@CharlesRezk I sort of remembered there was a section on obstruction theory. I have just looked it up and I probably had in mind Section 4 of Chapter 8, though I don't know if this is what you were looking for.

yes

I dont belong here. Let me dip

Does anyone have a thought on what property a category C would need such that the trivial model structure on C is cofibrantly generated? By the trivial model structure I mean the one where the weak equivalences are the isomorphisms and the (co)fibrations are all maps.

Suppose I have a coend in topological spaces and I want to verify that it is in fact a homotopy coend. Is there a reasonable condition I could check? I guess one should check that the diagram is cofibrant for the projective model structure but it looks like extremely annoying to check it directly

Oh Denis, we had a question maybe you know the answer: The definition of genuine cyclotomic spectra involves a functor that isn't homotopy-invariant

at least for orthogonal spectra

like, it's written in terms of the categorical fixed points

is there a way to do it without looking at those?

@HarryGindi G-spectra are totally homotopy invariant. I even wrote notes on my webpage about that :)

And of course you cannot work with genuine G-spectra without working with genuine fixed points, but that's a feature not a bug

Wait, so there are three kinds of fixed points or two?

In the integral htpy seminar, we saw geometric fixed points and homotopy fixed points, but are there also what Teena called (at the IHES event today) 'categorical fixed points'?

Are those fixed points of the associated borel spectrum?

or are they what you're calling the 'genuine fixed points'?

@HarryGindi They are what I prefer to call "genuine fixed points"

They also sometimes go under the name of "Lewis-May fixed points"

If you want, when $E$ is a G-spectrum we can define $E^H=\operatorname{map}_{\operatorname{Sp}^G}(\Sigma^\infty_+G/H,E)$. This might be either a lemma or a definition, depending on how you set up $G$-spectra

If you recall, we talked about this in the integral homotopy theory seminar: genuine fixed points were the adjoint to the functor we called "triv" (for which I think a more classical name is "inflation")

@lemiller i'm sorry for the extremely delayed reply, my participation here tends to go in phases. you are right that this requires some translation. what is certain is that the obstructions lie in cohomology groups computed in the derived $\infty$-category of $(E_,E_ E)$-comodules. i believe the fact that this is just (derived) derivations is discussed in both of the final goerss--hopkins papers (§4 of "moduli spaces..." and §2.4 of "moduli problems...").

i'm not sure i understand your question about replacing derivations with another representable functor. is that still meant to refer to obstruction theory, or is it just a general question? (andre--quillen cohomology is what shows up in the obstruction theory; i'm not sure there's any way around that.)

@NarukiMasuda what lecture/material are you referring to (in re categorified fourier transform)? you might be interested in this paper of ben-zvi--francis--nadler: arxiv.org/abs/0805.0157

@ScottBalchin I haven't thought about it at all, but my naive guess would be that this is true for every locally presentable category. Do you know whether or not that's the case?

@ThomasRot personally, i like "universal G-bundle" instead of "universal G-space". it's more descriptive, and also EG is not particularly universal in the category of G-spaces (unless you start to talk about naive/genuine G-spaces, etc.)

@NarukiMasuda this is not directly what you're asking for, but your friend might be interested in work of david jordan and collaborators, which connects factorization homology (of $E_2$-algebras in $Cat$ over surfaces) with skein modules and related ideas.