A Nerve functor into any $\infty$-comos $\mathcal{N}: Cat \to \mathcal{K}$

Question

I believe there is a notion of a nerve functor into any $\infty$-cosmos $\mathcal{K}$. My inclination is that it would be defined as the colimit of the constant functor that sends all objects to the terminal object in $\mathcal{K}$, which exists by the definition of an $\infty$-cosmos . Is this the correct definition? I cannot seem to find anywhere in Riehl and Verrity's work where this is explicitly defined. Thanks!

Answer 1

Your formula would, interpreted literally, produce a coproduct of terminal objects indexed by the connected components of your category. There is an interpretation of your proposal as a lax colimit which works when these exist, but they are not assumed in a general cosmos. It's probably clearer to define $\mathcal N(J)$ as the simplicial tensor of the terminal object with the ordinary simplicial set $N(J)$.

But those tensors are also not generally assumed to exist, and they shouldn't be. For instance, in the $\infty$-cosmos of Kan complexes, to get a tensor $*\otimes \Delta^1$ we would need a Kan complex $I$ such that maps out of $I$ were naturally isomorphic to maps out of $\Delta^1$, which is impossible. Of course, any Kan fibrant replacement of $\Delta^1$ works up to weak equivalence, but that's not strict enough for the $\infty$-cosmos approach. Weak tensors like this exist in any cocomplete $(\infty,2)$-category $\mathcal K$, and in this way such a thing does always admit a "nerve" like your $\mathcal N$. This won't necessarily be a strict functor, if such a thing even makes sense in your setting for $(\infty,2)$-categories.