@SaalHardali i thought about this at some point but it's been a while. I thought I had it in my head that you would need to replace the "swap the copies of C_2" automorphism with something else in GL_2(Z/p) at odd primes in order to get the Adem relations, so I'm surprised you're getting something out of the 'swap' at all. anyway- I'll try to sort out what I had in mind. bug me in a few days if you don't hear from me

I seem to recall someone (maybe @HarryGindi ?) telling me that if $\mathcal{X}$ is an $\infty$-topos then so is the tangent category $T_{\mathcal{X}}$. Does anyone have a citation for this? Maybe this only holds if $\mathcal{X}$ is the $\infty$-topos of spaces?

Oh I guess the nLab has a big section on this...

@DylanWilson Cool! Thanks.

Does anyone have an opinion on whether families of subgroups (closed under sub conjugation) should be considered as "open" subsets or "closed subsets" in equivariant stable homotopy?

@JonathanBeardsley Marc Hoyois has a note about this (proving a conjecture of Joyal), but I don't know if what you said is the right formulation. I think there are some hypotheses you need here, and Marc found sufficient ones.

mathematik.ur.de/hoyois/papers/loci.pdf

I think the following example fails: Let StrLoc be the classifying ∞-topos for strict local connective E_∞-rings. This is the category of sheaves in the étale topology on the opposite category of of finitely presented S-algebras

Hmm. Actually, not sure anymore about what I'm about to say

Anyway, everything I know for sure about this is in Marc's paper that I linked

I actually don't understand the notation Pr^{L,R} in that paper though. I'm a bit mystified.

@SaalHardali unfortunately it depends on whether you thing about it as coherent sheaves on a stratified variety of constructinle sheaves on a stratified space. In the latter it should be open I think, since free orbits represent the open stratum in the constructinle sense.

@HarryGindi Hm, this is interesting. Is it possible this is more general than just asking about the tangent bundle? It seems like possibly that question is answered by Section 35.5 here:ncatlab.org/nlab/files/JoyalOnLogoi2008.pdf ?

@JonathanBeardsley The tangent bundle is indeed easier: it is a left exact localization of the topos $\operatorname{Fun}(\operatorname{Space}^{fin}_\ast,X)$

@DenisNardin how can you tell that it's left exact?

@JonathanBeardsley This is HA.6.1.1.10.(2)

(and sorry for getting the wrong topos at first in my message)

@DenisNardin oh great. Yeah ok. I remember seeing it was an accessible localization of that category, but couldn't find the exactness statement. Thanks for the citation.

Is there any notion, in either 1-topos theory or ∞-topos theory, of replacing sets/spaces with a different topos, and redoing things? Like, if you have a category C and look at X-valued presheaves on C for X a topos, is it still a topos? Or is it something more general?

I guess I'm trying to understand @DenisNardin's comment about the topos $Fun(Space_\ast^{fin},X)$ in particular. Is that still a topos if $X$ isn't spaces?

I think some of the axioms are really easy to verify right off the bat, by just doing things like computing limits and colimits pointwise.

@JonathanBeardsley Yeah, if X is a lex localization of a presheaf category P(C), Fun(I,X) is a lex localization of P(I^{op}×X)

Or you can check the Giraud's axioms as you suggest, they can all be checked pointwise

Oh okay. Great. Yeah I thought it might be something like that.

Thanks :)

I'm sure this is somewhere in HTT, I'm just too lazy to check right now :D

Yeah no worries. I was sort of perusing HTT looking for it.

Maybe I'll try harder later too.

I feel moderately sure I've seen something like the phrase "X-valued presheaves" in there before.

In fact I think more generally, if X and Y are ∞-topoi, so is X⊗Y (where ⊗ is the tensor product in Pr^L), with basically the argument I wrote above (if X is lex localizaition of P(C) and Y lex localization of P(D) I think you can write X⊗Y as lex localization of P(C)⊗P(D)=P(C×D))

@DenisNardin yeah looks like this 4.8.1.9 in HA

I think you mean 4.8.1.19

Whoops, yeah.

So in HA it says that C⊗D≃RFun(C^{op},D), where RFun is the functors which admit left adjoints.

How does this get us to "presheaves valued in an ∞-topos are an ∞-topos?"

The corresponding statement in HTT, which is implied by the above, is that X-valued sheaves on a topological space are an ∞-topos.

Lurie has this other statement in HTT that says something called P(C;X), for C a small ∞-category and X an ∞-topos, is the product of X and P(C) in RTop, and that comment from HA tells us that this is the tensor product of X and P(C) in Pr^L, and is therefore an ∞-topos.

But I can't find anywhere where he explicitly says that P(C;X) is Fun(C,X).

@JonathanBeardsley it's not so bad to prove. if you're looking for a reference, a nice one for a more general claim is section 4.3.2 here: arxiv.org/pdf/1802.10425.pdf

(see corollary 4.25)

@DylanWilson Thanks! Yeah I think I'm definitely okay with checking the Giraud axioms, I just wasn't seeing how it followed from the tensor product statement.

Oh wait a sec @DylanWilson this is still the statement about sheaves right?

I feel very okay with that statement.

Also that looks like it's the statement about C-valued sheaves on X, rather than X-valued presheaves on C (and I believe the statement about C-valued sheaves on X is actually in HTT).

maybe I'm confused about which statement you were asking about then... I thought you wanted to know that "sheaves on a topos valued in a topos is a topos", the reference says "sheaves on a topos valued in a topos is their tensor product" and the HA remark says "tensor product of toposes is the product in the category of toposes"

Well, Denis mentioned that $Fun(S^{fin}_\ast, X)$ is a topos if $X$ is.

I'm sure I'm just missing something, but I can't see how to fit that statement into the statements you cited.

That category is not, at least on the face of it, a category of sheaves nor the category of functors out of a topos.

right, so that would be the example Psh((S_*^{fin})^{op}) \otimes X, right?

Oh I see now. So $Psh(C)\otimes X\simeq RFun(Psh(C)^{op},X)\simeq LFun(Psh(C),X)\simeq Fun(C,X)$?

Sorry, I think that little sequence was what I was missing.

yeah! sorry i missed what was missed :)

(also I think all those above categories are equivalent to Shv(Psh(C);X) in HTT's terminology)

oh, except maybe some of those C's should be opped

Hm... Really? I opped the Psh(C), that equivalence is in HA, then I was thinking that limit preserving functors out of the opposite of the free cocompletion should be the same as colimit preserving functors out of the free cocompletion?

and then colimit functors out of Psh(C) are the same data as functors out of C?

like, when X is Spaces, the left hand side would be Fun(C^{op}, Spaces) and the right hand side would be Fun(C,Spaces)

so you could either change the last two C's to (C^{op})'s, or the first two C's to C^{op}'s

Oh I see. I think I'm just so used to thinking about presheaves on spaces that I forgot Psh(C)=Fun(C^{op},S)

another way to formulate this is to say that Psh(C) is dualizable and its dual is Psh(C^{op}) so tensoring with one is the same as cotensoring with the other