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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?

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up vote 15 down vote accepted

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.)

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Thanks! I'm happy to assume that $\kappa$ is sufficiently large. Where can I find a proof of your first statement? – Mike Shulman Apr 25 '12 at 21:06
Also, my next question is about an analogous statement for relatively $\kappa$-compact morphisms. I hope it would be possible to deduce that from the absolute version, but maybe you know a reference for the relative case too? – Mike Shulman Apr 25 '12 at 21:14
One direction is fairly straightforward: if $\mathcal{C}$ is combinatorial, then there exists a cardinal $\kappa$ such that weak equivalences in $\mathcal{C}$ are closed under $\kappa$-filtered colimits. Then $\kappa$-filtered colimits are also homotopy colimits, so the functor $F$ from $\mathcal{C}$ to its underlying $\infty$-category preserves $\kappa$-filtered colimits, and in particular is accessible. It then follows that $F$ preserves $\tau$-compact objects for $\tau$ sufficiently large. – Jacob Lurie Apr 26 '12 at 2:55
The reverse implication seems a little trickier. One way to prove it is to first argue that it depends only on the Quillen equivalence class of $\mathcal{C}$ (in other words, only on the underlying $\infty$-category of $\mathcal{C}$). This lets you assume that $\mathcal{C}$ is a Bousfield localization of simplicial presheaves on some small simplicial category. From here it's not hard to reduce to the case where $\mathcal{C}$ is the category of simplicial sets, in which case the result is true for any uncountable $\kappa$. – Jacob Lurie Apr 26 '12 at 3:02
I'm afraid I don't know references. – Jacob Lurie Apr 26 '12 at 3:03

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