Commutation of limits with filtered colimits

Question

It is well known that in the category ${\bf Set}$ finite limits commute with filtered colimit. Now, how much general is the above result? Namely, are there theorems of the form:

If the category $\mathfrak{C}$ is ........, then finite limits commute with filtered colimit.

Answer 1

I think basically all theorems of that form stem from the theorem for $\textbf{Set}$. Here are some simple observations.

1. If $\mathcal{D}$ is a category that has limits of finite diagrams and colimits of small filtered diagrams and has the commutation property in question, then $[\mathcal{A}^\textrm{op}, \mathcal{D}]$ also has the commutation property.
2. If $\mathcal{D}$ is a category that has the commutation property in question and there is a conservative functor $\mathcal{C} \to \mathcal{D}$ that preserves limits of finite diagrams and colimits of small filtered diagrams, then $\mathcal{C}$ also has the commutation property.
3. If $\mathcal{D}$ is a category that has limits of finite diagrams and colimits of small filtered diagrams and has the commutation property in question, and $\mathcal{C}$ is a reflective (full) subcategory of $\mathcal{D}$ such that the reflector $\mathcal{D} \to \mathcal{C}$ preserves limits of finite diagrams, then $\mathcal{C}$ also has the commutation property.

Almost all examples I can think of of can be deduced from the above:

• Any Grothendieck topos, by 1 and 3.
• Any finitely accessible category, by 1 and 2.
• The category of sheaves of abelian groups on a Grothendieck site, by 1, 2, and 3.

I guess I should also mention AB5 categories, but the commutation property is basically equivalent to the AB5 axiom in the context of abelian categories.

The only example that seems to require non-trivial work to deduce from the theorem for $\textbf{Set}$ is the case of the $(\infty, 1)$-category of $\infty$-groupoids. Morally, it comes down to the fact that colimits of small filtered diagrams of simplicial sets are automatically homotopy colimits, which in turn comes down to the fact that the class of weak homotopy equivalences is closed under colimits of small filtered diagrams. On the other hand, limits of finite diagrams of simplicial sets are not automatically homotopy limits. What is true is that pullbacks of Kan fibrations are homotopy pullbacks. Since every morphism of simplicial sets can be factored as a weak homotopy equivalence followed by a Kan fibration, this allows us to deduce that homotopy colimits of small filtered diagrams preserve homotopy pullbacks. Homotopy terminal objects are obviously preserved, so it follows that homotopy limits of finite diagrams are preserved.

(There are some subtleties I am sweeping under the rug here. For instance, a 1-category with finitely many objects and finitely many morphisms is not necessarily finite as an $(\infty, 1)$-category. Then there is the problem of strictifying diagrams in the $(\infty, 1)$-category of $\infty$-groupoids, i.e. realising them as diagrams in the 1-category of simplicial sets. These subtleties make me wonder if it is really legitimate to say that the homotopy commutation property is derived from the commutation property.)