This question is motivated by my recent divertissements in the realm of nerves and realizations.
$\bf Set$-categories have the "classical nerve" sending $\bf C$ to the simplicial set $[n]\mapsto Fun([n],{\bf C})$; this could be the end of the story, but (un)fortunately categories can have much more structure: it would be stupid not to take into account the fact that $\bf C$ can be enriched over some other symmetric monoidal closed category $\cal W$, or can have additional structure we want to preserve (being a topos, being monoidal, being abelian, ...). Then we face various situations, letting $\bf A$ be from time to time a different kind of category which we want to send to a "coherently determined" simplicial set $N\bf A$:
- If ${\cal W}=\bf sSet$, we get Cordier's coherent nerve sending ${\bf A}\in {\cal W}\text{-}\bf Cat$ to $Fun({\frak C}[n],{\bf A})$, where ${\frak C}(-) \colon \Delta \to {\bf sSet}\text{-}\bf Cat$ sends [n] in a "standard" simplicial category; this is the "simplicial thickening".
- With suitable restrictions on $\bf Topoi$, we can get a "toposophic thickening" (click) of the nerve construction, which gives a simplicial set "naturally coherent" with the fact that $\bf A$ is not only a mere category, but a topos;
- If $\bf A$ is a dg-category as defined in Lurie's HA, ch I.1.3, then we can build th nerve $N_\text{dg}$, which albeit far from obviously arising from a cosimplicial object in the nerve-realization paradigm, seems to be "best-suited" to cope with the case $\bf A$ has an enrichment over chain complexes;
- The Duskin nerve provides a construction which is best-suited to cope with the case where $\bf A$ is monoidal: regard $\bf A$ as the one-object bicategory $B\bf A$, and then apply Duskin's machinery.
For the moment, forget about the "having more structure" case, which is unfortunately too difficult to axiomatize. (It's maybe possible to take a "doctrinal" point of view, but I don't want to get too far). My question is:
is there a "general theory" coping with the enriched case? Given a ${\cal W}$-category $\bf A$, what is the "canonical" way to define a nerve construction which "coherently takes into account" the $\cal W$-enrichment?
Partial answer: This seems to boil down to
- cocompleteness of ${\cal W}\text{-}\bf Cat$, to ensure that the equivalence $Fun(\Delta,{\cal W}\text{-}{\bf Cat})\cong {\rm Adj}(\widehat{\Delta},{\cal W}\text{-}{\bf Cat})$ holds true; is this cocompleteness automatic or it depends on some additional assumptions on $\cal W$?
- Presence of a "canonical" choice for a functor $\Delta\to {\cal W}\text{-}{\bf Cat}$, which generates by Kan extension a "realization", which has a right adjoint; again, I'm not sure I can find such a functor for any $\cal W$.
This right adjoint is precisely the "coherent nerve" I'm looking for.
