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

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?

• Some time ago I thought a bit about modifying the Bergner model structure to have more cofibrations. I was able to get some Reedy - ish categories to be cofibrant. It might be possible to work in a model structure like this to relax the cofibrancy assumptions on $C$, at the cost of imposing new fibrancy conditions on $D$. But actually -- even if $C$ is Bergner-cofibrant and $D$ is Bergner-fibrant, do we know that $sFun(C,D)$ comes out right? Usually I'd want my model structure to be monoidal to conclude things like that, which Bergner is not... – Tim Campion Sep 25 '20 at 21:27
• Also, are you aiming for the space of functors $C \to D$, or the $\infty$-category of functors $C \to D$? If I understand the notation $sFun(C,D)$ correctly, it will in general have non-invertible morphisms and be more closely related to the latter -- you'd have to throw away non-invertible morphisms to get something like the former. – Tim Campion Sep 25 '20 at 21:31

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 :