the source is going to have to be a symmetric monoidal Kan complex, right?

aka a commutative monoid

then the Day convolution is always closed monoidal if the target is, so yes

I think the internal hom is pointwise in the target

(in fact, the target more or less has to be closed symmetric monoidal for the Day convolution to exist)

I have a vague memory that the internal hom in Day convolution is a very strange colimit if written down explicitly, but I might be wrong.

@MingcongZeng @SaulGlasman so the source only need be $\E_n$-monoidal yes? but either way, if you want closed monoidal, it seems like you must be careful. that category, as a slice category, is already cartesian closed, so the usual closed monoidal equivalence would be giving a sort of strange equivalence of $Map(F\times G, Z)\simeq Map(F\otimes G,Z)$

If one presupposes that the internal hom just the right adjoint of the day convolution tensor product, it would seem to be something weird, as Mingcong says

Getting some interesting feedback on a similar question:

Is there a contsruction that takes the ??model category of model categories?? and produces the quasicategory of quasicategories?

Not even sure that thing exists.

Oy, I guess such a thing doesn't exist.

in chapter 4 of HTT, there's a rectification result for diagrams in a simplicial model category $M$ vs the quasicategory of functors into its underlying quasicategory $N^{hc}(M^{cf})$. is there an analogous result for (symmetric) monoidal functors into a (symmetric) monoidal model category, perhaps in HA?

@JonBeardsley Karol Szumilo essentially did that in his thesis, except you have to use cofibration categories rather than model categories.

and you get the quasicategory of (say) small quasicategories with finite limits rather than the quasicategory of quasicategories

*colimits

@RuneHaugseng oh neat! I hadn't seen this before.

The introduction is also really interesting. Kind of a brief history of homotopy theories of homotopy theory.

math.uni-bonn.de/people/szumilo/papers/cht.pdf

the arXiv version is newer, if I recall correctly

@RuneHaugseng suppose I have a monoidal category and take its never. where is the monoidal structure on the nerve coming from?

note i'm not inverting weak equivalences or anything. is the nerve a strict monoid in sSet with the Joyal model structure?

this has nothing to do with an ambient model structure, though

Ah sure. Right. It shouldn't. Just taking products to products.

yeah, although you'll need to be more careful if it's not a strict-monoidal category

If two functors left exact functors between abelian categories are isomorphic, do they have isomorphic right derived functors ?

Do you need some effaceability condition ?

everything is isomorphism invariant, no?

life is isomorphism invariant

thanks