Oh boy, I've got a doozy of a stupid question: There's a sort of slogan that I thought was true, namely that if G is a group, then given a G-object BG→C, the colimit (that is to say, homotopy orbits) is almost never a finite colimit. But if I'm not mistaken, this isn't the case for G=Z, since BG=S^1 is weak homotopy equivalent to a finite simplicial set

Is there a condition on a group G such that BG is actually a finite simplicial set?

I'm guessing that the thing I was thinking about was that for G a finite group, it's never true. It does seem like it would be true for free groups on finitely many generators by the same argument as S^1 (take a bouquet of circles)

Are those all of them?

@HarryGindi I think there are lots of BG's that are finite. tori of any dimension, higher genus surfaces, hyperbolic manifolds, to name a few

I'm trying to understand Dwyer & Kan's proof of the theorem that if $M$ is a group-like cofibrant simplicial monoid, then the map to its group completion $M \to M[M^{-1}]$ is a weak homotopy equivalence. (This is Proposition 9.5 of 'Simplicial localizations of categories'.) Their proof uses quasi-fibrations, with which I'm totally unfamiliar. Is there a more "modern" way to understand this proof?

@DylanWilson Ah, great, thanks!

@NarukiMasuda Prop 1.4.3.4 in HA says there is a t-structure where Sp(C)_{<0} consists of those X such that Ω^∞(X) is contractible.

Note sure if this is the same t-structure that Denis described. The image of C is clearly in Sp(C)_{≥0} though.

Oh actually it is the same, this is explicitly stated in the proof of said proposition.

@AlexanderCampbell The map whose geometric realisation Dwyer and Kan are considering is the one from the décalage $\mathrm{Dec}_0N(V)$ of $N(V)$ to $N(V)$. But unwinding the definition we see that $\mathrm{Dec}_0N(V) = \coprod N(V)_{/v}$. Hence $\mathrm{Dec}_0N(V) \to N(V)$ is a right fibration if $N(V)$ is a quasi-category, for which I seen no reason for it to be true.

But you can now try to answer the following question: For which simplicial sets $X$ are the fibres of $X_{/x} \to X$ homotopy fibres in the Kan-Quillen model structure?

(Sorry for not being able to provide a modern proof, but hopefully a modern question is better than nothing.)

@AdrianClough My interpretation of their construction is that $N(V,V)$ is the diagonal of the decalage of the bisimplicial nerve of $V$, rather than the decalage of the of the diagonal of the bisimplicial nerve of $V$.

One is reminded of the fact that a right fibration over a simplicial set whose homotopy category is a groupoid is in fact a Kan fibration, but I agree I don't see how to apply it in this case, alas.

It also reminds me of Quillen's Theorem B.

I'm thinking of stuff like categories of correspondences (and their universal properties), lax/op-lax co/limits, gray tensor product, 2-categorical unstraightning etc...

@SaalHardali I am not aware of such a survey. There was the idea to write a survey of $(\infty,2)$-categories at MSRI last semester, but it wasn't completed; the last I saw it didn't get far beyond the basics, and certainly didn't cover most of the topics you mention. In my humble opinion, the subject of $(\infty,2)$-categories has not yet been sufficiently developed for a survey to be written. I think your best bet for learning the subject is to read papers or to learn directly from the experts.

If you just want to get over your fear of 2-categories, you could start by consulting Steve Lack's 'A 2-categories companion' (arxiv.org/abs/math/0702535) and the references contained therein.

@AlexanderCampbell Thanks, just I got the feeling that a lot of stuff has been figured out recently so I thought maybe it was sufficiently developed. Thanks for the reference! :)

@AlexanderCampbell Woops, you are right, I wasn't being careful.

@AlexanderCampbell The statement that the homotopy category of $N(V)$ is equivalent to $\pi_0V$ seems plausible to me... and then perhaps $\mathrm{Dec}_0N(V)$ and $N(V,V)$ are homotopy equivalent over $N(V)$?

The first statement should follow from Theorem 5.7 of arxiv.org/abs/math/0607820v2.

I have a feeling this was asked here once before but I can't find it. Is there an non-hypercomplete $\infty$-topos whose hypercompletion is spaces?

@SaalHardali Parametrized spectra

@DenisNardin can you explain what is this topos and why it's an example? I didn't get it.

@S.carmeli One way of describing is the ∞-category of excisive functors from pointed finite spaces to spaces, which is a left exact localization of a presheaf category by standard Goodwillie calculus facts. Another, perhaps more enlightening, description is the ∞-category whose objects are pairs $(X,E)$ where X is a space and $E:X→Sp$ is a local system of spectra on $X$. Hypercompletion "contracts" the spectra to 0, so its hypercompletion is just spaces

In general this paper is a good treatment of this type of constructions