What does the homotopy coherent nerve do to spaces of enriched functors?

Question

Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a simplicial category (though this enrichment does not cooperate with the Bergner model structure, for instance). I can apply the coherent nerve functor and produce two quasicategories $N(C)$ and $N(D)$ which have a Kan complex of morphisms between them $Map(N(C),N(D))$. I'm interested in knowing if there's any way to control $Map(N(C),N(D))$ (even just up to homotopy) if I know what $sFun(C,D)$ is.

I could control the homotopy type of $Map(N(C),N(D))$ if I knew, for instance, that $C$ was also cofibrant, but this seems to be an extremely strong condition to require. In general, is there just no useful way to get information about $Map(N(C),N(D))$ from $sFun(C,D)$?

I should also add that in the example I'm most interested in, $C$ is just a discrete category (thought of as a simplicial category), so maybe that gives me extra control over its cofibrant replacement?

Answer 1

Have you looked in the paper by Jean-Marc Cordier and myself:

Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1 - 54?

We defined a simplicial set of coherent natural transformations between two simplicial functors $F,G:C\to D$. That may be useful as an intermediate setting. There are 'rectification' results if $D$ is complete/cocomplete and especially in the case in which $C$ is an ordinary category, one has an augmentation from its simplicial resolution $S(C)$ to $C$ itself which is well understood. (This may be how you can get at its cofibrant replacement as $S(C)$ is explicitly combinatorially specifiable and is better understood now than when we wrote that paper 22 years ago!))

In general I would not expect the simplicial category $sFun(C,D)$ to be that good a model for the other one. Some of our earlier papers perhaps :

Vogt's Theorem on Categories of Homotopy Coherent Diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986) pp. 65 - 90.

might also contain useful ideas. The explicit combinatorial arguments we gave can be useful in specific cases although a model / infinity category approach may suit your aims as well. I will not attempt to give more recent references than our papers as there are other regular contributors here who know that approach better than I do.