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added an Idea-section to Mackey functor (which used to be just a list of references). Also added more references.
Clark Barwick’s characterization almost says that $G$-equivariant spectra form a dependent linear homotopy type theory over finite $G$-sets. What is missing is the Frobenius reciprocity condition. Is it automatic? Or else what would imposing it imply?
(Just a quick thought,I haven’t really tried to follow it through yet. Have to go offline now, too.)
The base category is his $A^{\eff}$?
If
twisted generalized cohomology theory is ∞-categorical semantics of linear homotopy type theory,
and if twisted generalized cohomology theory is the cohomology intrinsic to a tangent $(infty, 1)$-category, your claim suggests we should be finding such constructions in Barwick’s paper?
I see there’s excisive approximation going on in section 7.
I’ll get back to this a little later to day when I have something else out of the way.
Meanwhile I changed my mind, maybe it’s better to regard this as being not about the pre-quantum data, but about the quantum data. I suppose we can say something like: take the $\infty$-category of1d-cobordisms with codimension 1-defects parameterized by G-sets and codimension 0-defects parameterized by correspondences of such. Then a $G$-spectrum is a symmetric monoidal $\infty$-functor from that to $Spectra$.
Regarded as spectral “sheaves with transfer” this way would make us ask for the corresponding “motivic Galois group” that makes this a category of modules. And Schwede-Shipley tell us that and how this indeed exists.
But instead of brainstorming here, I should actually read the article first in more detail. After I got something else out of the way…
So the idea of a Mackey functor came from induced and restricted representations. The Mackey formula becomes one of the conditions (the Mackey axiom) in Green’s definition (here).
Regarding Urs’s query in #2, isn’t this the difference between a Green functor and a Mackey functor? A Green functor (here) is a Mackey functor with some extra structure, including Frobenius relations.
Barwick’s follow-up paper
deals with spectral Green functors, so may be what you want.
Thanks! That’s very interesting.
I came across
In the $G$-equivariant context for a finite group $G$, the role of abelian groups in non-equivariant algebra is played by Mackey functors. The category of Mackey functors is a closed symmetric monoidal category with symmetric monoidal product, the box product. In addition to the expected generalization of commutative rings to simply commutative monoids for the box product, there is a poset of generalizations of the notion of commutative rings to the $G$-equivariant context: the incomplete Tambara functors. These interpolate between Green functors, the ordinary commutative monoids for the box product, and Tambara functors. The distinguishing feature for [incomplete] Tambara functors is the presence of certain multiplicative transfer maps, called norm maps. For a Green functor, we have no norm maps; for a Tambara functor, we have norm maps for any pair of subgroups $H \subset K$ of $G$.
So I’ve added that to Tambara functor, and begun Green functor.
@Urs If by Frobenius reciprocity you mean that $f_!(A\otimes f^*(B)) \simeq f_!(A)\otimes B$ for $f$ a map between finite $G$-sets, this certainly holds for equivariant spectra (ultimately because it holds for $Fin^G_{/(-)}$). In fact the functor $SH^G: (Fin^G)^{op} \to CAlg(Cat_\infty)$ is a “linear type theory with Beck-Chevalley condition” in the sense of dependent linear type theory. You can also put all finite groups together $FinGpd^{op} \to CAlg(Cat_\infty)$, this has all the same properties except that $f_!$ only exists when $f$ is $0$-truncated (i.e., a covering map).
Thanks for the input. My question in #1 is from over three years back. Right now I am too occupied with other things to get back to this. If you care, you should record what is worth recording on some $n$Lab page.
Perhaps this would slot in models and examples.
I wonder if someone could think up a general (and generalisable) result to prove with linear HoTT to rival Blakers-Massey proved in ordinary HoTT.
Perhaps this would slot in models and examples.
Yes!
Ok, so I’d need to know what this is:
the functor $SH^G: (Fin^G)^{op} \to CAlg(Cat_\infty)$
The codomain is commutative algebra objects in $(\infty, 1)$-categories (or symmetric monoidal $(\infty, 1)$-categories)?
So what is $SH^G$?
This is for “Stable Homotopy theory of $G$-equivariant spectra”, i.e. symmetric monoidal $\infty$-category (under smash product) of genuine $G$-spectra.
Ok, have at least mentioned it here, but as yet no fulfilment of the promise
For each example we also spell out some of the abstract constructions
So this is an example of those $G$-$\infty$-categories, discussed in that program – Parametrized Higher Category Theory and Higher Algebra – I tried to start a conversation on.
The $G$-$\infty$-category $\underline{Sp}^G$ of $G$-spectra - whose value on an orbit $G/H$ is the $\infty$-category $Sp^H$ of genuine $H$-spectra… (p. 5 of the General Introduction)
So I wonder if their $T$-$\infty$-categories can provide more models for linear HoTT.
For any orbital ∞-category $T$, we have a corresponding ∞-category $Sp^T$ (even $T$-∞-category) of $T$-spectra, which under our algebro-geometric analogy corresponds roughly to an ∞-category of “quasicoherent sheaves on $T$.”
This orbital property is
one requires the analogue of the Mackey decomposition theorem in T (“T is orbital”)
So Marc’s generalization above in #9 is their example 8 of that introduction:
The 2-category $\Gamma$ of finite connected groupoids and covering maps is atomic orbital. The corresponding homotopy theory of $\Gamma$-spectra is a variant of Stefan Schwede’s global equivariant homotopy theory
Right, the value of the functor $FinGpd^{op} \to CAlg(Cat_\infty)$ on a finite groupoid $X$ are spectral Mackey functors on $FinGpd/X$, aka spectra parametrized by the orbital category $\Gamma/X$, this being the usual ∞-category of $G$-spectra when $X=BG$.
That’s not global homotopy theory though. Global homotopy theory is what you get when you allow arbitrary maps between groupoids, not just covering maps: a global space (for finite groups) is a finite-product-preserving presheaf on the (2,1)-category of finite groupoids.
Thanks!
That example 8 continues:
To get exactly Schwede’s global equivariant homotopy theory (for finite groups) in our framework requires a larger orbital $\infty$-category of finite connected groupoids equipped with an incompleteness class.
That latter construction seems to be about restricting transfer maps
to place limits on the classes of transfers that exist in the corresponding $\infty$-category of spectra.
I wonder what this does to the Beck-Chevalley condition.
Ah I see, I should have looked at the paper!
I’m not sure how to translate Schwede’s global spectra in terms of Mackey functors. I’m guessing what they have in mind is that global spectra are spectral presheaves on connected finite groupoids with transfers only along covering maps, as opposed to all maps. The BC condition would still be encoded by the corresponding category of spans.
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