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Chetan Vuppulury (Apr 03 2021 at 04:09):Is there a classification/embedding theorem for abelian categories with closed monoidal structure? Something similar to the Freyd-Mitchell embedding theorem?
John Baez (Apr 03 2021 at 06:10):That's an interesting question! Since Freyd-Mitchell says any abelian category is sort of like a category of modules of a ring, I'd guess any monoidal abelian category is sort of like the category of modules of a Hopf ring, with the comultiplication in the Hopf ring providing the monoidal structure.
Mike Shulman (Apr 03 2021 at 13:34):The category of modules over a commutative ring is always a closed monoidal abelian category, right? Is it also the category of modules of a Hopf ring?
John Baez (Apr 03 2021 at 15:47):The modules of a Hopf ring should actually have duals, because a Hopf ring has an antipode. I say "should" because I guess I only know a weaker result: the finite-dimensional modules of a finite-dimensional Hopf algebra over a field have duals.
John Baez (Apr 03 2021 at 15:49):So, in my original reply I should have left out the antipode and worked with a "biring" rather than a Hopf ring... but "biring" also has another meaning, so I should have said "bialgebra over ", and then
Since Freyd-Mitchell says any abelian category is sort of like a category of modules of a ring, I'd guess any monoidal abelian category is sort of like the category of modules of a bialgebra over , with the comultiplication in this bialgebra providing the monoidal structure.
John Baez (Apr 03 2021 at 15:57):Nosing around for better information on this, I found a nice article:
It has a bunch of theorems, none of them analogous to a Freyd-Mitchell theorem unfortunately, but still nice. The basic one is Tannaka reconstruction:
Hopf algebras over a field can be characterized as those algebras whose category of finite dimensional modules is an autonomous monoidal category such that the forgetful functor to -vector spaces is a strict monoidal functor.
Here "autonomous" means that every module has a left dual, or maybe right dual, depending on whether we're talking about left or right modules (and I forget how the first left/right pair matches up with the second one here).
John Baez (Apr 03 2021 at 15:59):But one nice thing is that this paper strips off the trappings of linear algebra and gives some pure category theory results. Like this:
Theorem 4.1 Let T be a monad on a monoidal category C. Then there is a bijective correspondence between
(i) bimonad structures on the monad T
and
(ii) monoidal structures on the Eilenberg-Moore category of the monad T such that the forgetful functor to C is a strict monoidal functor.
John Baez (Apr 03 2021 at 16:08):Here's another nice one:
Theorem 4.7 - Let B be a monoid in a closed braided monoidal category A. Suppose that the unit of A is a regular generator of A and that − ⊗X and X⊗ − preserve colimits in A for all X∈ A. Then there is a bijective correspondence between
(i) right closed monoidal structures on the category BMod of B-modules such that the forgetful functor BMod A is strict monoidal and right closed, and
(ii) Hopf monoid structures on B.
This shows that we can eliminate the finite-dimensionality conditions I mentioned earlier if we settle for "closed monoidal" instead of "monoidal with duals" (= autonomous).