Question

In an ordinary category $C$, one says that an object $X$ is $\kappa$-compact if the representable functor $Hom(X,-)\colon C \to Set$ preserves $\kappa$-filtered colimits. We say $C$ is locally presentable if it is cocomplete and "generated" by $\kappa$-compact objects for some $\kappa$.

In an $(\infty,1)$-category $C$, one says that an object $X$ is $\kappa$-compact if the representable functor $Hom(X,-)\colon C \to \infty Gpd$ preserves $\kappa$-filtered $(\infty,1)$-colimits. We say $C$ is locally presentable if it is cocomplete and "generated" by $\kappa$-compact objects for some $\kappa$.

There are many equivalent, also analogous, definitions in both cases.

An $(\infty,1)$-category is locally presentable if and only if it admits a presentation by some locally presentable, cofibrantly generated model category. However, the only proof of this fact that I have seen (in A.3.7.6 in Higher Topos Theory) uses a different equivalent definition of both notions (as an accessible localization of some presheaf category). Thus my question:

Is there any relationship between an object $X$ being $\kappa$-compact in a locally presentable model category and being $\kappa$-compact in the $(\infty,1)$-category that it presents?

Answer 1

If $\mathcal{C}$ is a combinatorial model category, then for all sufficiently large regular cardinals $\kappa$, an object of the underlying $\infty$-category is $\kappa$-compact if and only if it can be represented by a $\kappa$-compact object of $\mathcal{C}$. The meaning of "sufficiently large" might depend on $\mathcal{C}$.

If $\kappa$ is not sufficiently large, then you generally don't have such an implication in either direction. For example, every finitely presented group is a compact object when viewed a (discrete) simplicial group, but need not be a compact object in the associated $\infty$-category (this requires that the classifying space of the group be finitely dominated). On the other hand, any space which is finitely dominated (i.e., a homotopy retract of a finite cell complex) is a compact object in the $\infty$-category of spaces, but cannot be represented by finite simplicial set unless its Wall finiteness obstruction vanishes.

(These counterexamples are not particularly compelling: you could get around the first one by restricting your attention to cofibrant objects, and the second one is very particular to the cardinal $\omega$. So perhaps there is something better to say.)