I would like to know for what kind of model category finite homotopy limits commute with sequential homotopy colimits. Would cofibrantly generated and finitely locally presentable be enough? It seems to be true in simplicial sets.
I found a proof of something close in Stephan Schwede's paper "Spectra in model categories and applications to the algebraic cotangent complex", but I wasn't able to deduce the statement above from his.
Jacob Lurie has a proof of the analogous statement for colimits and limits of quasi-categories, in "Higher topos theory".
Anything related would be much appreaciated!
Best Answer
In combinatorial model categories finite limits commute with (sufficiently large) filtered homotopy colimits. Suppose, for simplicity, that the combinatorial model category is simplicial and generating cofibrations have $\lambda$-presentable domain and codomain. In this case $\lambda$-filtered colimits are homotopy colimits. Suppose, in addition, that the underlying locally presentable model category is $\lambda$-locally presentable. Then $\lambda$-filtered colimits commute with $\lambda$-small limits. Finite limits are $\lambda$-small for all $\lambda$.
Lets say that the category $J$ indexing the homotopy limit is finite if it has finitely many objects and morphisms, and the diagram $EJ$ of simplicial sets serving as a cofibrant replacement of the constant diagrams of points indexed by $J$ in the projective model structure on ${\cal S}^J$ has a finite simplicial set in each entry. For example, a finite group is not a finite category by this definition. A finite homotopy limit is a homotopy limit over a finite diagram.
Suppose that $\cal M$ is a simplicial $\lambda$-combinatorial model category and $F\colon J\to \cal M$ a finite diagram. Then $\mathrm{holim}_J F$ may be computed as a weighted limit with the weight $EJ$, in other words this is an end construction: $$ \mathrm{holim}_J F = \mathrm{hom}(EJ, F), $$ which is a finite weighted limit commuting with $\lambda$-filtered colimits, hence, commuting with $\lambda$-filtered homotopy colimits. In particular, if $\lambda=\aleph_0$, then filtered homotopy colimits commute with finite homotopy limits.