Pullbacks of categories

Let $\mathfrak{Cat}$ be the 2-category of small categories, functors, and natural transformations. Consider the following diagram in $\mathfrak{Cat}$: $$\mathbb{D} \stackrel{F}{\longrightarrow} \mathbb{C} \stackrel{G}{\longleftarrow} \mathbb{E}$$

There are several notions of pullback one could investigate in $\mathfrak{Cat}$:

• The ordinary pullback in the underlying 1-category $\textbf{Cat}$: these exist and are unique, by ordinary abstract nonsense. Explicitly, $\mathbb{D} \mathbin{\stackrel{1}{\times}_\mathbb{C}} \mathbb{E}$ has objects pairs $(d, e)$ such that $F d = G e$ (evil!) and arrows are pairs $(k, l)$ such that $F k = G l$. This evidently an evil notion: it is not stable under equivalence. For example, take $\mathbb{C} = \mathbb{1}$: then we get an ordinary product; but if $\mathbb{C}$ is the interval category $\mathbb{I}$, we have $\mathbb{1} \simeq \mathbb{I}$, yet if I choose $F$ and $G$ so that their images are disjoint, we have $\mathbb{D} \mathbin{\stackrel{1}{\times}_\mathbb{C}} \mathbb{E} = \emptyset$, and $\emptyset \not\simeq \mathbb{D} \times \mathbb{E}$ in general.

• The strict 2-pullback is a category $\mathbb{D} \mathbin{\stackrel{s}{\times}_\mathbb{C}} \mathbb{E}$ and two functors $P : \mathbb{D} \mathbin{\stackrel{s}{\times}_\mathbb{C}} \mathbb{E} \to \mathbb{D}$, $Q : \mathbb{D} \mathbin{\stackrel{s}{\times}_\mathbb{C}} \mathbb{E} \to \mathbb{E}$ such that $F P = G Q$, with the following universal property (if I'm not mistaken): for all $K : \mathbb{T} \to \mathbb{D}$ and $L : \mathbb{T} \to \mathbb{E}$ such that $F K = G L$, there is a functor $H : \mathbb{T} \to \mathbb{D} \mathbin{\stackrel{s}{\times}_\mathbb{C}} \mathbb{E}$ such that $P H = K$ and $Q H = L$, and $H$ is unique up to equality; if $K' : \mathbb{T} \to \mathbb{D}$ and $L' : \mathbb{T} \to \mathbb{E}$ are two further functors such that $F K' = G L'$ and $H' : \mathbb{T} \to \mathbb{D} \mathbin{\stackrel{s}{\times}_\mathbb{C}} \mathbb{E}$ satisfies $P H' = K'$ and $Q H' = L'$ and there are natural transformations $\beta : K \Rightarrow K'$ and $\gamma : L \Rightarrow L'$, then there is a unique natural transformation $\alpha : H \Rightarrow H'$ such that $P \alpha = \beta$ and $Q \alpha = \gamma$. So $\mathbb{D} \mathbin{\stackrel{s}{\times}_\mathbb{C}} \mathbb{E} = \mathbb{D} \mathbin{\stackrel{1}{\times}_\mathbb{C}} \mathbb{E}$ works, and in particular, strict 2-pullbacks are evil.

• The pseudo 2-pullback is a category $\mathbb{D} \times_\mathbb{C} \mathbb{E}$, three functors $P : \mathbb{D} \times_\mathbb{C} \mathbb{E} \to \mathbb{D}$, $Q : \mathbb{D} \times_\mathbb{C} \mathbb{E} \to \mathbb{E}$, $R : \mathbb{D} \times_\mathbb{C} \mathbb{E} \to \mathbb{C}$, and two natural isomorphisms $\phi : F P \Rightarrow R$, $\psi : G Q \Rightarrow R$, satisfying the following universal property: for all functors $K : \mathbb{T} \to \mathbb{D}$, $L : \mathbb{T} \to \mathbb{E}$, $M : \mathbb{T} \to \mathbb{C}$, and natural isomorphisms $\theta : F K \Rightarrow M$, $\chi : G L \Rightarrow M$, there is a unique functor $H : \mathbb{T} \to \mathbb{D} \times_\mathbb{C} \mathbb{E}$ and natural isomorphisms $\tau : K \Rightarrow P H$, $\sigma : L \Rightarrow Q H$, $\rho : M \Rightarrow R H$ such that $\phi H \bullet F \tau = \rho \bullet \theta$ and $\psi H \bullet G \sigma = \rho \bullet \chi$ (plus some coherence axioms I haven't understood); and some further universal property for natural transformations.

  By considering the cases $\mathbb{T} = \mathbb{1}$ and $\mathbb{T} = \mathbb{2}$, it seems that $\mathbb{D} \times_\mathbb{C} \mathbb{E}$ can be taken to be the following category: its objects are quintuples $(c, d, e, f, g)$ where $f : F d \to c$ and $g : G e \to c$ are isomorphisms, and its morphisms are triples $(k, l, m)$ where $k : d \to d'$, $l : e \to e'$, $m : c \to c'$ make the evident diagram in $\mathbb{C}$ commute. The functors $P, Q, R$ are the obvious projections, and the natural transformations $\phi$ and $\psi$ are also given by projections.

  Question. This seems to satisfy the required universal properties. Is my construction correct?

  Question. What are the properties of this construction? Is it stable under equivalences, in the sense that $\mathbb{D}' \times_{\mathbb{C}'} \mathbb{E}' \simeq \mathbb{D} \times_\mathbb{C} \mathbb{E}$ when there is an equivalence between $\mathbb{D}' \stackrel{F'}{\longrightarrow} \mathbb{C}' \stackrel{G'}{\longleftarrow} \mathbb{E}'$ and $\mathbb{D} \stackrel{F}{\longrightarrow} \mathbb{C} \stackrel{G}{\longleftarrow} \mathbb{E}$?

• Finally, there is the non-strict 2-pullback, which as I understand it has the same universal property as the pseudo 2-pullback but with "unique functor" replaced by "functor unique up to isomorphism".

  Question. Is this correct?

General question. Where can I find a good explanation of strict 2-limits / pseudo 2-limits / bilimits and their relationships, with explicit constructions for concrete 2-categories such as $\mathfrak{Cat}$? So far I have only found definitions without examples. (Is there a textbook yet...?)

Solution 1:

Zhen, I am not sure I understand your question.

1. Every stict 2-limit is obviously also a 1-limit in the underlying 1-category, so these are not really different concepts (a 2-limit is a strengthen version of a limit; BTW, since Cat is 2-complete then every 1-limit in Cat is automatically a 2-limit).

2. Your construnction of a 2-pseudo pullback is fine. However, it is easy to verify that it is not stable under equivalence of categories ([Added] in the sense that: $C$ is a limit of $F$ and $D$ is equivalent to $C$ does not imply that $D$ is a limit of $F$). All of the mentioned limits are defined in terms of "strict" adjunctions (or, more acurately, in terms of strict universal properties), i.e. there is a natural isomorphism: $$\Delta(C) \rightarrow F \approx C \rightarrow \mathit{lim}(F)$$ To obtain a concept that is stable under equivalences, you have to replace this natural isomorphism by a natural equivalence of categories (plus perhaps some additional coherence equations). Of course, every strict 2-limit is also a "weak" 2-limit in the above sense (because every isomorphism is an equivalence), so again in a complete 2-category you will not get anything new.

[Added] 3. Let $\mathbb{W}$ be a 2-category, and $X$ a 1-category. There are three types of 2-cateogrical cones in $\mathbb{W}$ of the shape of $X$:

• $\mathit{Cone}$ --- objects are strict functors $X \rightarrow \mathbb{W}$, 1-morphisms are strict natural transformations between functors, and 2-morphisms are modifications between natural transformations

• $\mathit{PseudoCone}$ --- objects are pseudo functors $X \rightarrow \mathbb{W}$, 1-morphisms are pseudo natural transformations between functors, and 2-morphisms are modifications between natural transformations

• $\mathit{LaxCone}$ --- objects are lax functors $X \rightarrow \mathbb{W}$, 1-morphisms are lax natural transformations between functors, and 2-morphisms are modifications between natural transformations

A limit of a strict functor $F \colon X \rightarrow \mathbb{W}$ is a 2-representation of: $$\mathit{Cone}(\Delta(-), F)$$ where $\Delta$ is the usual diagonal functor. A pseudolimit of $F$ is a representation of: $$\mathit{PseudoCone}(\Delta(-), F)$$ And a lax limit is a representation of: $$\mathit{LaxCone}(\Delta(-), F)$$ In each case if you take equivalent functors, then you get equivalent representations. However, in each case the notion of equivalent functors is different. Perhaps your problem is that you are using the equivalence from $\mathit{PseudoCone}$ in the context of $\mathit{Cone}$

[Added^2] I have missed one of your questions:

Finally, there is the non-strict 2-pullback, which as I understand it has the same universal property as the pseudo 2-pullback but with "unique functor" replaced by "functor unique up to isomorphism".

If by a non-strict pullback you mean a weak (pseudo)pullback in the above sense, then the universal property is much more subtle --- it does not suffice to say that there is a functor $f \colon X \rightarrow \mathit{Lim}(F)$ that is unique up to a 2-isomorphism (just like in the definition of a limit you do not say that there is an object which is unique up to 1-isomorphism), you have to say that for every cone $\alpha \colon \Delta(X) \rightarrow F$ there exists $f \colon X \rightarrow \mathit{Lim}(F)$ such that for any cone $\beta$ on $X$ with its $g \colon X \rightarrow \mathit{Lim}(F)$ and every family of 2-morphism $\tau \colon \alpha \rightarrow \beta$ that is compatible with $F$ there exists a unique 2-morphism $f \rightarrow g$ such that everything commutes.

However, if by a non-strict pullback you mean a lax pullback, then the construction is similar to your construction of a pseudopullback --- without requirement that your $f$ and $g$ are isomorphisms.

You have aslo asked:

Where can I find a good explanation of strict 2-limits / pseudo 2-limits / bilimits and their relationships, with explicit constructions for concrete 2-categories such as $\mathfrak{Cat}$? So far I have only found definitions without examples. (Is there a textbook yet...?)

I do not know of any good textbook, but can provide you with two examples.

There is a simple general procedure to construct strict/pseudo/lax limits and colimits in $\mathbf{Cat}$. You shall notice that to give a monad is to give a lax functor $T \colon 1 = 1^{op} \rightarrow \mathbf{Cat}$. Then the lax colimit of $T$ is the Kleisli category for the monad $T$, and the lax limit of $T$ is the Eilenberg-Moore category for the monad $T$. This idea may be pushed a bit further: you may think of a lax functor $\Phi \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$ as of a kind of a "multimonad". Then its multi-Kleisli resolution is given by the Grothendieck construction $\int \Phi$ (this construction gives a fibration precisely when $\Phi$ is a pseudofunctor). And similarly its multi-Eilenberg-Moore category is given by a suitable collection of (ordinary) algebras. In these construction if you impose a reguirement on cartesian morphisms / algebras to be isomorphisms, then you get a pseudocolimit / pseudolimit of $\Phi$, and if you impose identities instead of isomorphisms you get a colimit / limit.

What is more, the Grothendieck construction works also for the bicategory of distributors; and because there is a duality on the bicategory of distributors you may construct (lax/pseudo/strict) limits in this bicategory via the Grothendieck construction as well.