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!