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Stub Frobenius reciprocity.
Added Lawvere’s Frobenius condition for hyperdoctrines to Frobenius reciprocity.
Thanks. I added the topos-theory usage. And I’m pretty sure that it’s about cartesian closedness of the inverse image functor, not the existential quantification, so I fixed that.
It would be nice to have a comment on how the section “In representation theory” relates to that “In category theory” if it does, or else a warning that it does not. Hm, does it?
Mike wrote
I’m pretty sure that it’s about cartesian closedness of the inverse image functor, not the existential quantification, so I fixed that.
Argh, yes, you’re quite right. I don’t know how I managed to write that.
I don’t know any representation theory, but from scanning the Wikipedia page on induced representations it looks as though Frobenius reciprocity in that context means the adjunction $i_! \dashv i^*$ coming from an inclusion $i \colon H \hookrightarrow G$ of a subgroup, in the indexed category $G \mapsto k \text{-} Vect^G$. But I won’t add this until we hear from someone who knows what they’re talking about…
Some minor edits to Frobenius reciprocity.
What I meant by my previous comment (I don’t think it was clear) is that it seems that in representation theory the term ’Frobenius reciprocity’ refers to the existence of the induction-restriction adjunctions; but in category theory (especially with hyperdoctrines) it refers to a property of the analogous adjunctions. So I don’t know why Lawvere chose the term.
Slightly tangentially, I have been playing around with pullback/pushforward and ’transfer’ as discussed by Turaev in his HQFT book. If one has a group homomorphism $f: G\to H$ you can pullback crossed $H$-algebras to crossed $G$-algebras, and provided $Ker f$ is finite, you can push them forward. If $f$ is a cofinite inclusion you can also do an induction or transfer process that is left adjoint to pullback. (This is the analogue of the restriction/induction adjoint pair of representation theory, but because it is not simply representations in Vect, the usual Frobenius reciprocity situation does not hold and the pushforward and induction processes do not coincide. Has anyone seen anything like this before?
Is Lawvere’s the original usage of the term in category theory?
I wonder if this would be a good question for the categories email list. I would really like to know why that condition is called “Frobenius”.
I had assumed, though for no compelling reason, that the term was Lawvere’s. He introduced hyperdoctrines in Adjointness in foundations, I think, and in Equality in hyperdoctrines (here), not long afterwards, he uses the term ’Frobenius reciprocity’ without giving any source. Anything else I could say would be a guess, though. I too would like to know what the reasoning behind the name was.
Mike Shulman has kindly uploaded a file I sent him, consisting of handwritten string diagram calculations for the proof of the second proposition (bottom of the page) of Frobenius reciprocity.
I don’t know what the history is behind either use of ’Frobenius’ (either as in Frobenius algebra or Frobenius reciprocity). But I have removed some discussion that had appeared on that page:
The relationship between the two usages is not clear. In fact, since the category-theoretic usage is a special case of being a Hopf adjunction, it seems as though it might have been misnamed, since Frobenius algebras and Hopf algebras are similar, but different. The word “Frobenius” is also sometimes used in category theory to denote a condition which is in some way analogous to the characteristic property of a Frobenius algebra.
because the two propositions taken together indicate to me a reasonable relationship between the two usages. Whether such a relationship was in Lawvere’s mind when he wrote Adjointness in Foundations, I have no idea. I’d like to know what he was thinking.
Thanks! If Lawvere did know of this relationship at some level (which seems conceivable), it might finally actually explain the use of “Frobenius” for $f_!(C\times f^*B)\cong (f_!C)\times B$ as originating from Frobenius algebras (and having nothing to do with the representation-theoretic “Frobenius reciprocity”).
When I see $f_{!}(C\times f^*B)\simeq (f_{!}C)\times B$ the first thing that jumps into my mind is something along the lines of the Grothendieck-Riemann-Roch theorem. This type of thing keeps appearing for me while thinking about Fourier-Mukai transforms on different cohomology theories.
There’s a MathOverflow question about formulas of that type.
As far as Frobenius terminology, may I add to the confusion by bringing up the question of relation of the notions of quasi-Frobenius and pseudo-Frobenius rings (both notions are e.g. in Carl Faith’s 2-volume Algebra, mainly vol. 2).
Zoran, I don’t have that book and I’m nowhere near a university library. Can you tell us what those notions are exactly? Do they have much to do with Frobenius algebras?
Let me come back to the (old) question discussed above on what exactly the relation between the representation theoretic and the hyperdoctrinal use of “Frobenius reciprocity” is and how it is related to Fourier-Mukai transforms.
So it is true that most texts on representation theory say that Frobenius reciprocity means just the existence of the adjunction $(f_! \dashv f^\ast) = (Ind \dashv Res)$. An exception is the PlanetMath page which claims without further ado that this is equivalent to the projection formula
$Ind(Res(W) \otimes V) \simeq W \otimes Ind(V) \,.$Indeed what is true is that with an adjunction $(f_! \dashv f^\ast)$ between closed monoidal categories given, then this projection formula is equivalent to $f^\ast$ being a strong closed functor. This is one of the basic statements highlighted in May’s Isomorphisms between left and right adjoints, some of which is extracted a bit at Wirthmüller context.
And that then also clarifies the relation to Fourier-Mukai-type transforms mentioned in #12 above, via some discussion as in Abstract integral transforms
I’ve added some brief cross-links to these entries now, accordingly, and in particular added some comments to Frobenius reciprocity. But I don’t have time today to do this justice.
Just a small comment: the term itself was introduced by Lawvere in the “Equality in hyperdoctrines” paper, not yet in “Adjointness in foundations”. I’ve updated the reference accordingly.
I improved the proof of Proposition 1.1. In the original proof $\pi$ was not explained. A commutative square was drawn without clarifying that one of the sides was, at least in the absence of some other definition of $\pi$, being defined as the composite of the other three. Furthermore hom-tensor adjointness and $f_!$-$f^*$ adjointness were being deployed in a rapid and inexplicit way. I think it’s worth having a proof that’s really easy to follow.
checking whether I can point the reader to the section In cartesian categories for discussion of Frobenius reciprocity for base change between slices of finitely complete categories, I found I could not.
Have now added brief mentioning of the relevant pasting law argument (here) but I am not claiming that there would not be a more thorough edit needed to do justice to this section.
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