@AaronMazel-Gee Alas the only reference I know is Lewis-May-Steinberger. The proof is essentially the same as the proof of non-equivariant Atiyah duality though

I think the reference is actually Dold-Puppe (LMS even cites this in the intro to their section on duality)

but I'm a little confused. I feel like everyone always says you have to go genuine to get Atiyah duality but... isn't Borel completion $\mathsf{Sp}^G \to \mathsf{Sp}^{hG}$ symmetric monoidal, so shouldn't it preserve dualizability? (Note: the composite back into $\mathsf{Sp}^G$ is only lax symmetric monoidal, but I think the functor above is symmetric monoidal. Also, $\mathsf{Sp}^{hG}$ has the defect that dualizable objects need not be compact, so maybe that's the point of going genuine?)

@DylanWilson You don't need to go genuine to have Atiyah duality for Borel cohomology: as you say the argument works fine. The problem is that you would like to have it for, say, Bredon cohomology and that is impossible unless your coefficient system is upgraded to a Mackey functor (i.e. you go genuine as opposed to "naive")

Which is of course just a long-winded way of saying that Borel spectra are a symmetric monoidal Bousfield localization of genuine spectra, so Borel completion preserves all dualizability diagrams

ok phew. so, from the point of view of dualizability we have: (i) Borel G-spectra are good, (ii) ordinary stabilization of G-spaces bad, (iii) genuine G-spectra good?

Indeed

and (iv) literature = confusing

Also, to quote Mark Behrens, "Borel G-spectra are not naive, they're genuine!"

because they embed into genuine G-spectra

**mark

thanks :(

** :)

I vote we ban the use of the word naive. Especially because everyone (including the people who invented the terminology) uses it inconsistently when applied to G-spectra

I agree. Do you have a replacement adjective?

Borel and genuine seem okay. And the third option (which I think is "naive" as it was originally used?) is used infrequently enough that "presheaves of spectra on the orbit category" seems okay

maybe even Borel and no adjective? like just use "G-spectra" for the genuine thing, and say genuine once at the beginning of the paper to make it clear what you mean?

That's what I currently do, but outside of equivariant homotopy theory people seem to be confused by it

hm... well it takes time for terminology to stabilize. we'll see what happens in a few years

Idea I just got: Maybe we should call the "naive" G-spectra "Bredon G-spectra", because they represent Bredon cohomology

This has a nice parallelism with Borel G-spectra

If I have a G-manifold M, embedded in a representation space V, I have found in web.math.rochester.edu/people/faculty/doug/otherpapers/… (pg. 97) claim that the degree of the map S^V \to S^V we get from equivariant Pontrjagin-Thom coincides with the Euler characteristic of M. It claims that there is a classical proof of why this is so. What is the argument?

@Dedalus Are you asking about the non-equivariant statement?

Because it seems to me that this has nothing to do about equivariant homotopy theory

Regardless, the way I'd prove it is: use Atiyah duality to identify the map with the composite $\mathbb{S}→DΣ^∞_+M∧Σ^∞_+M→\mathbb{S}$, then use the fact that $C_\ast(-):\mathrm{Sp}→D(\mathbb{Z})$ is symmetric monoidal and it induces the degree isomorphsim $[\mathbb{S},\mathbb{S}]→[\mathbb{Z},\mathbb{Z}]$ to conclude

I am asking about the classical statement, yes. I was curious as to whether there is a version not using spectra, which was elementary.

I'm pretty sure that the classical argument is essentially equivalent but messier, and the fundamental insight is a version of Atiyah duality

You need to know what the PT map and the Thom diagonal do on homology

I was thinking if one could use the Euler class somehow, but this probably won’t work.

Well, I guess some argument of the kind might work. The Euler class gives you good control on the PT map, but I don't know how you'd go from there to the behaviour of Thom diagonal without essentially reproving Atiyah duality

web.ma.utexas.edu/users/a.debray/lecture_notes/… It came up as an exercise here, pg. 33. Atiyah duality has been stated, so maybe that is what one should use. Still, I thought there should be an ”easy” proof lying around.

@DenisNardin Sorry, I am trying to understand your argument. Why would the symmetric monoidality of C_* and the fact that it induces the degree isomorphism allow me to conclude? I suppose it has to do with the fact that C_*(D \Sigma M)) should be dual to C_*M in D(Ab)?

Well, I'm just using that the Euler characteristic of a perfect comples E is just the composition $\mathbb{Z}→E⊗DE→\mathbb{Z}$

And I suppose the dual is just C^*(M)

So the composition $\mathbb{S}→M⊗DM→\mathbb{S}$ is just the Euler characteristic of $C_*(M)$

Is there an easy way to see that the Euler characteristic coincides with the composition you said? This is a very nice argument!

Hmm.. let's see. I would try to show that it is additive on fiber sequences and that it is true for complexes concentrated in one degree

Then you can easily go by induction on the length of the complex

Is there a concrete description of the map C_*(S^0 \to C_*(M) \otimes C_*(DM)? Say, after using Thom isomorphism to identify DM and some suspension of the Thom bundle

Do you want an algebraic description (it's just the unit map of End(C_*(M))) or a geometric one (in which case it is the map S^V→M_+∧M^{\nu} obtained by composing the PT map S^V→M^{nu} with the Thom diagonal M^{\nu}→M_+∧M^{\nu})

I think you gave an answer in either case. Thanks!

This note is helpful to keep track of all this structure

Thank you. Atiyah duality seems to be a very powerful statement.

great, thanks @DenisNardin and @DylanWilson! i would be quite happy with the terminologies borel, bredon, genuine (or with "genuine" left implicit)

This might not be on topic, but: I saw upstream someone wrote, "It's defined... for $(\infty,\infty)$-categories, so you can simply reflect that into $(\infty,2)$-categories."

Presumably this works with 2 replaced with other natural numbers

Is this kind of thing (i.e. proving results for (∞, n)-categories for all n at once) a standard/common/useful application of (∞, ∞)-categories?

@ArunDebray to some extent, you could say (oo,oo)-category theory subsumes that of (oo,n)-cats for all n.

I've also heard "coarse $G$-spectra" to refer to $\mathrm{Sp}^{BG}$