# Compact objects in the $\infty$-category presented by a simplicial model category

Let $$\mathsf{M}$$ be a simplicial model category presenting an $$\infty$$-category $$\mathcal{M}$$. I'm interested in a general statement relating compact objects in $$\mathcal{M}$$ (in the $$\infty$$-categorical sense) with the compact objects in $$\mathsf{M}$$. Here's roughly what I expect to be true but if i'm missing some assumptions or if some are redundant feel free to phrase the correct statement as an answer.

Suppose further that $$\mathsf{M}$$ satisfies that weak equivalences between fibrant objects are stable under filtered colimits. Then is the following true

1. Let $$X$$ be a compact cofibrant object in $$\mathsf{M}$$. Is $$X$$ compact as an object in $$\mathcal{M}$$?

2. Is every compact object in $$\mathcal{M}$$ a retract of (the image in $$\mathcal{M}$$) of some compact object in $$\mathsf{M}$$?

• Here's a related question in the stable setting: mathoverflow.net/questions/289950/… – David White May 31 '19 at 12:27
• I suppose your boldface condition rules out $Top$ as an example, but it's worth recalling that in $Top$, the only compact objects are the finite discrete spaces. – Tim Campion Jun 3 '19 at 3:06
• @TimCampion Oh right! Seems like in $Top$ what makes the statement work is that finite CW complexes look compact when mapping into a filtered diagram of cofibrations. That might certainly complicate the general statement I was hoping for :( – Saal Hardali Jun 3 '19 at 10:47

If $$X$$ is such that $$X \times \Delta^n$$ is compact for every $$n$$ then yes. This happens, for example, if the cotensor functor $$(-)^{\Delta^n}$$ preserves filtered colimits, a condition which is quite common. Sufficient conditions of a similar nature are described in Proposition 5.3.1 of this paper. That said, you should expect the answer to your question to be negative in general (though I don't have a counter-example of the top of my head, at least not with $$X$$ cofibrant).
• Yes, though verifying the assumption there is not completely obvious. You need to check that any map $f: X \to Y$ of finite simplicial sets can be factored as a cofibration $f: X \to Z$ with $Z$ finite, followed by a categorical equivalence $Z \to Y$. One way to do this is to choose a finite simplicial set $I$ which is categorically equivalent to $\Delta^0$ and which contains a non-degenerate edge $\rho:\Delta^1 \hookrightarrow I$ (e.g., take $I=\Delta^0\coprod_{\Delta^{\{1,3\}}}\Delta^3\coprod_{\Delta^{\{0,2\}}} \Delta^0$) and set $Z = X \times I \coprod_{X \times \{\rho(1)\}} Y$. – Yonatan Harpaz Jun 18 '19 at 11:53