@JonathanBeardsley to have an "underlying \infty-category", all you need is a relative category (via the barwick--kan model structure), not even a model category -- let alone a simplicial model category. i think it's irrelevant to your question that you happen to be examining a relative category whose objects are simplicially enriched categories

or rather, "yes" -- the underlying \infty-category of the bergner model category structure on simplicially enriched categories is indeed Cat_\infty

@HarryGindi are you asserting that sCat_Bergner is compatibly enriched over sSet_Joyal? i'm not sure i believe that

Is a dg Z-module the same thing as a chain complex of abelian groups?

(It seems like that should be true, but I want to double-check)

@AaronMazel-Gee No, not even that.

It has a simplicial structure that is badly behaved, but it's closer to being enriched over sset_Joyal than over sset_Quilen, but it is really just rubbish

As you were saying the nice part of that category is the model category

@Arun yeah

@MikeMiller ok, thanks!

(as a more precise statement: the Leibniz rule collapses to saying d(r a) = r (da) for a dg R-module, recovering complexes of R-modules)

@JonathanBeardsley Serious work on my part will start next week, I hope to have it finished by the end of September maybe.

@Pieter Thanks for the info! In retrospect my original comment was kind of snooty. Thanks for working on it, I'm looking forward to it!