Is there something like a genuine equivariant Kan extension? For example, in spectral Mackey funtors of Guillou and May, the Day convolution produces a naive smash product, which encodes no norms. Do we have a categorical way of building the genuine smash product, or even the "many" smash products corresponding to equivariant linear isometry operads?
@MingcongZeng This is being worked out in our parametrized homotopy theory project. For the Kan extension, look at math.mit.edu/~jshah/thesis.pdf . For the Day convolution, it is forthcoming but not yet put on the arxiv
Will the genuine Day convolution produce all smash products from equivariant linear isometry operads like arxiv.org/abs/1511.07363, or can it do even more like produce smash products out of any N_\infty operads?
@DenisNardin I don't quite get the part that if you can produce smash product for the complete multiplicative universe you can do it for any universe. But is this now a theorem that given any N_infty operad you can produce a smash product that the category of its commutative ring is Quillen equivalent to modules over the operad?
So one requirement for producing a right-induced model structure using an adjunction $F:C\rightleftarrows D:G$ where $C$ is a model category and $F$ is left adjoint to $G$ seems to be that $G$ preserves sequential colimits, but this seems quite strong...
user351585
8:01 PM
Or more generally, filtered colimits, at least if you're using the right-induced model structure from Schwede-Shipley on the category of algebras over a monad, which assumes that the monad preserves filtered colimits (and so you can use the standard result that the forgetful functor from algebras to the underlying category also preserves filtered colimits). But I think preserving filtered colimits is not unreasonably strong: given a ring map R -> S, the forgetful functor from Mod(S) to Mod(R)...
user351585
...certainly preserves filtered colimits, and I suspect the forgetful functors from Rings to Ab and from CommRings to Ab do as well--although these one (involving rings) don't preserve coproducts at all!
Yeah fair enough. I guess I want it to hold for the adjunction between Top and Haus (where by Top I mean the category of all topological spaces). This is for a undergraduate project a student of mine is working on.
But I don't even really know how to think about this, although I suspect it's not true.
I.e. I really really doubt the inclusion functor from Haus to Top preserves filtered colimits.
user351585
9:00 PM
Oh I see, you want to know if there's a nice transfer model structure on Haus. I have no idea personally but I think there's real value in somebody thinking through, and writing up, which point-set properties can be assumed on topological spaces and still yield a reasonable Serre/Quillen-like model structure, and which for the point-set problems which lead to terrible problems in trying to construct a model structure, an explanation of where the problem lies.
user351585
(Although perhaps someone has already done this, and I don't know about it.)
user351585
This kind of thing is probably unfashionable but I think it is valuable and certainly also publishable if it's written well
@DenisNardin In arxiv.org/abs/1511.07363v2 Blumberg and Hill prove that for any equivariant linear isometry operad one can build a symmetric monoidal category of G-spectra such that the category of commutative rings is Quillen equivalent to the category of algebras over the operad
Ah no, we do a completely different thing. For one, we work with quasicategories, so we cannot do that kind of jury-rigging. In our setting commutative rings are always E_∞-rings
We have a notion of G-symmetric monoidal category where you can talk about algebras for all N_∞-operads (and more), and there is a G-Day convolution that will put a G-sm structure on functor categories
Well, I don't know what you mean by "complete smash product'. The smash product is the same as the usual one, but you have a notion of smashing a G-set of objects now
An example is that in Guillou-May's spectral Mackey functor and the smash product there, the Eilenberg-Mac Lane spectrum of a Green functor is a commutative ring on the nose, however if we use HHR definition of orthogonal G-spectra, then it requires a Tambara functor as input to make the Eilenberg-Mac Lane spectrum a commutative ring
Yeah so I am curious about whether we can realize all N_infy operads as smash products, and then thinking maybe a G-Day convolution might help in construction
But in the end I still have no idea at all what a G-Day convolution should look like
The problem is that the multiplicative norm N^G(-) is not, in general, given by smashing G-copies of your object. This works for cofibrant objects in the HHR model, but it is very subtle
That's why we ended up including the functor N^G(-) (and N^T(-) for T a finite H-set) as part of the data of a G-symmetric monoidal structure
