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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 $\mathsf{M}$?

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    $\begingroup$ Here's a related question in the stable setting: mathoverflow.net/questions/289950/… $\endgroup$ – David White May 31 '19 at 12:27
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    $\begingroup$ 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. $\endgroup$ – Tim Campion Jun 3 '19 at 3:06
  • $\begingroup$ @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 :( $\endgroup$ – Saal Hardali Jun 3 '19 at 10:47
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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).

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  • $\begingroup$ That's cool! Just to make sure I understood correctly the linked proposition. In particular the Joyal model structure on simplicial sets satisfies the conditions of 5.3.1. as well, is this correct? $\endgroup$ – Saal Hardali Jun 18 '19 at 8:05
  • $\begingroup$ 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 $Z = X \times I \coprod_{X \times \{\rho(1)\}} Y$. $\endgroup$ – Yonatan Harpaz Jun 18 '19 at 11:53

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