I think my question about strictifying a monoidal Kan complex to a strictly monoidal Kan complex can be answered in the affirmative by passing to spaces, strictifying, and then taking the singular simplicial complex. I believe that Sing(-) should preserve strict monoids in Top because it's a right adjoint.
Hrm... okay here's another one. Suppose I've got a simplicial model category $C$. Then I can form the "underlying $\infty$-category" by taking the simplicial nerve of $C^\circ$, the full subcategory of bifibrant objects (which is a fibrant simplicial category). Suppose I take more objects than just the bifibrant ones, but I pick them carefully so that they don't mess up the Kan enrichment. Will this have the same simplicial nerve?
I've only added in "equivalent" objects anyway, and the resulting thing will still be a quasicategory. But it seems possible that something could go wrong...
In particular, I want to take the full subcategory of $C$ on bifibrant objects and one other cofibrant object $x$ such that mapping into or out of $x$ from a bifibrant object (or itself) is always a Kan complex.
