If I have a monoidal simplicial category, can I make it a strict monoidal simplicial category without changing its underlying quasicategory?

Assuming there is general nonsense that says a pseudomonoid in simplicial categories can be replaced by a strict monoid which is simplicially monoidally equivalent (in the standard sense of enriched category theory) then the nerve won't change I don't think...

@JonathanBeardsley Under which monoidal product for sCat?

If we weren't working simplicially but just 2-categorically, would the question be: "Every pseudomonoid in (2,1)-cat wrt the cartesian product is triequivalent to a strict monoid in 2-cat"?

If that's the correct rephrasing in 2-categories, it seems false to me, but I have no proof either way

Say I have a functor-up-to-coherent-homotopy (or an $A_\infty$-functor) between simplicial categories. What is the least painful way of turning this into a functor of $\infty$-categories? Is there a nerve construction for weak simplicial sets which produces quasi-categories?

@TomBachmann I think that Geoffroy Horel has some papers on this as well.

@HarryGindi does sCat not just have a Cartesian product which is a simplicial functor in both coordinates? I feel like this should just be some coherence theorem for "enriched pseudo-algebras" in something by an Australian

Or like, pseudo-algebras for the free monoid monad on the 2-cat of simplicial categories

In fact I feel like one might be able to just mimic the classical proof that strictifies an ordinary monoidal category. I'll have to check to make sure nothing goes wrong though.

@JonathanBeardsley iirc, the Mac Lane coherence theorem for monoidal categories is pretty hard and very combinatorial

I think that the way Lurie probably handles it is using A_n algebras, and the coherence theorem would be something like "every A_5 algebra in infty-cat is an A_infty algebra"?

seems to me like it is untrue though

Maybe if you said something like "if X is an n-truncated A_{n+3} algebra, it is naturally an A_infty algebra?"

algebra in Cat_infty

maybe not n-truncated, maybe you want n-coskeletal

At any rate, being monoidal in the ordinary enriched sense seems like a much much stronger condition than being an A_infty algebra.

And even being a pseudomonoid only means it is homotopy-associative rather than coherently associative, which is usually what is meant by a monoid in Cat_infty