MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.