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at compact object in an (infinity,1)-category I have added the definition and stated the examples: the $\kappa$-compact objects in $(\infty,1)Cat$/$\infty Grpd$ are the essentially $\kappa$-small $(\infty,1)$-categories/groupoids.
How are $\kappa$-compact objects in a locally presentable $(\infty,1)$-category related to $\kappa$-compact objects in a combinatorial model category that presents it? Does one property imply the other? Are they equivalent? I’m willing to take $\kappa$ to be arbitrarily large.
I have put a pointer to your MO discussion with Jacob Lurie on this question into the entry compact object in an (infinity,1)-category .
By coincidence, right now I have started wanting to know more about the class of objects $X \in Sh_\infty(SmthMfd)$ such that $Hom(X,-)$ preserves the $\infty$-colimit
$\mathbf{B} U := \lim_{\to_n} \mathbf{B} U(n)$over the smooth moduli stacks of unitary bundles.
I realize that I haven’t thought about such issues enough.
Thanks. If I have a chance to work out the details in his rather sketchy answer, I’ll put them on the nLab page too.
I am of course interested in this for purposes of modeling univalent universes with object classifiers. If the corresponding statement for relatively $\kappa$-compact morphisms is true for $\kappa$ sufficiently large, then that means the universes we construct in model categories, which classify fiberwise $\kappa$-small fibrations, actually do present object classifiers of relatively $\kappa$-compact morphisms in the $(\infty,1)$-topos being presented. Which is what we expect, of course.
How is the $\infty$-colimit
$\mathbf{B} U := \lim_{\to_n} \mathbf{B} U(n)$different to the ordinary colimit?
I ask because I was considering a similar problem (compact objects) in the mere (2,1)-category of Lie groupoids and anafunctors, in particular with regards to a similar colimit of classifying spaces. I conjecture (based on some other work) that a proper Lie groupoid such that the orbit space $X_0/X_1$ is compact is sufficient to preserve the colimit. But perhaps I should be thinking of colimits in a (2,1)-sense, no just an ordinary colimit.
Every hocolim is computed as an ordinary colimit over a suitable diagram, so the question is how much one has to bend over backwards to massage this into a “suitable diagram”. And here the answer is indeed: not much at all.
The hocolim is computed as the colim over a projectively cofibrant diagram. A cotower diagram is projectively cofibrant if all morphisms are cofibrations and the first and hence all objects are cofibrant. So it’s sufficient here to use an injective model structure on simplicial presheaves and then everything goes through very naively.
Sorry if that reply was too terse. It’s past 2am here and I should call it quits. Let’s talk about it tomorrow :-)
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