@DylanWilson any progress on the tate frobenius adem relations for odd p?

@JonathanBeardsley The actual reason why it's not an equivalence is because that is not a cylinder, since $\Delta^1$ is not an interval object for the Joyal model structure. The usual choice for that is $J$, the nerve of the "free-living isomorphism".

Are there alternative sources to infinity-topoi than HTT? HTT is formidable, but it takes time to read it.

Is there a high level "infinity categorical" explanation for why orthogonal spectra in G-spaces give genuine equivariant G-spectra rather than naive? Personally i'm not sure i even have a good high level understanding of why orthogonal spectra are such a good model for ordinary spectra in the first place.

Maybe something like a half derived approach could help me understand this. Something like defining an $\infty$-category of orthogonal spectra where the space level is already derived.

@SaalHardali I don't really have an answer, but some suggestions. Firstly, it may be easier to think about G-objects in symmetric spectra, where G is a finite group. (This also gives genuine spectra, see e.g. the paper of Hausmann.) Then one thinks about what stops you from doing the same thing for a general oo-category. And I think the point is that in constructing G-symmetric spectra, you pick a specific notion of weak equivalence which cannot be formulated in abstract terms.

@SaalHardali sorry, I still haven't gotten a chance to sort that out! there are a few different places signs might be appearing, and signs are not my strong suit...

@DylanWilson Mine neither :(. If you ever figure out anything interesting let me know :)

@TomBachmann hmmm interesting. I'm still hoping there's something less techincal to say here...

re 'orthogonal G-spectra': you would invent these if you tried to invert smashing by representation spheres. You've got a diagram Orth^{op}--->Cat sending every object to G-spaces and every embedding V-->W to looping by W-V; we want the limit of this thing (which computes the colimit of the adjoint diagram in Pr^L). To compute the limit, we can take the corresponding cartesian fibration and look at cartesian sections.

if we instead look at all sections, we get the notion of an orthogonal spectrum

we want the nice subcategory of those which correspond to 'omega spectra', and this is nice enough territory where that will be a localization

precisely a localization at the 'stable equivalences'

if, instead, you wanted to emphasize the intuition of 'equivariant semiadditivity' or 'having transfers', then you'd invent spectral mackey functors. The connection between the two comes from relating the existence of transfers to the self-duality of orbits, and observing that you can compute the dual of an orbit using atiyah duality by embedding the orbit into a representation sphere... but for that to work you need such spheres to be invertible, whence the first approach.

Does Orth here mean orthogonal G-representations?

yes, sorry, I was making up notation as I went :)

But isn't the question actually why orthogonal G-spectra looks like a limit over only trivial representations, but somehow gives the limit over all representations?

ack! I misread the questions. I didn't know you could model things that way... presumably they're doing something extra funny with the weak equivalences? lemme check out this hausmann paper....

oh, I see you said the same thing about weak equivalences

Well I have always been confused about this, so happy to hear any suggestions how to think about it ^^. I think similar games happen with things like genuine G-operads (use the usual category of operads on G-spaces, but then put some clever notion of weak equivalence) and, ultimately, with G-commutative rings.

But "someone filled with the weak equivalences" has never rung particularly illuminating with me.

okay, I think I'm starting to get a picture of what's happening with this 'G-objects in orthogonal/symmetric spectra'. In the case of G-objects in symmetric spectra, it looks like we're doing the following

step 1: just take the usual stabilization of G-spaces

step 2: observe that the universe/Atiyah duality gives you a guess for the self-duality of a finite G-set, using the embedding into a permutation representation

step 3: invert the map that compares tensoring with cotensoring by the dual (and put the shift on the other side so you don't have to have it already be invertible)

step 4: repeat this for every subgroup of G

does that sound right?

@SaalHardali In fact they give both, depending of which equivalences you are taking. Something similar happens with \Gamma-spaces in the equivariant context: you can model G-commutative monoids with product-preserving functors out of Fin_{G,*} if you're willing to choose some weird weak equivalences

The latter observation is due to Shimakawa, I believe

The statement for orthogonal spectra is due to Schwede, I believe

@DylanWilson Are you suggesting that genuine G-spectra is a localization of the stabilization of G-spaces?

@TomBachmann yeah I think that's right. this goes back to Hill-Hopkins: first stabilize, then invert the canonical comparison maps from \coprod_T ---> \prod_T. Then Denis took that further in his G-stability paper with indexed semiadditivity/stabilization more generally

that's probably a more useful description than my 'outline' anyway...

@Dylan This "take all sections" is a really nice perspective, even in a non-equivariant context, thanks! Really illuminates where some of the point-set definitions of a spectrum (which do not require it to be an Omega-spectrum) fit in the homotopy-invariant language.

@DylanWilson interesting!

@Dedalus here is a sort of "cliffsnotes" for HTT (i think it is a precursor): arxiv.org/abs/math/0306109

in some sense some of these weird equivariant structures are connected to the enrichment. if you enrich equivariant objects in G-spaces, rather than just spaces, you get a lot of unexpected consequences because non-G-equivariant isomorphisms between objects have an impact

@DylanWilson somehow I'm not sure if I believe this. You are claiming that the right adjoint of the functor Fun^\times(Fin_G, Sp) -> Fun^\times(Span_G, Sp) is fully faithful. On objects which have homotopy groups concentrated in degree zero, this is saying that the forgetful functor from Mackey functors to coefficient systems is fully faithful. Is that really true?

For example, a morphism of Mackey functors from A (burnside ring mackey functor) to any other mackey functor is the same as a section over G/G. The same is true for the cofficient system Z, which also underlies a mackey functor. But Z and A are not the same...

@DylanWilson Can you make it more precise? I'm not sure I understand what kind of localization makes the map ∐_T→∏_T invertible (in my paper it's more about forcing ∏_T to also be a coproduct, which is monadic but not a localization)

I guess you could invert all possible maps ∐_T X→∏_T X , but I'm not sure the end result would be what you want

@TomBachmann @DenisNardin you're both right- I was confused... let me see if I can sort out what is true...

[What I think is true is that given a functor F: Orb_G -> Cat, it is a property whether the "Wirtmüller morphisms" exist and are isomorphisms. Now given the functor G/H \mapsto Fun(Orb_H, Sp), I think the initial functor under it in which the wirthmüller morphisms are isomorphisms is genuine G-spectra. This is just reformulating denis's results, I guess.]

@TomBachmann Yes, I do things in a different order, but I believe this is essentially what I do in that paper

hmm... well now I just dunno how to square this with the existence of a weird model structure on G-objects in symmetric spectra that presents G-spectra

maybe I need to understand what Tyler is saying, and somehow an enrichment everywhere is hiding something

I think the point is that for any G-representation V you can represent S^{-V} as the S¹-desuspension of a finite G-space

So you can write every G-spectrum as colim_n Σ^{-n}Σ^∞E_n

No wait that first statement cannot be literally true

weird, I didn't know that. what does that look like for S^{-\sigma} when G=C_2?

yeah

But we are claiming the second statement is true, so something like that must work

The second statement just says that the localizing subcategory generated by the suspension spectra is everything, doesn't it? This just follows from self-duality of the generators.

@TomBachmann I feel that there must be some other argument, because that statement is true also for G compact Lie

I wonder if it's possible to see this as a stabilization of the weird model structure Denis referred to that builds G-commutative monoids?

that would separate out the stability concern from the transfers concern

The only argument I know for the compact lie case is by some induction on dimension. You know that the dual involves desuspending by the tangent representation, which kills the center or something like that.

wait! I'm confused about Tom's original confusion

which one? ^^

I think I did want to invert all the maps \coprod_TX --->\prod_TX, like Denis was saying. that wouldn't produce a full subcategory, so the objection wouldn't apply

It's not clear to me that this would produce anything like G-spectra though

DK-localizations tend not to preserve any limits or colimits

What does invert mean if not pass to the subcategory of local objects?

it means formally invert- it's the universal target of a functor that inverts those maps

Literally categorical localization: if W is a set of arrows you take the pushout of W←W×Δ¹→C

like if you take an ordinary category with weak equivalences and use it to present an infty category, this is what you're doing

Hmok

Sorry somehow almost all localizations I even think about are bousfield localizations.

@DylanWilson Why do you think that works?

trying to check now... I can get Sp^G as a retract that also essentially surjects, but I guess I need the morphism spaces not to blow up

okay, I think it works using the Nikolaus-Scholze way of computing mapping spaces in a localization

like you'll basically be 'G-additivizing' the mapping spaces, if I'm not making an error...

anyway- sorry for the mess, I've gotta head out! let me know if you two sort it out!

What's the nikolaus-scholze way? ^^

@EdoardoLanari oh thank you of course.

Well I just announced this chat room on the ALGTOP listserv (again). So we might get an influx of new participants (which would be great!). For anyone logging on who is having trouble talking in here (because you're lacking in "reputation") or is having trouble with making LaTeX work in here, feel free to email me.

Well, I'm here because of your announcement. Thanks!

@DanGrayson Welcome!

I apologize in advance to all the new people here for the furious torrent of elementary ∞-category questions I post in here (and which people like Rune and Denis are generous enough to answer).

i have two sheaf theory questions. i've been looking through the back two chapters of HTT, but i'm not so familiar with it all so perhaps someone here can easily answer. basically, i want to restrict to "nice" topological spaces, in two ways. for one, i want my ∞-topoi to have enough points (so that, if i understand correctly, taking stalks is conservative). what is the condition on a topological space making this so, is it "sobriety"?

and then also, sieves are somewhat more unfamiliar, and i'd like to restrict to just hypercovers, or even ordinary covers if i can get away with it. from §6.5.4, it looks like i can restrict to covers for topological spaces of finite homotopy dimension (or i guess even "locally" should work), and then the hypercover-type sheaf condition coincides with the sieve-type one (i.e. the ∞-topoi are hypercomplete). is this correct?

and furthermore, can i impose additional conditions so that the cover-type sheaf condition also coincides with these?

@DanGrayson welcome!

@AaronMazel-Gee as far as I can tell, the sheaf condition using sieves and using covers are just the same. If C_0 is the sieve generated by the cover U_1,...,U_n then the sheaf condition at the sieve C_0 is equivalent to the Cech nerve of the cover. The story with sieves is because they organize better into a category (unlike covers, they form a post) so while locally you can always translate, the global picture with sieves is nicer somehow. I hope I'm not mistaken here of course.

*to the Cech nerve of the cover being equivalent to the sections on the union

@S.carmeli interesting, i'm not seeing how to prove that. (and i'm surprised, because of the relevance of hypercovers.) is there some specific assertion that such-and-such subcategory of the sieve is final?

@AaronMazel-Gee And Shachar is right: sieves and covers are the same thing. The way to understand that is that sieves are just (-1)-truncated presheaves on slices of your site, and if {U_i→X} is a family of arrows the Čech complex (seen as a presheaf of spaces) is just the (-1)-truncation of ∐_i Hom_X(-,U_i)

Hypercovers are the truly mysterious objects - I swear I have no idea what they actually mean!

@AaronMazel-Gee No sober spaces are those that can be reconstructed from their topoi, this has nothing to do with hypercompletion

The hypercomplete topos of a topological space always has enough points though, if that's what you were thinking

I seem to recall someone telling me the following thing a long time ago, but I'm not sure it's true. Given a natural transformation of functors F→G in Fun(C,∞Cat) then their image under the Grothendieck construction, as a functor over C of Cartesian fibrations, preserves Cartesian morphisms. Is that true?

@JonathanBeardsley Yes. It's in fact an iff

Nat(F,G) is equivalent to the space of maps of categories over C^{op} that preserve cartesian morphisms

@AaronMazel-Gee hypercovers are different from sieves=covers. Actually, I wondered at some moment if there's a notion of a "recursieve" that recover hypercovers via some poset of full subcategories, and will give hypersheaves. Something like a sieve, and a sieve for each member of it, and a sieve for each member of each member e.t.c. but I didn't know why I want that so I stopped thinking. It might be fun to have such a thing.

thank you both for the responses. i see that there's a bijection between sieves and open covers (or maybe just "irredundant" open covers). but what is the relationship between cosimplicial limit versus the limit over the entire sieve -- it sounds like you're saying they are equal? this is the finality/initiality statement i was trying to ask before.

i think i'm missing something basic here.

@AaronMazel-Gee There's no bijection between sieves and open covers... but the descent conditions are the same. Let me try to elaborate

You probably know the definition of sieve as a subcategory of $C_{/c}$. I want to claim that there's a bijection between sieves and functors $C_{/c}\to {ø,*}$ which sends an element $d→c$ to $*$ iff it is in the sieve

i see that, certainly

Note that this is just the Grothendieck construction on the presheaf!

ah, indeed!

I.e. Sieves are the same thing as cartesian fibrations whose fibers are (-1)-truncated

The reason I'm saying this is that the descent condition is clearer in terms of the presheaf: if $F:C^{op}→Space$ is a presheaf and $S:C_{/c}^{op}→Space_{\le -1}$, then the limit of F over the Grothendieck construction is just $\operatorname{Nat}(S,F|_{C_{/c}})$

do you mean cartesian? i think above it should've been $(C_{/c})^{op} \to {\varnothing,\ast}$

@AaronMazel-Gee Yes indeed, sorry

Why am I saying this? Well, because I gave you one way of presenting the presheaf $S$, but it is by no means the only one

Suppose I have $d={d_i→c}$ elements generating the sieve $\int S$. I claim that I can represent $S$ as the colimit of the Čech nerve of $d$

In particular this implies that $\operatorname{Nat}(S,F|_{C_{/c}})$ is just the limit representing the standard sheaf condition for the covering

sorry, just a sec. it sounds like you're making a general claim about natural transformations -- what is it?

ahh i think it's just what you said to john above!

I'm making a general claim about presheaves. Let $G:D^{op}\to \operatorname{Space}$ be a presheaf, then $G=\operatorname*{colim}_{d\in \int G} \operatorname{Hom}_D(-,d)$

This is essentially a variant of the Yoneda lemma (the lke of the Yoneda embedding along itself is the identity)

Thus I am claiming $\operatorname{Nat}(S,F|_{C_{/c}})=\lim_{[d→c]\in \int S} F(d)$

Are you with me so far?

given $F,G : X \to Cat$, we have $Nat(F,G)$ = maps of cartesian fibrations over $X^{op}$ = cartesian sections of the pullback = the limit of the composite $Gr^-(F) \to X^{op} \to ...$ i feel like this should land in $Cat$, not $Cat^{op}$?

yes, specifically your Nat is among presheaves on $C_{/c}$ is that right?

Yes, exactly

Good. Now I'm going to show that $\operatorname{Nat}(S,F|_{C_{/c}})$ can be computed in a different way

this is just global sections over S

okay, great

Or rather, I'm going to give you a different interpretation of what "generating the sieve" mean

Let's do some general nonsense again. If $G:D^{op}\to \operatorname{Space}$ is a presheaf, its (-1) truncation is the presheaf $G_{\le-1}$ sending $d$ to $*$ if $G(d)\neq \varnothing$ and to $\varnothing$ otherwise

sorry, real quick -- i can finish my thought above for cocartesian fibrations: given $F,G : X \to Cat$ i see that $Nat(F,G) = lim ( Gr(F) \to X \xrightarrow{G} Cat)$. is there a likewise description via cartesian fibrations?

err haha i guess we can have two conversations at once, the beauty of the written word

@AaronMazel-Gee Yes, I believe a similar formula holds

Although if I try to get the formula right now I'm going to screw this up

ohh it must be that $Nat(F,G) = lim(Gr^-(F)^{op} \to X \to Cat)$

ah, you beat me

totally

And I got it wrong, as predicted :). But you got it right, so all's good

okay, back to your thread -- i agree that $(-1)$-truncation is pointwise

Good. I'm going to give you a formula for the (-1)-truncation of G

Precisely I claim that $G_{\le -1}(-)$ is the colimit of the simplicial diagram $G(-)$, $G(-)×G(-)$ etc.

The "Čech nerve" of $G$

That's because you can check it pointwise, and if $G(d)\neq \varnothing$, you can do an extra degeneracy argument

Ok, but what does (-1)-truncation have to do with generating sieves? Well, I'm glad you asked

err, it's the cech nerve of $X \to \tau_{-1}X$

@AaronMazel-Gee Well, if you want, yeah :)

what does (-1)-truncation have to do with generating sieves?

So let ${d_i→c}$ be a bunch of objects of $C_{/c}$. Then the (presheaf associated to the) sieve generated by them is $G:C_{/c}^{op}\to {\ast,\varnothing}$ given by $G(d→c)=\ast$ if $d→c$ factors through $d_i$ for some $i$ and $\varnothing$ otherwise

Said it differently $G(d→c)=\ast$ if $\coprod_i \operatorname{Hom}_{C_{/c}}(d,d_i)\neq \varnothing$ and $\varnothing$ otherwise

ah yes, the sieve is the (-1)-truncation of the coproduct (as presheaves)

That is $G$ is the (-1)-truncation of the presheaf $\coprod_i \operatorname{Hom}_{C_{/c}}(-,d_i)$

Indeed.

That is I can represent $G$ as the colimit of a Čech nerve, which is a bit painful to write down

But the upshot is that the usual sheaf conditon for the family $\{d_i\to c\}$ is exactly the sheaf condition for the generated sieve

OHHH this IS the cech nerve

And this concludes our argument

fantastic, thank you. let me see if i can summarize...

(warning: properly understanding the above argument might cause one to decide to never use pretopologies again and always reason in terms of sieves because it's just so much more elegant... :))

given an open cover, we claim that the limit of a presheaf over the sieve coincides with the usual totalization over the cech nerve. earlier, i can consider the cech nerve as a simplicial object as a presheaf of sets (hence spaces), and its geometric realization is the sieve. so equating presheaves $F : C^{op} \to X$ with colimit-preserving functors $F : PShv(C) \to X$ (say X is presentable, i'm sure i'm being inefficient here) we get the desired equivalence.

Yeah sounds more or less right

brilliant. thanks, denis!

i think i meant to say limit-preserving functors $PShv(C)^{op} \to X$, or something. anyways, the key point is this geometric realization.

@DenisNardin ah okay yeah i guess i can see this, because the Cartesian lifts are precisely the functors that live in the image of F that assemble into a natural transformation

@JonathanBeardsley one way of saying this is that the Grothendieck construction defines an equivalence $Fun(C^{op},Cat) \xrightarrow{\sim} Cart_C$ to the category of cartesian fibrations and cartesian-morphism-preserving functors over C

@AaronMazel-Gee yeah totally, I just couldn't find the concrete definition of Cart_C anywhere

I was trying to reconstruct it from the definition of sSet^+_/C in HTT

Are the morphisms that preserve Cartesian lifts basically the fibrations in the model structure?

yeah, it's the underlying $\infty$-category of that simplicially enriched category

Right

for the record, i'm still curious to know when the $\infty$-topos of sheaves on a given topological space has enough points

Like, I could not find a reference that said "The morphisms in Cart_C=N(sSet^+)_C° are precisely the maps over C that preserve Cartesian lifts."

@JonathanBeardsley i think not necessarily. the point is that you mark those morphisms that "want" to be cartesian over $C$, and you always preserve marked morphisms.

and the fibrant objects are equivalent (as a category) to the category of cartesian fibrations over C. the inverse is given by marking the cartesian edges.

Okay I see. So the maps preserve markings. And everything, up to equivalence, is a marked thing in which the markings are precisely the Cartesian edges.

God. I'm 99% sure I've been through this before, except last time I was talking to Rune. I can't keep any of this stuff in my head.

PS @DenisNardin I'm finally coming back to trying to write down the stuff you explained to me in Germany over a year ago, about E-orientations and the tangent category

@AaronMazel-Gee In general no. The classical counterexample is HTT.6.5.4.8

In fact the ∞-topos of a topological space is hypercomplete iff it has enough points (since its hypercompletion has enough points, HTT pag 686 point (6))

Looking at the situation one naturally starts thinking "ok, but there must be some way to measure the difference between a topos and its hypercompletion, no?". To which I can only say "Godspeed"

Does the functor "H" from $\mathbb{F}_p$-algebras to $H\mathbb{F}_p$-algebra spectra have a left adjoint?

@kiran If you take connective $H\mathbb{F}_P$-algebra spectra, then $\pi_0$ is your adjoint. Otherwise I don't think an adjoint exists (as your functor does not preserve limits in the non-connective world)

@DustinClausen I see now. Thanks! (I'm good with first question too. )

Given a Cartesian fibration $E\to C$ corresponding to a functor $F:C^{op}\to Cat_\infty$ I'd like to say that objects of $E$ are exactly pairs $(c, e\in F(c))$ and a morphism $(c,e\in F(c))\to (d, f\in F(d))$ is a pair $(\phi, \alpha)$ for $\phi:c\to d$ in $C$ and $\alpha:e\to \phi^\ast(f)$ in $F(c)$. Can I just say this? Or is there some way to read this off from straightening/unstraightening in HTT?