$\operatorname{Ind}(C^I) = \operatorname{Ind}(C)^I$?

Question

$\DeclareMathOperator\Ind{Ind}$Let $C$ be a (small) category, and $I$ a finite category, is it true that the natural functor $\Ind(C^I) \to \Ind(C)^I$ is an equivalence of categories? where $\Ind$ denotes the category of ind-objects (so the free completion under filtered colimits) and the exponentials are for categories of functors.

This is proved by Lurie in Higher topos theory (proposition 5.3.5.15) when $C$ is an infinity category and $I$ is a finite poset. He gives an example to show that this cannot be generalized to the case of $I$ a finite simplicial set — but finite category is much more restrictive and his example doesn't rule this out at all.

I'm mostly interested in the case where $C$ is a 1-category and already has finite colimits, but I'm curious of any interesting things that can be said more generally.

Answer 1

So, the question appeared more subtle than I initially thoughts so I have written a short paper with more references and the details of what I'm going to say below. It is available here.

Here is the summary:

First, Makkai's theorem cited by Ivan is indeed false. Building on Ben Wieland's comment (thank you very much for this!), one get the following counter-example.

Let $I=B(\mathbb{Z}/2\mathbb{Z})$ the category with one object $*$ and one non-trivial automorphism satisfying $f^2=1$ and let $C$ be the subcategory of $I \times \omega$ containing all the objects, and all the arrows $(*,n) \to (*,m)$ when $n < m$ but only the identity when $n = m$.

In particular, functors $I \to C$ are all constant, as $C$ has no non-trivial endomorphisms, but there is a non-trivial functor $I \to Ind(C)$: indeed the colimits of the chain of the $(*,n)$ is in the Ind completion, and as a presheaf, it is the pullback along the projection $C \to I$ of the unique representable presheaf, so it comes with a $\mathbb{Z}_2$-action and hence is a non-trivial functor $ I \to Ind(C)$, which doesn't belong to $Ind(C^I)$.

As $C$ has no non-trivial endomorphisms it is Cauchy complete, hence $C$ identifies with the full subcategory of $\omega$-presentable objects of $Ind(C)$ and hence the exponential $Ind(C)^I$ is a counter-example to Makkai's theorem.

Now this construction can be generalized to any category and any ordinal, and this leads to the following theorem:

Theorem: Let $I$ be a category and $\kappa$ be a regular cardinal. The following condition are equivalent.

1. $I$ is $\kappa$-small, has no non-identity endomorphisms, and its posetal relfection is well-founded.
2. for all category $C$, the functor $Ind_\kappa(C^I) \to Ind_\kappa(C)^I$ is an equivalence.
3. for all $\kappa$-accessible category $A$, the category $A^I$ is $\kappa$-accessible with its $\kappa$-presentable objects being the functors from $I$ to $\kappa$-presentable objects of $A$.

In particular, one recovers Lurie's version of the theorem (I mean for 1-categories), as well as Meyer's result mentioned by Peter Haine. Also Makkai theorem is false even for uncountable $\kappa$. But this isn't the end of the story. If we are interested in category with colimits, then everything works more nicely and we get a positive answer to my questions:

Theorem: Let $I$ be a category and $\kappa$ be a regular cardinal. The following conditions are equivalent.

1. $I$ is $\kappa$-small.
2. for all category $C$ with $\kappa$-small colimits, the functor $Ind_\kappa(C^I) \to Ind_\kappa(C)^I$ is an equivalence.
3. for all locally $\kappa$-presentable category $A$, the category $A^I$ is locally $\kappa$-presentable, with its $\kappa$-presentable objects being the functors from $I$ to $\kappa$-presentable objects of $A$.

The proofs of these and additional details are in the paper linked above.

It seems Makkai's theorem was used in a few places in the literature - but from what I can tell, all uses I've seen are covered by these two theorems.

Answer 2

Yes. This follows directly from 5.1 in Strong conceptual completeness for first-order logic by Makkai (1988, Annals of Pure and Applied Logic, doi:10.1016/0168-0072(88)90019-X).

Answer 3

This short answer is a complement to Simon's detailed answer. In Simon's answer he considers arbitrary $\kappa$-accessible categories $A$ (in the first theorem) and locally $\kappa$-presentable ones (in the second theorem).

There is an intermediate case arising when a $\kappa$-accessible category $A$ has colimits of all $\lambda$-indexed chains, for some fixed infinite cardinal $\lambda<\kappa$. (This presumes that $\kappa$ is uncountable.) A $\lambda$-indexed chain in $A$ is a functor $\lambda\longrightarrow A$, where the ordered set $\lambda$ is interpreted as a category.

Theorem: Let $\kappa$ be a regular cardinal, $\lambda<\kappa$ be a smaller infinite cardinal, and $A$ be a $\kappa$-accessible category with colimits of $\lambda$-indexed chains. Let $I$ be a $\kappa$-small category. Then the category $A^I$ is $\kappa$-accessible, and the $\kappa$-presentable objects of $A^I$ are the functors from $I$ to the $\kappa$-presentable objects of $A$.

References: this is Theorem 6.1 from my new preprint "Notes on limits of accessible categories", https://arxiv.org/abs/2310.16773 .

Prior art: the Pseudocolimit Theorem of Raptis and Rosicky, https://arxiv.org/abs/1403.3042 , http://www.tac.mta.ca/tac/volumes/30/19/30-19abs.html , Section 2, is a somewhat similar and related result. This is based on Proposition 3.1 from the paper of Chorny and Rosicky, https://arxiv.org/abs/1110.0605 , https://doi.org/10.1016/j.jpaa.2012.01.015 .

In fact, the idea goes back, at least, as far as the unpublished 1977 preprint of Friedrich Ulmer "Bialgebras in locally presentable categories" (Univ. of Wuppertal), Theorem 3.8 and Corollary 3.9.