@skd I don't have any answer to this question but it's interesting to me.

@GeoffroyHorel Since you're here, I might as well confirm the answer to Denis' question the other day: your model structure on simplicially-internal categories is not a simplicial model structure in any known way, is it? But probably it's also not known that no such simplicial enrichment exists, right?

@JonathanBeardsley please let me know if you make any progress on this question!

@TimCampion This model structure is simplicial. It is transferred from Rezk's model structure which is simplicial. The structure of a simplicial model category can be lifted along the right adjoint functor. The non-existence of a simplicial enrichment is for the category of simplicially enriched categories. Having a space of objects make things a lot easier.

@GeoffroyHorel Wow, thanks -- I'm glad I asked! Now I see that you transfer the projective model structure. Do you know if a similar construction works with the Reedy or injective model structures?

Ï don't know but I would bet against it. I think at the time I wrote this I had a counter-example (or at least a reasonable candidate for a counter-example) but I never actually wrote down the details anywhere. Now I have no idea what this conter-example was :)

Congrats @TylerLawson annals.math.princeton.edu/articles/12861

@CharlesRezk god this is so cool... so excited about it.

Anyone know about "deformation functors," and could maybe help me understand some of the basic ideas around them, w/r/t deformation theory