# Is there lore about how endofunctors of Cat interact with the formation of presheaf categories?

This is a request for references about a peculiar categorical construction I've run into in some work I've been doing, and about which I'd like to learn as much as I can.

Let $\mathrm{Cat}$ be the category of small categories, and let $\mathrm{PSh}(C)$ be the category of presheaves of sets on a category $C$. Suppose we are given a "reasonable" endofunctor $\Xi\colon \mathrm{Cat}\to \mathrm{Cat}$. I want to consider a certain "intertwining" functor $$V\colon \Xi\mathrm{PSh}(C) \to \mathrm{PSh}(\Xi C)$$ defined by the formula $$(VX)(\gamma) = \mathrm{Hom}_{\Xi\mathrm{Psh}(C)}(A\gamma, X),$$ where $X$ is an object of $\Xi\mathrm{PSh}(C)$, $\gamma$ is an object of $\Xi C$, and $A\colon \Xi C\to \Xi\mathrm{PSh}(C)$ is the functor obtained by applying $\Xi$ to the Yoneda functor $C\to \mathrm{PSh}(C)$.

Note: it's unreasonable to expect for a randomly chosen $\Xi$ that the category $\Xi \mathrm{PSh}(C)$ is even defined, since $\mathrm{PSh}(C)$ is a large category, and $\Xi$ is given as a functor on small categories. And even if it is defined, it's unreasonable to expect that $V$ is well-defined, since $(VX)(\gamma)$ may not be a set. But here are some reasonable examples:

• Let $\Xi C= C\times C$. Then $V\colon \mathrm{PSh}(C)\times \mathrm{PSh}(C)\to \mathrm{PSh}(C\times C)$ is the "external product" functor, which takes a pair of presheaves $(X_1,X_2)$ on $C$ to the presheaf $(c_1,c_2) \mapsto X_1(c_1)\times X_2(c_2)$ on $C^2$.

You can generalize this by considering $\Xi C= \mathrm{Func}(S,C)$, where $S$ is a fixed small category.

• Let $\Xi C = C^{\mathrm{op}}$. Then $V\colon \mathrm{PSh}(C)^{\mathrm{op}} \to \mathrm{PSh}(C^{\mathrm{op}})$ is a sort of "dualizing" functor, which sends a presheaf $X$ on $C$ to the presheaf $c\mapsto \mathrm{Hom}_{\mathrm{PSh}(C)}(X, Fc)$ on $C^\mathrm{op}$; here $F\colon C\to \mathrm{PSh}(C)$ represents the Yoneda functor.

• Let $\Xi C=\mathrm{gpd} C$, the maximal subgroupoid of $C$. Then $V\colon \mathrm{gpd}\\,\mathrm{PSh}(C)\to \mathrm{PSh}(\mathrm{gpd}C)$ is such that $(VX)(c)$ is the set of isomorphisms between $X$ and the presheaf represented by $c$.

The sorts of questions I have include the following.

1. What makes a functor $\Xi$ reasonable? Is it enough if it's accessible?

2. I think $V$ should be the left Kan extension of the Yoneda functor $B\colon \Xi C\to \mathrm{PSh}(\Xi C)$ along $A$. Is this true? When can I expect to have $VA\approx B$?

3. How does $V$ of a composite $\Xi \Psi$ relate to the composite of the $V$s of each term?

4. Given a functor $f\colon C\to D$, you get a bunch of functors between the associated presheaf categories. How does $V$ interact with such functors?

There's really only one or two examples of $\Xi$ that really I need to understand this for, and I don't want to spend time working out a general theory of this thing. It would be most convenient if someone can point me to a reference which talks about this construction. Even one that deals with particular instances of it would be helpful.

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I've answered some of these. Q2: V is always a left Kan extension, and VA=B exactly when A is a full embedding (easy!). Q4: if $\Xi$ is not merely a functor, but a 2-functor, then V commutes with the functors induced by restricting presheaves along f. ($\Xi$ is a 2-functor in my first example, but not in the other two examples.) – Charles Rezk Mar 15 '10 at 2:24

This is really just a comment, but it's too long to fit.

Many people have come up against the problem that PSh isn't an endofunctor of Cat, because even if C is small, PSh(C) usually isn't. There's a standard way to solve this problem, as follows.

• Replace Cat (small categories) with CAT (locally small categories)
• Replace PSh (presheaves) with psh (small presheaves, i.e. small colimits of representables)

Then psh is genuinely an endofunctor of CAT. If C is small then psh(C) = PSh(C). But if C is not small then psh(C) is a proper subcategory of PSh(C).

In fact, psh is not only an endofunctor of CAT, but a monad. It's free small-cocompletion. That is, it takes a category and freely adjoins colimits.

The unit of this monad is the Yoneda embedding. Given this, and given that the Yoneda embedding plays a part in your considerations, I wonder whether the multiplication of the monad plays a part too.

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That is an interesting thought. – Charles Rezk Mar 10 '10 at 18:11
Incidentally, it is tempting to calculate $V$ in the case $\Xi=PSh$; this is illicit in the way I set things up, but maybe not with your suggestion. Anyway, if you run the formula, then $V$ takes a functor $G: PSh(C)^{op}\to Set$ to the "closest available" representable functor $Psh(C)^{op}\to Set$, i.e., the one represented by $GF$, where $F: C\to Psh(C)$ is Yoneda. – Charles Rezk Mar 10 '10 at 18:15
What are algebras for this monad? – David Carchedi Apr 2 '10 at 14:57