Higher Degree Data in a Cosimplicial Quasicategory and Delooping

Question

If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference.

My question is regarding accessing data in non-trivial cohomological degree of a cosimplicial space. Ultimately, for me, this comes down to questions about looping and delooping cosimplicial spaces and/or quasicategories.

1) Given a cosimplicial quasicategory $C^\bullet$, can I take the homotopy pullback of $\ast\to C^\bullet\leftarrow \ast$ (with some chosen base point in $C^\bullet$), to obtain a notion of $\Omega C^\bullet$?

2) If so, can I then compute what should correspond to data in cohomological degree 1, and hence what would be in $\pi_{-1}Tot(\Omega C^\bullet)$ (using e.g. a Bousfield Kan sort of spectral sequence $\pi^s\pi_t\Rightarrow\pi_{t-s})$, of the connected component containing the base point, by somehow "delooping" $\Omega C^\bullet$ and attempting to compute $\pi_0Tot(B\Omega C^\bullet)$? Of course, without stability, it doesn't really make sense to talk about $\pi_{-1}$, that's just where this "information would live" if it did.

My main issue is what the construction of "delooping" should be in the situation of a 1-fold looped cosimplicial quasicategory? Also, does the category of cosimplicial quasicategories admit an initial object?

The motivation for this question comes from homotopical descent theory.

Thanks, as usual!

Answer 1

So, as far as I can tell, this question has a lot of high-falootin' vocabulary in it, but is actually pretty basic. It just took me a while to think about the right way. I should mention that my understanding was greatly clarified by talking to Adeel in the Homotopy Theory Chat Room.

So, basically, the $(\infty,1)$-category of cosimplicial pointed $(\infty,1)$-categories is complete. Thus, that loop space construction I give exists. What's more, taking loops of the cosimplicial $(\infty,1)$-category is done levelwise, so we end up with a cosimplicial pointed $(\infty,0)$-category which is also an associative group (both in the bigger category, and levelwise). What's important to note here is that since it has a base point, this loop space construction now loses all information about the other connected components. So, we're now essentially looking at automorphisms of the base point, and that this construction yields a cosimplicial space.

Now, it follows immediately that the delooping of this loops cosimplicial space exists. What's less obvious is that now that we've passed to the realm of spaces, we just deloop to get a space (rather than all of the non-invertible morphisms that we may have had in our connected component of the base point). One really ultimately needs to be careful about what one means by "delooping" here. Anyway, all the degree stuff works out as I indicated in the question, it's just that we lose any of the non-groupoid sorts of information (as we must, in my mind, if we want to do any kind of "taking homotopy groups" anyway).