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 $\mathsf{M}$?