@CharlesRezk Has it ever been shown that there does not exist a simplicial enrichment of the category of simplicially enriched categories compatible with the Bergner model structure? I'm imagining no-go theorem like the Bourke--Gurski theorem that the category of Gray-categories does not admit a monoidal biclosed structure compatible with Lack's model structure.
this question has been bugging me for a while, and i've either not understood explanations given to me by others, or i've not received a clear answer
is there a sheaf of E_oo-rings on M_cub = (A^2 / G_m) whose global sections are tmf?
problems immediately arise: for one, it's not a deligne-mumford stack. but it is an artin stack, so maybe instead of working with the etale site of M_cub one could work with the smooth site?
in the classical setting, the cuspidal curve y^2 = x^3 has additive degeneration. i don't know how to construct an associated derived formal group in this case
however, i believe the analogous question for the simpler case of bo/ko has a positive solution; namely, that bo is the global sections of a sheaf of E_oo-rings on the (deligne-mumford) moduli stack of possibly singular quadratic curves (i.e., curves of the form y = x^2 + bx + c)
@Pieter Do you know what is the scope of the project? Is it just going to be HTT+HA+SAG+miscellanea in stacks project form, or is it even more ambitious?
And thank you for working on this! Something like this has been needed for a while now
That's up to Jacob, and impossible to say at this point I think. Finishing SAG, and recombining everything up to that point into a coherent whole is already pretty ambitious.
I don't know what would come after that, that's too far in the future for me to have an idea of.
user351585
@skd A small point: you might have to invert 6 if you want every elliptic curve to be isomorphic to one of the form y^2 = x^3 + Ax + B (I assume the A and B in this family of equations are the coordinates in your A^2?)--otherwise you can look at the stack of Weierstass curves instead, which is still not DM, and has the same problem you already described
I want to say that a cocartesian fibration p: E -> B with weakly contractible fibres is both cofinal and coinitial (i.e. its op is cofinal). On the one hand, the inclusion E_b -> E_/b is cofinal (using the cocartesian maps) so if the fibres are weakly contractible then E_/b is weakly contractible which makes p coinitial by Theorem A.
On the other hand I can compute a colimit over E by first doing a left Kan extension along p and then a colimit over B. For a functor F : B -> C the left Kan extension p_! p^* F is given at b by the colimit over E_b of the constant functor with value F(b), so if E_b is weakly contractible then p_! p^* F is equivalent to F (and exists no matter what colimits you've got in C), hence the colim of p^*F is the same as that of F, which is equivalent to p being cofinal.
@RuneHaugseng I wrote in this chatroom an argument a month or so ago saying that a cocartesian fibration with weakly contractible fiber is a localization map (i.e. it identifies the target as the source with some arrows inverted). Localization maps are both cofinal and coinitial
(the argument was not totally unlike yours, basically pullback along p has to be fully faithful since p_!p^*=1 by direct computation and you can identify the essential image by looking at p^*p_!)
@aaaaaaaaaaaaaaa yeah, i should have said that M_cub[1/6] is A^2 \ G_m. as you said, that's M_cub is not isomorphic to A^2 \ G_m over the integers. but, as you said, it's still not apparent if there is a sheaf of E_oo-rings on M_cub[1/6] whose global sections is tmf[1/6]
@skd i think this falls into the category of "that would be great, but nobody has any idea how to prove it". in particular the simpler question of whether there's some kind of sheaf on the smooth site of M_ell is not known either
@skd My one cent, less than two cents: the way that the semistable reduction theorem (and the higher-dimensional version of Abramovich-Kaku) seem to get used again and again in building compactifications of moduli spaces is to choose integral models for curves so that the integral models have semistable reduction--for example, replacing additive reduction with multiplicative reduction, without changing the isomorphic type of the curve over the generic point
user351585
Both the toroidal and Satake-Bailly-Borel compactifications of Shimura varieties throw in points classifying varieties with various types of semistable reduction (or extensions of such varieties by algebraic toruses)--just saying, the arithmetic geometers really try to restrict attention to the case of semistable reduction. Algebraic topologists might be able to do the same: instead of working with things like the stack of Weierstrass curves...
user351585
...maybe some kind of semistable reduction theorem can be used to replace a moduli problem considered in spectral algebraic geometry with one where all the underlying classical (non-spectral) varieties have semistable reduction, so that you get DM stacks instead of Artin stacks, and then constructing sheaves of ring spectra on them is much much easier
Here is one thing I suspect is obvious, but I am not seeing: (Everything is commutative in what I am saying) Let R[G] be a group ring for some finite group G and suppose that S is a ring which is a R[G]-module as well. Suppose that there is an element a \in S such that Tr(a) = \sum_{g \in G} ga = 1. Let now x \in S and consider \sum_{g \in G} (ga)*(g^{-1}x) . I am reading the claim that this is equal to x. Presumably, this is because the sum is equal to \sum_{g \in G} (ga)*x, but I do not see why... What am I missing?
@aaaaaaaaaaaaaaa that's very interesting, thanks! i've not heard of the semistable reduction theorem before; do you know of a good survey?
user351585
4:50 PM
@skd For the theorem itself and a history of its variants and generalizations, the introduction to Abramovich-Kaku is good: arxiv.org/pdf/alg-geom/9707012.pdf
user351585
For its use in moduli theory, there is a good and humorous (really!) account in Kato and Usui's book "Classifying spaces of degenerating polarized Hodge structures"
user351585
I'm sorry, I just mistyped Kalle Karu's name twice--it's Karu, not Kaku :/
No problem, I agree with you that it'd be cool to see a spectral stack model for tmf and not just Tmf and TMF, that's an old and good problem
user351585
11:02 PM
@Dedalus Perhaps the argument is that \sum_{g\in G} (ga)*(g^{-1}x) = \sum_{g\in G} g*g^{-1}*ax = \sum_{g\in G} ax = (\sum_{g\in G} a)x = Tr(a)x? This assumes that the G-action on S satisfies a*(gx) = (ga)*x, though, which I don't see in your assumptions
