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(Homotopy) sifted colimits commute with finite (homotopy) products in the category of sets (respectively spaces).
Is it possible to point out a bigger class of categories for which this is true?
Jacob Lurie points out in a comment on MathOverflow (http://mathoverflow.net/questions/181188/commutation-of-simplicial-homotopy-colimits-and-homotopy-products-in-spaces) that this is false for arbitrary presentable ∞-categories.
On the other hand, it seems like this might be true for cartesian closed presentable ∞-categories, because the argument for sets seems to go through in this case.
Also, could it be true for algebras over a finitely accessible ∞-monad? The forgetful functor from algebras to spaces creates limits and sifted colimits, so commutativity should follow from commutativity in spaces.
In general, is it possible to describe a more general class of categories that covers the above examples?
HTT Lemma 5.5.8.11: If formation of products preserves sifted colimits separately in each variable, then sifted colimits commute with finite products.
For example, it’s true in any infinity-topos.
Yes, in ∞-toposes almost all properties of spaces still hold, including the one under discussion, but I was hoping for a bigger class, e.g., some version that would allow for algebraic structures.
You could try to work out the $(\infty, 1)$-categorical version of locally sifted-presentable categories.
Great, this looks exactly like the answer I would like to have!
Is it true that in the 1-categorical situation sifted colimits commute with finite products in a locally sifted-presentable category?
At least in some other paper they claim that in a locally fintely presentable category filtered colimits commute with (a) finite limits and (b) small products, so the above statement seems plausible.
I guess I’m not even sure what the ∞-analog of a locally finitely presentable category is. Is it a combinatorial model category such that there is a generating set of cofibrations whose domains and codomains are compact objects (i.e., hom out of them preserves filtered colimits), or, I guess, equivalently there is a set of homotopy generators consisting of compact objects?
Is it true that in the 1-categorical situation sifted colimits commute with finite products in a locally sifted-presentable category?
Yes. Use the same argument that shows that filtered colimits preserve finite limits in a locally finitely presentable category.
At least in some other paper they claim that in a locally fintely presentable category filtered colimits commute with (a) finite limits and (b) small products, so the above statement seems plausible.
(b) is false in $Set$.
I guess I’m not even sure what the ∞-analog of a locally finitely presentable category is.
It is defined in Higher topos theory. I do not know if every locally finitely presentable quasicategory can be presented by a combinatorial model category of the kind you describe.
(b) is false in Set.
Most likely I am misinterpreting something, but in the paper
Continuous categories revisited J. Adamek, F. W. Lawvere, J. Rosicky http://tac.mta.ca/tac/volumes/11/11/11-11abs.html
the authors claim in 1.2.(iv) that for all locally finitely presentable categories products distribute over filtered colimits. They attribute this to Grothendieck and Verdier.
I’m somewhat mystified here. Take a countable product of copies of the identity functor $\hom(1, -): Set \to Set$; this is $\hom(\mathbb{N}, -)$. This functor cannot preserve the colimit of the filtered diagram of inclusions of finite subsets $F$ into $\mathbb{N}$ (since this colimit is $\mathbb{N}$, it would mean $colim_{F \subset \mathbb{N}} \hom(\mathbb{N}, F) \to \hom(\mathbb{N}, \mathbb{N})$ is an isomorphism, but that’s impossible since $1_\mathbb{N}$ is not in the image).
Most likely I am misinterpreting something, but in the paper
Continuous categories revisited J. Adamek, F. W. Lawvere, J. Rosicky http://tac.mta.ca/tac/volumes/11/11/11-11abs.html
the authors claim in 1.2.(iv) that for all locally finitely presentable categories products distribute over filtered colimits. They attribute this to Grothendieck and Verdier.
The colimit in the left-hand side of 1.2 (iv) is indexed by the product of the filtered categories. I think that’s what they mean by “distribute” rather than “commute”.
You could try to work out the $(\infty, 1)$-categorical version of locally sifted-presentable categories.
These are called projectively generated ∞-categories in HTT.
Thanks, projectively generated ∞-categories seem to be the right notion.
In fact, Lurie in section 5.5.8 gives a reference to Rosický’s paper “On homotopy varieties”, which develops the same theory for the case of model categories.
So in fact the theory that Zhen Lin alluded to seems to exist already.
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