@TylerLawson thanks, I was thinking as a module... what if you forget about 1? Then it's not so obvious, because at least xi_1 / tau_0 isn't killed by any of the generators

I think you answered that too, though

the second example works for that too. i suppose i could have been more concise rather than being silly

do you know if there's a nice expression for the kernel of "act on xi_1"?

motivation: trying to compute TAQ(F_p), so if there's a standard way to do that, that'd be great too

as a module? that I don't. i calculated out through Q^9 on it once and didn't really see a pattern

a student of mine informed me today that TAQ(F_p) was calculated by Lazarev based on work of basterra-mandell

oh thanks, I'll look into that

(cohomological version) it's free over their Steenrod operations R^n on a generator in degree 1 modulo a relation that it's annihilated by R^3

unfortunately these are the weird Steenrod operations where R^n x is only defined for n > |x|

lazarev also points out that the map from TAQ-cohomology to ordinary cohomology has image precisely consisting of the Bockstein, which is kind of amusing

@Nat thanks! I think Prop 4.2 does it (at least by naturality from the orientation).

@SaulGlasman @TylerLawson It is in Lazarev's paper in the structured ring spectra volume. I think it is also on the arxiv. I did not understand his argument and was surprised that the answer is so small.

unfortunately he deduces it from THH(F_p), and I was hoping to go the other way

does anyone know a conceptually satisfying description of the difference between a grothendieck context (e.g. six functors in algebraic geometry) and a wirthmuller context (e.g. six functors for local systems of spectra)?

uhhhhh something about the difference between what $p_!$ looks like?

@QiaochuYuan I always think about it as Grothendieck contexts being built on pushforwards whereas Wirthmuller contexts being built from pullbacks

I feel like @AaronRoyer was saying something about this a while ago

well there you go

That wasn't really grammatical

Replace "whereas" with "versus"

Haha. I wouldn't have noticed if you hadn't said anything.

Then those comments were for everybody except you, I suppose =P

i not read word so good

Well, I write word not so good, so

i'm also doing my usual thing of having many tabs and a tex file open, as well as having the tv on, so i may not be 100% on board

uhoh. craig's here...

things are about to get out of hand

@Aaron: okay, let me see if I know what you mean by that. in the context of, say, sheaves of sets, pushforward is somehow the "easy" operation to define, and everything else requires some work. in the context of, say, local systems, it's pullback that is somehow the "easy" operation to define, and everything else requires some work. is that what you mean?

that weirds me out a lot. i always thought i was supposed to think of sheaves as analogous to functions, but if pushforward is the easy operation on sheaves then they should be more like distributions

(but since constructible sheaves are built out of local systems it still seems reasonable to think of them as analogous to functions)

This has been my understanding, which may very well be wrong: In the motivic story there are the following operations: (f^*, f_*) for any morphism, (f_!, f^!) for separated and finite type morphisms, and smooth morphisms get an extra adjoint (f_#, f^*)

then there are natural transformations relating these. and when the dust settles

we get: f_! is f_* for proper guys

and, if f is smooth

f_* is a shift of f_# by like... \Omega_f or something (relative diff'ls)

and f^! is a shift of f^*

so I feel like "Grothendieck context" vs "Wirthmuller context" is secretly "proper context" vs. "smooth context"

because the Wirthmuller isomorphisms, even in equivariant homotopy theory, behave as you'd expect if someone had taken the time to build six operations for some "genuine" relative homotopy theory over a base that was allowed to be a stack (e.g. BG)

and if we said BG was a smooth stack

(which maybe we should if G is a compact Lie group.)

This also nicely explains the shifts present in equivariant Poincare duality

@Dylan Po Hu basically did the six operations for local systems of spectra on orbifolds.

right! I should've mentioned that

speaking of proper referencing, everything I just said I learned from section 2 of Marc's paper here: math.mit.edu/~hoyois/papers/lefschetz.pdf

Everything Dylan described also exists for local systems of spectra, so I don't think there's a major conceptual difference between the two contexts. The confusing thing is that some people use the shrieks with a different meaning than in the AG context.

@MarcHoyois Do we get six operations for local systems on, say, the etale site? Or do we need to go to constructible sheaves?

Yeah the pushforward is not going to preserve locally constant sheaves...

well that one isn't

but does that preclude some other pushforward from being defined?

does {local systems} ---> {constructible sheaves} have an adjoint? I would believe the answer "no" to there being no reasonable pushforward, but it's just weird that we get six operations for local systems of spectra

but not, say, local systems of abelian groups?

Going back to the shriek thing, the shriek adjunctions on the Wirthmuller context and the Grothendieck context nLab pages do not match the AG use. For instance $f_*$ always has a right adjoint (sometimes denoted by $f^\times$), but it is $f^!$ only if $f$ is proper. Similarly, $f^*$ may have a left adjoint in a variety of situation (sometimes denoted by $f_\sharp$), but it is $f_!$ only if $f$ is etale or something.

@DylanWilson Hmm well the inclusion of local systems ought to preserve finite colimits, so that doesn't preclude a right adjoint from existing, but as these categories are small I don't know how you would construct it.

fair.

But once you have six operations for local systems of spectra you also have them for local systems of chain complexes.

ah- of course! always derive...

Of course, as you know, one reason for local systems appearing is that homotopy invariant sheaves on nice topological spaces are locally constant. Not so in AG.

I thought something like this was true for the etale site though?

er... I think I'm confusing half-remembered facts

so ignore me.

and I gotta go- see you Marc!

Alright, see ya!

okay, i thought about it for a bit and i'm pretty convinced that sheaves behave at least in some respects like distributions, not like functions

(but that constructible sheaves behave like functions)

(and perverse sheaves i guess behave like L^2 functions in that they have this self-duality going on)

it's somehow easiest to see what's going on if you consider, say, the behavior of sheaves of objects in a category where you've made no assumptions about the existence of limits vs. colimits, vs. the behavior of, say, vector bundles

in the first case all you can do is take pushforwards and you can't even take stalks

in the second case all you can do is take pullbacks

I guess when I say "no assumptions about the existence of limits or colimits" I should also say "presheaves." meh

now, quasicoherent sheaves. still not sure which side these guys fall on

i think quasicoherent sheaves are functions again. man, that's confusing. maybe this would all be clear to me if i knew more topos theory

ah, maybe not. i don't know what quasicoherent sheaves are

no, okay, quasicoherent sheaves are functions. that's justified by tannaka reconstruction results

I'm inclined to agree that pullback is the fundamental operation for quasicoherent sheaves

but I'm not sure about the analogy/distinction you are making there

after all, if you use the espace étalé construction of sheaves on a topological space, then pullback becomes the fundamental operation

@QiaochuYuan Are you familiar with math.uchicago.edu/~may/PAPERS/FormalFinalMarch.pdf?

@QiaochuYuan For what it's worth, arxiv.org/abs/1501.01999 might also be interesting (you can't get much more recent than that)

math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf could someone possibly explain what is meant by theorem 11.2 e.g. I don't know immediately how to parse a map from an object to a category object

Am I just to read this as saying that there is a 2-category of stacks and a Yoneda equivalence of categories Sheaves(M) \iso Stacks(M,Sheaves) where Sheaves is the stack of sheaves?

that looks like the only reasonable interpretation of the statement to me

Anyone know about this? Kind of interesting: arxiv.org/pdf/1403.4130v2.pdf

If I want to construct a Thom spectrum from a map of topological spaces $X\to BGL_1(R)$ (not a map of $\infty$-groupoids), do I need the map to be a fibration?

I do not think so, at least in the case $R=\mathbb{S}$ it doesn't seem necessary

I think you need the universal map $EGL_1(R)\to BGL_1(R)$ to be a fibration, but that's essentially by definition as far as I understand

Maybe it depends on your construction of the Thom spectrum

Yeah. I feel like I never hear anyone say anything about this. However, it's not true I don't think that an arbitrary continuous map $X\to BG$ produces a presheaf of spaces on $BG$, is it?

It does, it's just not a great model

I mean, it does, but I feel like we need to replace the map by a fibration.

To be clear, I'm not working in the $\infty$-category of Kan complexes or anything here.

The presheaf is easier to describe if the map is a fibration

Hmmm.

I wish I could figure out what's necessary from May-Sigurdsson but there's so much notation I can't understand.

Which is frustrating, b/c I think there's a LOT in there. And I think I'm doing things that they've already done.

just a quick sanity check: in the category of abelian groups, the addition map is actually a map of abelian groups, right? Since it corresponds to the fold map (one can of course just check the axioms as well)

abelian groups are precisely the groups for which the multiplication map is a group map

yes, that is what I figured

since otherwise we won't have that the inverse goes to inverses

@Zhen: the espace etale construction is specific to sheaves of sets, isn't it? i have in mind (pre)sheaves of objects in an arbitrary category, and at that level of generality it's clear that pushforward is the fundamental operation

and yeah, maybe I should just read this may paper carefully...

I don't much believe in sheaves of things other than sets or n-groupoids, but if you restrict attention to presheaves then, yes, pushforward is fundamental

but in some sense pushforward is really pullback

because it's induced by a functor going in the other direction

well, yes, but that's how pushforward of distributions works too. functions pull back and disributions are dual to those

exactly analogously, you can think of an open subset of a space as a function to sierpinski space, and (pre)sheaves as "things that it makes sense to test against functions to sierpinski space"

of course i'm more than willing to agree that in the categorical setting there's richer phenomena than functions vs. distributions vs L^2 functions or whatever, I'm just trying to figure out what I should be looking at if I find myself with some six functors-looking formalism to predict whether it will behave like a grothendieck or a wirthmuller context

well, if you really want to, you can think of sheaves as being distributions

after all, a sheaf is the same thing as a geometric morphism to the object classifier

i don't know why that means i should think of sheaves as being distributions

the Sierpiński space is the classifying space for open sets

the ground field is the classifying object for linear functionals (but this is tautological)

Anyone familiar with May's construction of "iterated Thom spectra"?