@IanColey i'm trying. it's going poorly

in a parallel spirit to the "don't be afraid to promote your own work" from a little while back, i just wanted to say how enjoyable it has been reading @ClarkBarwick, Saul Glasman, and @PeterHaine's exodromy book. really, really inspiring stuff!!

@AaronMazel-Gee Thanks for reminding me of that! I've only read the first couple sections of the Raskin paper... I should probably revisit it at some point. In a similar vein, there's a pretty interesting appendix in Antieau-Mathew-Morrow-Nikolaus (arxiv.org/abs/2003.12541) which talks about some of this graded/filtered cyclotomic stuff as well.

Is there a nice discrete symmetric monoidal category that outputs Z x BU under Segal's infinite loop space machine? Of course, the topological category of complex vector spaces and isomorphisms will do, but I want to category to be discrete. There's also Mandel's "inverse K-theory" functor that describes such a discrete category, but I was hoping that maybe "nature" gives us a more easily describable discrete category that works.

Does anyone know whether or not the stable model structure on Γ-spaces that gives us connective spectra is a Bousfield localization of the pointwise model structure?

(pointwise = fibrations and w.e.'s are just determined pointwise, but cofibrations are determined by lifting)

I feel like there is evidence for this one way or the other coming from the fact that the cofibrations in the stable model structure are the same as the cofibrations in the pointwise model structure?

Yeah, right... that seems to suggest it should be exactly the left Bousfield localization at the stable equivalences.

So I guess my next question then is: if we consider the pointwise model structure on Γ-spaces, which I believe is simplicially enriched, and take its underlying ∞-category, and then stabilize in the ∞-categorical sense, do we get the underlying ∞-category of the stable model structure on Γ-spaces?

Yeah I think probably follows from the stuff in Marco Robalo's thesis where he relates ∞-categorical stabilization to model category theoretic stabilization.

@JonathanBeardsley exactly: a left bousfield localization is exactly when you keep the cofibrations and enlarge the weak equivalences. then, the identity functors define a quillen adjunction $M_{given} \rightleftarrows M_{loc}$. on underlying $\infty$-categories, this gives (what is sometimes called) a "left localization adjunction" $M[W_{given}^{-1}] \rightleftarrows M[W_{loc}^{-1}]$: an adjunction in which the right adjoint is fully faithful.

(by "stabilize in the $\infty$-categorical sense", i'm guessing you just mean "impose the very special segal condition" (or whatever it's called)?)

(and note that if a fully faithful functor admits an adjoint, then that adjoint is a localization)

@kiran objects: vector bundles over the standard simplices, morphisms fiberwise iso's (pullback squares)

More specifically, I want the morphisms to be in Delta in the base. If X is a stack of 1-groupoids given to us as a fibration of categories over Top, we can pull it back to Delta and then look at the total space of that fibration. This is a way of describing colim_n (X(\Delta^n)) which is one definition of the space associated with a stack, so for the stack of finite-dimensional vector spaces we get the space of finite-dimensional vector spaces, that is, the disjoint union of the BU(k)'s.

The symmetric monoidal structure is fiberwise direct sum.

@DanielBruegmann Ah cool thanks! It seems like I can use your method more generally right? As in, if I have a symmetric monoidal topological groupoid, it defines a stack of 1-groupoids on Top, I pull back to Delta and take the total space of that fibration of categories, and the symmetric monoidal structure comes along for the ride. Is that right?

@DanielBruegmann How does this work? If I have a vector bundle over Δ^n and another over Δ^m how do I sum them?

Hey @DenisNardin I never got an email back about the seminar on hermitian k-theory

is that still happening?

@HarryGindi You're in the mailing list but the semester doesn't start until november

ah ok

cool, for some reason I thought it started in october

alright great

You can check on Marc's website

=]

sounds good. excited =]

@HarryGindi It's been postponed for two weeks due to COVID

cool seeya then

@DenisNardin @kiran Ops, it seems I accidentally lost the (1-categorical) symmetric monoidal structure there. How about the following alternative: Let's model the symmetric monoidal structure using fibrations over Gamma. I think there's a fibration to Top times Gamma which describes the stack of vector bundles. We pull this back to Delta times Gamma. The projection down to Gamma is a fibration, so the map to Gamma is a fibration. It's fiber over 1_+

should be the category I described earlier (so this is only an answer if you loosely interpret the question to allow for symmetric monoidal categories to be presented by 1-categorical fibrations over Gamma). The fibration over Top times Gamma describes the stack of vector bundles together with its symmetric monoidal structure given by direct sum. Is there some way to edit recent comments?

@DanielBruegmann You cannot edit the comments. But I still do not understand your construction, can you elaborate?

@ReubenStern Thanks :) That's very kind of you!