Working equivariantly for the action of a monoidal category

I recently got an email question from Sergey Arkhipov with a question, which I couldn’t answer to my own satisfaction, so I thought I would throw it open to the peanut gallery.

One construction I’ve used a lot in my recent work is the equivariant derived category for the action of a group G on a space X (in basically whatever category you like). This is basically the poor man’s way of understanding sheaves on the quotient stack of that space by the group.

But, of course, one could forget that there was ever a space there, and just remember that you have a category of sheaves on X, which the group G acts on. So, questions:

  1. Is there a construction of the equivariant derived category which makes no reference to the space and just uses the category of sheaves?
  2. If there a generalization of this construction where the action of G can be replaced by one of an arbitrary monoidal category?

The first question is in that class of things I’m sure I could do myself if I forced myself to sit down and do it: the answer is something like replacing the category with the category of locally constant sheaves on BG valued in your category. The second, I’m less sure about.

15 thoughts on “Working equivariantly for the action of a monoidal category

  1. Suppose that you’re in a Ben-Zvi-Francis-Nalder-ish situation: i.e., say that C is an ordinary category (say, of “spaces”) which has finite products, and that we have a contravariant functor (“taking category of sheaves, with pullbacks on morphisms”)

    Sh: C –> Stable presentable symmetric monoidal (infinity,1)-categories (k-linear if you want, or whatever)

    which sends finite limits to finite colimits.

    Then say that you have an object X in C, and a group object G in C which acts on X. The “usual way” to define the equivariant Sh_G(X) would be to take something like the limit over the cosimplicial set whose simplices are the Sh(G^n x X), and where the coface maps come from pullbacks along the action and multiplication maps. Under the Ben-Zvi-Francis-Nadler hypothesis above, this clearly can be done directly in terms of the tensor categories. This answers 1, I guess. And maybe 2, if you intepret “arbitrary monoidal category” as, say, a cogroup object in the target category of Sh.

    I should maybe mention that the “usual way” of defining Sh_G I talked about above is only really reasonable when you have enough descent: say when Sh is a sheaf for a topology on C which includes all the action and projection maps (for which it’s probably sufficient to just include the map from G to the terminal object).

    I should also mention that I’m saying this “on the fly” without having really thought about it, so GRAIN OF SALT, EVERYONE! :)

  2. Let C be the ordinary category of (nonequivariant) sheaves on X. Then as you say the group G acts on C. An equivariant sheaf on X is an object F of C together with isomorphisms X -> gX for each g in G, satisfying some conditions. So the category of equivariant sheaves is the homotopy limit / fixed point category C^hG = holim_G C.

    If C is instead some (infinity, 1)-categorical version of sheaves on X, the same construction should work, with the proviso that “satisfying some conditions” becomes “together with higher coherence data”. If C is only a *derived* category, then you may run into the standard “gluing in the derived category” problem–you’ve thrown away the kind of information you need to talk about that higher coherence data.

  3. Reid- like any sane person, I intend to keep the DG structure on sheaves, so don’t worry about that end of things. It’s only for historical reasons that we don’t say “the equivariant DG category.”

  4. There is a very clean answer to this question, as long as you’re working with a notion of sheaf that has a good product theorem,
    relating the category of sheaves on the product with the categorical tensor product of sheaves on the factors. (I think this is along the lines of anon’s comment above). Namely in this case the equivariant derived category of a G-space X can be recovered as the invariants of G acting on the derived category of X (where of course by derived category we mean some intelligent
    replacement thereof, as a dg, A_oo or other kind of oo-category, and we use Lurie’s DAG 2 for the theory of monoidal categories in such a universe).

    To elaborate, let’s write D(G) for sheaves on G. Then D(G) is a monoidal (oo-)category. Moreover the action of G on a point gives us a distinguished module category
    D(Vect) for D(G). The claim is that D(X/G)=Hom_{D(G)-mod}(D(Vect),D(X)), the obvious analog
    of invariant vectors in the module D(X), assuming the above product theorem. Such theorems can be found in my paper with Nadler and Francis for quasicoherent sheaves and my most recent with Nadler for all or coherent D-modules. But I don’t think this works for just holonomic D-modules, or constructible sheaves: eg you need to resolve the structure sheaf of the diagonal in the plane in terms of external products of constructible sheaves on the line..

    More concretely, the equivariant category D(X/G) is (essentially by definition) the limit of a natural
    diagram of derived categories on X, G x X, G x G x X, etc — i.e. the Borel construction of the quotient
    space. This is a version of descent – the quotient X/G is the colimit of the simplicial
    space with simplices as above, and sheaves on this colimit is the limit (or totalization)
    of the cosimplicial category of sheaves on the simplices. Now using the product theorem you can rewrite this diagram in terms of D(G) and D(X), and you find you’re just writing the limit that defined invariants or Hom as above.

    To say this for a more general monoidal category you need to
    know what you mean by “the trivial module” to take invariants — ie you need something like an augmented monoidal category. Otherwise there’s no problem.

  5. David-

    Perhaps I missed something here, but isn’t that how constructs the co-equivariant derived category of a coaction of a comonoidal category?

  6. The original question was about generalizing from group actions to actions by monoidal categories. I assume this covers group actions because the category of G-sets acts on the category of sheaves. (or G-modules on the category of sheaves of abelian groups, etc) What’s an example that’s not from a group?
    The suggestion of sheaves on BG with values in the original category suggests that monoidal categories are too general and you only want topoi. (But your last comment suggests a lot of generality.)

    G-modules don’t form a topos. Is there some linear notion of a topos, the kind o
    f structure you’d get from sheaves of abelian groups?

  7. I assume this covers group actions because the category of G-sets acts on the category of sheaves.

    The monoidal category I had in mind for a group is the one where the group elements are objects, the product is the tensor product, and all the morphisms are identities.

  8. To expand on 5 (I was writing from my phone, and thus a bit terse):

    What all the commenters above seem to have done is to turn the action of a group on a category into a coaction of sheaves on that group as a comonoidal category. Then they’ve explained how to work equivariantly with respect to such a coaction. Which is actually quite different from the question I asked. Now maybe the answer to my question really is that I should have included “co-“‘s in the title of this post. But of course then I would have to ask “when can I turn the action of a monoidal category into the coaction of a comonoidal one?” which seems to have gotten no attention.

  9. Ben asked in post 6: “Is there some linear notion of a topos, the kind of structure you’d get from sheaves of abelian groups?”

    Yeah, there is: the notion of a grothendieck abelian category. The definition is “internal”, and the analog of the “external” characterization of topoi comes from the Gabriel-Popescu theorem or whatever.

  10. Ben – My mind is very clouded by too many great TFT talks, but I think I was giving the equivariant derived category for a monoidal action as claimed.. first of all grant me that for a monoidal category C with an augmentation (ie with a C-module structure on Vect), we have the notion of C-invariants Hom_C(Vect, – ). Claim: for D(G) with pushforward to a point, this functor of C-invariants reconstructs the equivariant derived category.

    To see this we need to reinterpret the descent diagram as calculating Hom_C(Vect,-) — of course functors from Vect to any M is M.. now you need to rewrite the functors D(X)==> D(G) tensor D(X)
    that we’re coequalizing as describing D(G)-invariant homs..
    you’ll point out that should be given by two functors
    D(X)===> (dual of D(G)) tensor D(X), and I’ll argue back
    that we’re using the self-duality of D(G).. and maybe I’ll be wrong, but that’s the picture I had in mind.

    In fact I’ll make the stronger statement (with Francis and Nadler):
    if we take quasicoherent sheaves, then the functor M–> M^G
    from D(G)-categories to categories over BG is an equivalence of
    2-categories.. more generally M –> M^H for any subgroup H of G is an equivalence from D(G)-categories to D(H\G/H)-categories.. this is relevant to Sergey’s questions, though perhaps not to the issue at hand.

  11. David-

    If you’re thinking about D(G) then you’re using a comonoidal category, not a monoidal one (which G itself is). I mean, when your structure map is D(X) \to D(G) \otimes D(X), that’s a coaction.

  12. Ben – D(G) is a monoidal category, and D(X) is a module category via D(G) tensor D(X) –> D(X).. that’s the structure I was referring to.

    You’re right, “augmentation” is the wrong terminology, but the pushforward to the point makes D(pt) into a module category over D(G)..

  13. David-

    Let me try to summarize and clear my head.

    D(G) is monoidal in 2 different ways AND comonoidal. D(X) is both a module category for one of the monoidal structures, and a comodule for the comonoidal one. The equivariant derived category is a very natural homotopy limit in terms of the coaction, and you’re claiming that it also has a description in terms of the action due to the adjunction between the action and coaction. I can belive that, though it still seems like the upshot is that for monoidal action to have an equivariant derived category, it should secretly be comonoidal.

  14. I agree in principle, but of course in this categorified setting all reasonable maps have adjoints, so the distinction between monoidal and comonoidal seems moot. Also I would say the notion of invariants (or equivariants) is only more naturally comonoidal if you think of sheaves as categorified versions of functions (so they pull back) – but you could equally think of them as categorified versions of measures, which push forward — I think of D(G) as a group algebra of G, so it’s perfectly natural to think of defining invariants (or equivariants) of a G-action in these terms.. But in any case the key point for me is that it’s not clear you can define the equivariant derived category in the fashion we’re discussing, either monoidally or comonoidally, using only constructible sheaves, I think you need a better “function theory” like D-modules or O-modules.

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