prove theequivalence for these three classes of finite bilimits.

bicolimA−→wbilimV

Q←−G ' wbilimV

Q←−bicolimA−→

G

where the finite category Q and the 2-functors V and G stand for the finite2-categories and 2-functors which define biequalizers, finite biproducts andbicotensors with a finite category, in each case.

We may take these bilimits to be pseudoequalizers, finite pseudoproductsand pseudocotensors with a finite category, because these exist in Cat , and areequivalent to any choice of the corresponding bilimits (see observation 4.8).Pseudoproducts are just 2-products (see subsection 5.3), and pseudocotensorsare just cotensors (see observation 5.17). It then suffices to prove that the2-filtered pseudocolimit (of which we have a construction in subsection 5.16)commutes with each of these classes of finite pseudolimits:

pscolimA−→wpslimV

Q←−G ' wpslimV

Q←−pscolimA−→

G

where Q, V and G stand as before for the data that defines pseudoequalizers,finite 2-products, or cotensor products with a finite category. Since this isestablished in theorems 7.10 (and observation 7.15), 7.19 and 7.23, the resultfollows. �


8. Ends

In ordinary category theory, apart from the usual conical limits offunctors A −→ B, we have the notion of the end of a functor in twovariables Aop ×A −→ B. The theory of ends and coends can simplify manycalculations in category theory, as shown for example in the comprehensive[21].

During the study of the 2-categorical theory for this work (in particular,[15]), we noticed that some results about weighted 2-limits could be provenquickly if we were able to apply a version of the calculus of ends and coendsto the 2-categorical setting. In fact, there is a close relationship betweenends and weighted limits, that extends to the 2-dimensional case.

To end this thesis, we develop basic definitions and results on a 2-categoricalversion of ends, which we find useful to prove propositions involving weighted2-limits and weighted 2-colimits. Definitions and results on ends in enrichedcategory theory can be found in [9].

Definition 8.1. Given functors Aop ×AF //

G// B , a dinatural

transformation between them Fα +3 G is a collection of arrows

F (A,A)αA // G(A,A) , one for each object A ∈ A, such that for all

Af // B in A the following diagram commutes

F (A,A)αA // G(A,A)

G(idA,f)

''F (B,A)

F (f,idA)77

≡

F (idB ,f) ''

G(A,B)

F (B,B) αB// G(B,B)

G(f,idB)

77

When one of F or G is a constant functor ∆B, we call α a wedge.

Definition 8.2. Given a functor Aop ×A F // B and an object Z ∈ B, a

wedge Zα +3 F is a collection of arrows Z

αA // F (A,A) , one for each

object A ∈ A, such that for all Af // B in A the following diagram

commutes

ZαA //

αB

��

≡

F (A,A)

F (idA,f)

��F (B,B)

F (f,idB)// F (A,B)


Definition 8.3. The end of a functor Aop ×A F // B is a wedge

Lα +3 F such that every other wedge Z

β +3 F factorizes uniquely

through it: if we have arrows ZαA // F (A,A) for each A ∈ A such that

F (idA, f)βA = F (f, idB)βB for all Af // B in A , there exists a unique

Zh // L with αAh = βA for all A ∈ A:

F (A,A)F (idA,f)

((Z

≡

≡

βA00

βB ..

h // L ≡

αA88

αB &&

F (A,B)

F (B,B)

F (f,idB

66

This universal wedge L is denoted

∫AF (A,A).

We can dualize the definitions.

Definition 8.4. The coend of a functor Aop ×A F // B is a wedge

Fα +3 L such that every other wedge F

β +3 Z factorizes uniquely

through it: there exists a unique Lh // Z with hαA = βA for all A ∈ A.

This universal wedge is denoted

∫ A

F (A,A).

Observation 8.5. When the relevant limits exist, a collection of arrows

ZαA // F (A,A) is an end when the following diagram is an equalizer

Zα //

∏A

F (A,A)s //

t//∏

f :A→BF (B,B)

where α, s and t are the unique morphisms such that

πAα = αA

π(f :A→B)s = F (idA, f)

π(f :A→B)t = F (f, idB)

Analogously, coends are certain coequalizers of coproducts.This says that ends exist whenever products and equalizers exist and arepreserved by representable functors.

Proposition 8.6. Given functors AF //

G// B , we have a functor

Aop ×AHom(F−,G−) // Ens


The end of this functor is the set of natural transformations F ⇒ G, i.e.∫A

Hom(FA,GA) = Nat(F,G)

Proof. We propose of course Nat(F,G)αA // Hom(FA,GA)

as the function taking the A-component of a natural trans-

formation: Fλ +3 G 7→ FA

λA // GA . This gives a wedge

Nat(F,G)α +3 Hom(F−, G−) because, given a A

f // B in A,

we have

(Hom(idFA, Gf)αA)(λ) = Gf ◦ λAand

(Hom(Ff, idGA)αB)(λ) = λB ◦ Ffand these are equal for all such f precisely when λ is a natural transformationF ⇒ G.

By the same reasoning, dinaturality of a Zβ +3 Hom(F−, G−) says that

αA(z), with varying A, gives the components of a natural transformation,

for every z ∈ Z. Then the canonical function Zh // Nat(F,G) assigns to

each z ∈ Z the natural transformation with components (αA(z))A. �

We want to obtain a 2-categorical analogue of the result in proposition8.6.

Definition 8.7. Given a 2-functor Aop ×A F // B and Z ∈ B, a 2-wedge

Zα +3 F is a family of morphisms Z

αA // F (A,A) for each A ∈ A such

that the following diagram commutes

A(A,B)F (A,−) //

T (−,B)

��

≡

B(F (A,A), F (A,B))

B(αA,idF (A,B))

��B(F (B,B), F (A,B))

B(αB ,idF (A,B))// B(Z,F (A,B))

This is: for all A

f))

g55⇓ε B in A we have

F (A, f)αA = F (f,B)αB

(the condition for an ordinary wedge) and

T (A, ε) ∗ idαA = T (ε,B) ∗ idαB


Definition 8.8. Given a 2-functor Aop ×A F // B a 2-end is a 2-wedge

Lα +3 F such that every other 2-wedge Z

β +3 F factorizes uniquely

through it: if we have morphisms ZαA // F (A,A) for each A ∈ A such

that F (A, f)βA = F (f,B)βB and T (A, ε) ∗ idβA = T (ε,B) ∗ idβB for all

A

f))

g55⇓ε B in A, then there exists a unique Z

h // L with αAh = βA for

all A ∈ A:

F (A,A)F (idA,f)

((Z

≡

≡

βA00

βB ..

h // L ≡

αA88

αB &&

F (A,B)

F (B,B)

F (f,idB

66

This universal 2-wedge L is denoted

∫AF (A,A).

Observation 8.9. Also in this 2-categorical setting, 2-ends are particular2-limits, by a very similar formula:

(8.10)

∫AF (A,A)

α //∏A

F (A,A)s //

t//∏A,B

{A(A,B), F (B,B)}

is an equalizer, and then 2-ends are preserved by representable 2-functors,just like any 2-limit.

Proposition 8.11. Given 2-functors AF //

G// B , we have a 2-functor

Aop ×AHom(F−,G−) // Cat

The 2-end of this 2-functor is the category of 2-natural transformationsF ⇒ G and modifications between them, i.e.

(8.12)

∫A

Hom(FA,GA) = [A,B](F,G)

Proof. The proof is very similar to the 1-dimensional case.

We propose [A,B](F,G)αA // Hom(FA,GA) as the functor taking

the A-component of a 2-natural transformation or a modification:

Fλ +3 G 7→ FA

λA // GA , and λϕ µ 7→ λA

ϕA +3 µA .


Given A

f))

g55⇓ε B in A we have

(Hom(idFA, Gf)αA)(λ) = Gf ◦ λA(Hom(Ff, idGA)αB)(λ) = λB ◦ Ff

and

(Hom(idFA, Gf)αA)(ϕ) = idGf ◦ ϕA(Hom(Ff, idGA)αB)(ϕ) = ϕB ◦ idFf

so that

Hom(idFA, Gf)(λ) = Hom(Ff, idGA)(λ)

Hom(idFA, Gf)(ϕ) = Hom(Ff, idGA)(ϕ)

for all such f precisely when λ is a 2-natural transformation F ⇒ G and ϕis a modification. Then

(8.13) Hom(idFA, Gf)αA = Hom(Ff, idGA)αB

Given a 2-natural transformation Fλ +3 G , we have

(Hom(idFA, Gε) ∗ idαA)λ = Hom(Fε, idGB) ∗ idαB )λ

because this is equivalent to

Gε ∗ idλA = idλB ∗ Fε

which is true by 2-naturality of λ. Then we also have that

(8.14) Hom(idFA, Gε) ∗ idαA = Hom(Fε, idGB) ∗ idαBEquations 8.13 and 8.14 say that α defines a 2-wedge.

To see that it is universal, following the same reasoning in reverse we getthat if we have a 2-wedge

Zβ +3 Hom(F−, G−)

and Zf // Z ′ in Z, then βA(Z) is the A-component of a 2-natural trans-

formation, and βA(f) is the A-component of a modification, for each A. Thuswe define the unique functor

Z h // [A,B](F,G)

as assigning to each Z the 2-natural transformation F ⇒ G with components(βA(Z))A, and to each f the modification h(Z) h(Z ′) with components(βA(f))A. �

Of course, we also have the following


Definition 8.15. The 2-coend of a 2-functor Aop ×A F // B is a 2-wedge

Fα +3 L such that every other 2-wedge F

β +3 Z factorizes uniquely

through it: there exists a unique Lh // Z with hαA = βA for all A ∈ A.

This universal 2-wedge is denoted

∫ A

F (A,A).

Formulas involving 2-ends and 2-coends, just like 1-categorical ends andcoends, are very useful for calculations. We give an application to illustratethis.

Proposition 8.16. Given 2-functors Bop F // Cat and B G // Cat , we

can consider both wcolim−→

F G and wcolim−→

G F , and these are canonically iso-

morphic.

Proof.

[Bop, Cat ](F, Cat(G−,Z)) ∼=∫BCat(FB, Cat(GB,Z))

∼=∫BCat(GB, Cat(FB,Z)

∼= [Bop, Cat ](G, Cat(F−,Z))

where we use the formula 8.12 and the 2-natural isomorphism

Cat(X , Cat(Y,Z)) ∼= Cat(Y, Cat(X ,Z))

This says that

[Bop, Cat ](F, Cat(G−,Z)) ∼= [Bop, Cat ](G, Cat(F−,Z))

and thus any object representing either of these 2-functors represents theother. By the 2-Yoneda lemma (theorem 2.34),

wcolim−→

F G ∼= wcolim−→

G F

�

Another useful proposition is the following co-Yoneda lemma

Proposition 8.17. Given a 2-functor Bop F // Cat , we have an isomor-phism, 2-natural in A

FA ∼=∫B{B(A,B), FB}

Proof.

Cat(X ,∫B{B(A,B), FB}) ∼=

∫BCat(X , {B(A,B), FB})

∼=∫BCat(B(A,B), Cat(X , FB))

∼= [B, Cat ](B(A,−), Cat(X , F−))

∼= Cat(X , FA)


where we use the defining property of cotensor products, preservation of2-ends by representable 2-functors, and the formula 8.12. By the 2-Yonedalemma (theorem 2.34), we obtain

FA ∼=∫B{B(A,B), FB}

�

Finally, we remark that even though 2-ends and 2-coends are useful forcomputations, in categories with certain 2-limits and 2-colimits they give nonew universal objects.

Observation 8.18. A limit of a 2-functor B F // C weighted by a 2-functor

B W // Cat is a 2-end, provided the 2-end and the cotensor products ap-pearing below exist:

[B, Cat ](W, C(C,F−)) ∼=∫DCat(WD, C(C,FD))

∼=∫DC(C, {WD,FD})

∼= C(C,∫D{WD,FD})

Then we have

wlim←−

W F ∼=∫D{WD,FD}

because this 2-end represents the 2-functor defining the 2-limit (see definition4.3).

We already know that a 2-end can be expressed as a 2-limit by the formula8.10. We now give a proof of this fact by calculus of 2-ends.

Observation 8.19. The 2-end of a 2-functor Aop ×A F // B is a 2-limitweighted by an hom 2-functor whenever the 2-limits appearing below exist:∫

AF (A,A) ∼=

∫A

∫B{A(A,B), F (A,B)}

∼=∫A,B{A(A,B), F (A,B)}

∼= wlim←−

A(−,−) F

where we use

F (A,A) ∼=∫B{A(A,B), F (A,B)}

as a consecuence of the co-Yoneda lemma 8.17 applied to F (A,−), and thefact that 2-ends commute with 2-ends (this is called the “Fubini theorem”for iterated 2-ends; see [21, remark 1.16] for a 1-categorical version of thisstatement, or [14, section 2.1] for the general formulation in the enrichedsetting).


9. Bibliography

[1] Igor Bakovic. “Fibrations of bicategories”. In: Preprint available athttp://www. irb. hr/korisnici/ibakovic/groth2fib. pdf (2011).

[2] Jean Benabou. “Introduction to bicategories”. In: Reports of the Mid-west Category Seminar. Springer. 1967, pp. 1–77.

[3] GJ Bird, Gregory Maxwell Kelly, A John Power, and Ross Street.“Flexible limits for 2-categories”. In: Journal of Pure and Applied Algebra61.1 (1989), pp. 1–27.

[4] Francis Borceux. Handbook of Categorical Algebra 1, Basic CategoryTheory, vol. 50 of Encyclopedia of Mathematics and its Applications.1994.

[5] Matıas Ignacio Data. “Una construccion de bicolımites 2-filtrantes decategorıas”. Universidad de Buenos Aires, 2014.

[6] M Emilia Descotte. “Una generalizacion de la Teorıa de Ind-objetos deGrothendieck a 2-categorıas”. Universidad de Buenos Aires, 2010.

[7] M Emilia Descotte and Eduardo J Dubuc. “A theory of 2-pro-objects”.In: Cahiers de topologie et geometrie differentielle categoriques (alsowith expanded proofs as arXiv preprint arXiv:1406.5762) 55 (2014).

[8] M Emilia Descotte, Eduardo J Dubuc, and Martın Szyld. “On thenotion of 2-flat 2-functors”. In: to appear ().

[9] Eduardo J Dubuc. “Kan extensions in enriched category theory”. In:(1970).

[10] Eduardo J Dubuc and Ross Street. “A construction of 2-filtered bicol-imits of categories”. In: Cahiers de topologie et geometrie differentiellecategoriques 47.2 (2006), pp. 83–106.

[11] Delphine Dupont. “Interchange of filtered 2-colimits and finite 2-limits”.In: arXiv preprint arXiv:0904.1553 (2009).

[12] Thomas M Fiore. Pseudo Limits, Biadjoints, and Pseudo Algebras:Categorical Foundations of Conformal Field Theory: Categorical Foun-dations of Conformal Field Theory. 860. American Mathematical Soc.,2006.

[13] Jonas Frey. “Notes on 2-categorical limits”. In: (2010).[14] Gregory Maxwell Kelly. Basic concepts of enriched category theory.

Vol. 64. CUP Archive, 1982.[15] Gregory Maxwell Kelly. “Elementary observations on 2-categorical

limits”. In: Bulletin of the Australian Mathematical Society 39.02 (1989),pp. 301–317.

[16] Gregory Maxwell Kelly. “Structures defined by finite limits in theenriched context, I”. In: Cahiers de topologie et geometrie differentiellecategoriques 23.1 (1982), pp. 3–42.

[17] Gregory Maxwell Kelly and Ross Street. “Review of the elements of2-categories”. In: Category seminar. Springer. 1974, pp. 75–103.

[18] John F Kennison. “The fundamental localic groupoid of a topos”. In:Journal of pure and applied algebra 77.1 (1992), pp. 67–86.

[19] Stephen Lack. “A 2-categories companion”. In: Towards higher cate-gories. Springer, 2010, pp. 105–191.

[20] Tom Leinster. “Basic bicategories”. In: arXiv preprintmath.CT/9810017 589 (1998).

BIBLIOGRAPHY 97

[21] Fosco Loregian. “This is the (co) end, my only (co) friend”. In: arXivpreprint arXiv:1501.02503 (2015).

[22] Saunders Mac Lane. Categories for the working mathematician. Vol. 5.Springer Science & Business Media, 1978.

[23] Saunders Mac Lane and Ieke Moerdijk. Sheaves in geometry and logic:A first introduction to topos theory. Springer Science & Business Media,2012.

[24] Ieke Moerdijk. “The classifying topos of a continuous groupoid. I”.In: Transactions of the American Mathematical Society 310.2 (1988),pp. 629–668.

[25] Franciscus Rebro. “Building the bicategory Span2(C)”. In: arXivpreprint arXiv:1501.00792 [math.CT] (2015).

[26] Emily Riehl. Categorical homotopy theory. 24. Cambridge UniversityPress, 2014.

[27] Emily Riehl et al. “Weighted Limits and Colimits”. In: available atmath. harvard. edu/ eriehl (2009).

[28] Ross Street. “Correction to “Fibrations in bicategories””. eng. In:Cahiers de Topologie et Geometrie Differentielle Categoriques 28.1(1987), pp. 53–56. url: http://eudml.org/doc/91390.

[29] Ross Street. “Fibrations in bicategories”. In: Cahiers de topologie etgeometrie differentielle categoriques 21.2 (1980), pp. 111–160.

[30] Ross Street. “Limits indexed by category-valued 2-functors”. In: Journalof Pure and Applied Algebra 8.2 (1976), pp. 149–181.

http://eudml.org/doc/91390

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