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 $\mathsf{M}$?
> 
> 
> 



Compact objects in the $\infty$-category presented by a simplicial model category
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](https://arxiv.org/abs/1507.01564). 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).


