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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 7th 2012

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeNov 7th 2012

Somebody wrote a lot of detailed material in the entry what is nice. It however leaves much work to make it coherent. The section “explanation” should maybe be called “The case of Lie groups”. The section after that is giving the geometric view point by looking at the sections of equivariant bundles. However, it is extremely important what kind of sections: in the context of Lie groups, the standard attached to word induced representation is to have not continuous or smooth but rather $L^2$-sections. Also I would like to put somewhere the interpretation via the tensor product (say for algebraic groups, one looks at representations at the level of enveloping algebra of a Lie algebra so the inducing functor from $U\mathrak h$ to $U\mathfrak{g}$ is simply the extension of scalars . In so massive entry it is difficult to organize where to put such a remark, unless one spends time thinking how toseriously reorganize.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 21st 2012

at induced representation I have made a Definition-section with two subsections. The first one, “traditional definition” contains the material that was there already, and a new one General abstract formulation in homotopy type theory adds an abstract point of view.

(I added a similar comment also to infinity-representation.)

I long meant to expand on this, but no time so far. This addition was promted by discussion over at the HoTT site.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeNov 22nd 2012
• (edited Nov 22nd 2012)

I like the addition of the version in HoTT !

On the other hand, I think it would be good just to omit the word “general” in title and leave “abstract formulation…”; I mean it is very general in one direction, and brings many new cases, but it is not likely of course including really all existing versions of induced representations (algebraic, functional, Hopf algebraic etc.), not all of which could fit into infinity topos setup. Namely, the things which can be delooped are rather rare; many things, say noncommutatve noncocommutative Hopf alebras, where also there is induced representation will likely not have classifying spaces even in generalized sense (unless of course, we can go to some sort of noncommutative higher category theory, but this one is expected to have somewhat weaker semantics than HoTT). Of course, it is quite general. “Abstract formulation” is suggestive enough and who knows what is HoTT will already also know which level of generality that is. The real problem is now that the section on “traditional point of view” has only the group case so far, and even only discrete groups. Unfortunately, I will be able to help there only much later.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeNov 22nd 2012

So this ’general abstract’ formulation is making further sense of Lawvere’s original linkage of representation theory to logic:

As a final example of a hyperdoctrine, we mention the one in which types are finite categories and terms arbitrary functors between them, while $P(A)=S^A$, where $S$ is the category of finite sets and mappings, with substitution as the special Godement multiplication. Quantification must then consist of generalized limits and colimits…By focusing on those $A$ having one object and all morphisms invertible, one sees that this hyperdoctrine includes the theory of permutation groups; in fact, such $A$ are groups and a “property” of $A$ is nothing but a representation of $A$ by permutations. Quantification yields “induced representations” and implication gives a kind of “intertwining representation”. Deductions are of course equivariant maps. (Adjointness in Foundations, p. 14)

Is there an analogue of this:

Frobenius reciprocity is saying that coinduction is isomorphic to induction (ie induction is both left and right adjoint to restriction) provided that $f:G \to H$ is an inclusion.

But that relied on a finiteness assumption.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeNov 22nd 2012

I have vague memory that in modular representation theory (that is finite characteristic case) they have much use from distinguishing restriction and corestriction functors, I am not sure about induction vs, coinduction. The groups there are quite usual, like GL(n).

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 22nd 2012

re #4

it would be good just to omit the word “general” in title

Okay, sure. I was emplyong the common phrase “general abstract” for category-theoretic formulations. Notice though that it captures everything that the entry originally and still is discussing at all: representations of gorups. But of course its not “fully general” in that one can consider representations of other things, too. I’ll not object if you remove the word “general”.

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeNov 22nd 2012

OK, let us remove general only once in the stage when we discuss things beyond groups. For now it is more general :)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeNov 22nd 2012

re #5:

Thanks, David! I hadn’t seen that. Have added the reference now and a pointed to it at induced representation.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeNov 22nd 2012
• (edited Nov 22nd 2012)

Frobenius reciprocity is saying that coinduction is isomorphic to induction (ie induction is both left and right adjoint to restriction) provided that f:G→H is an inclusion.

Let’s see, just to be sure: there are these two meanings of Frobenius reciprocity, the representation-theoretic and the category-theoretic one. For the former, the above sentence holds pretty much by definition, I gather. (?) Or are you saying that the sentence above starts with the latter meaning of Frobenius reciprocity? That I would need to further think about…

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeNov 22nd 2012

I always get confused about these things. Are the representation-theoretic and the category-theoretic meanings of Frobenius reciprocity not interlinked?

I gathered from Simon and Todd here that finiteness was needed for Frobenius to imply that induction and coinduction are isomorphic, so not holding by definition.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeNov 22nd 2012

Are the representation-theoretic and the category-theoretic meanings of Frobenius reciprocity not interlinked?

Given the Lawvere-type interpretation of the induced representation as the dependent sum in a locally cartesian closed category of representations they are interlinked. But I never saw somebody admit that this is the reason why the term is used that way, historically. And even if, what I usually see is that “Frobenius reciprocity” in representation theoretic terms just means that “the induction functor is adjoint to the restriction functor”, whereas the category-theoretic meaning would be a second step: given that it is adjoint, it in addition satisfies a certain relation that exhibits the restriction functor as a cartesian closed functor.

That’s why I am asking. I was wondering if you had a deeper insight here. I remember we had this kind of discussion before and I think it ended inconclusively.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeNov 22nd 2012

Hi David,

sorry, I realize that I might be talking nonsense: I didn’t actually follow your links (still haven’t) , being in a haste (still am). Maybe if you have (or somebody else reading this here has…) a second you could add a remark on that relation to Frobenius reciprocity.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeNov 22nd 2012

In Equality in Hyperdoctrines Lawvere’s interested in hyperdoctrines in which two kinds of identity hold, the possession of which allows for a satisfactory theory of equality. The first of these conditions is

…formally similar to, and reduces in particular to, the Frobenius reciprocity formula for permutation representations of groups,

these forming a hyperdoctrine. The second condition is Beck-Chevalley. So does that explain the origin? Frobenius reciprocity generally holds in the case of representations, but in the specific case of permutation representations it takes on a particular form, and it’s that form that Lawvere uses to designates a special property of some hyperdoctrines.

My other small wonder concerns later where he writes of a morphism, $f$, between groups

If $f$ is of finite index the analogous constructions for linear representations yield isomorphic results, which is perhaps why there seems to be no established name for “universal quantification” in representation theory.

So ’coinduction’ is the established name now, leading to the question of whether induction and coinduction can coincide ever in the realm of $\infty$-representations.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeNov 23rd 2012
• (edited Nov 23rd 2012)

Added a section Examples and applications - Hecke algebra.

Here is the Hecke algebra of an $H$-representation $E$ with respect to some $i : H \to G$, expressed in homotopy type theory:

$Hecke(E,i) = \underset{\mathbf{B}G}{\prod} \left[ \underset{\mathbf{B}i}{\sum}E, \underset{\mathbf{B}i}{\sum} E \right] \,.$

:-)

(David, I’ll get back to you re #14. I was about to, but then got distracted by Hecke…)

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeNov 24th 2012

Could we set up a form of higher hyperdoctrine for these higher representations, some functor from group objects in an ambient $(\infty,1)$-topos to the $(\infty,1)$-category of $\infty$-actions?

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeNov 24th 2012

Oh, am I missing the point that we’ve already got something better, i.e., a dependent type theory, with pullbacks of slices, etc.?

What can I say exactly about the relationship between a hyperdoctrine and a dependent type theory?

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeNov 24th 2012
• (edited Nov 24th 2012)

we’ve already got something better, i.e., a dependent type theory, with pullbacks of slices, etc.?

Yes!

What can I say exactly about the relationship between a hyperdoctrine and a dependent type theory?

The notion of hyperdoctrine is essentially the attempt to axiomatize how slices behave in a locally cartesian closed category, to be used in cases where we may not actaually have one. If we have a locally cartesian closed category already, its codomain fibration is a hyperdoctrine, but we don’t really need to say that, we can just stick with our locally cartesian closed category. And that’s (the semantics of a) dependent type theory.

• CommentRowNumber19.
• CommentAuthorDavid_Corfield
• CommentTimeMay 22nd 2016

At induced+representation#general_abstract_formulation where it has

The general case of $\infty$-groups in $\infty$-toposes is further discussed in sections 3.3.11-3.3.13 of [dcct]

I guess that’s now 5.1.14-5.1.16. Is the new section numbering likely to remain stable?

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeMay 23rd 2016

Thanks for catching this. Not sure if this will remain stable in the future. But so generally I shouldn’t be pointing to section numbers like this. I’ll remove the specific section numbers.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeMay 23rd 2016

I was there due to thinking about the (co)monads on $Act(H)$ induced from a map of $\infty$-groups, $f: H \to G$, by

$(\sum_f \dashv f^* \dashv \prod_f) \colon Act(H) \stackrel{\overset{\sum_f}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{\prod_f}{\to}}} Act(G) \,,$

and whether they could be seen as a form of necessity/possibility.

Is there a reason why the traditional treatment (here and elsewhere) just looks at $H$ a subgroup of $G$?

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeMay 23rd 2016
• (edited May 23rd 2016)

whether they could be seen as a form of necessity/possibilit

They certainly could, and I think it will be interesting to do so.

Is there a reason why the traditional treatment (here and elsewhere) just looks at $H$ a subgroup of $G$?

One thing to keep in mind is that in general the $\infty$-adjunction applied to a rep on an ordinary space may end up giving something whose underlying object is higher groupoidal. In traditional discussions one will have to find conditions that serve to avoid this.

For instance in the extreme case that $G = \ast$, and $H$ acting on the point, the above left adjoint produces the groupoid $\mathbf{B}H$, regarded as an $\infty$-representation of the trivial group.

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeMay 23rd 2016
• (edited May 23rd 2016)

In traditional discussions one will have to find conditions that serve to avoid this.

Life really is easier in an untruncated world. Who would want to go away from there?

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeMay 23rd 2016
• (edited May 23rd 2016)

Yes, one of the big themes in higher stuctures is that everything becomes simpler, more basic, more elementary, instead of more complicated. In a century from now it will be common wisdom. The kids will be taught homotopy theory and will look upon what is remembered of the contemporary state of the art with the same respect with which we regard roman numerals and epicycles.

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeMay 24th 2016
• (edited May 24th 2016)

At the level of 1-groups, for the dependent product what do we have?

For subgroups

• $\mathbf{B} H \to \mathbf{B} G$: essentially a fiber product of the thing $H$ acts on with coset space $G/H$.

What of epis?

• $\mathbf{B} G \to \ast$: fixed points of the $G$-action.

What of

• $\mathbf{B} G \to \mathbf{B} H$, so $H$ a quotient of $G$: something like what’s fixed in the $G$-action by the associated normal subgroup?

• CommentRowNumber26.
• CommentAuthorDavid_Corfield
• CommentTimeMay 24th 2016

So p. 150 of Basic Bundle Theory and K-Cohomology Invariants is useful here.

For $\rho: H \to G$, and $H$-action $Y$, the coinduced action (dependent product) is $Map_H(G, Y)$, $G$ regarded as an $H$-space by left multiplication via $\rho$. (Couple of misprints there, I think.)

• CommentRowNumber27.
• CommentAuthorDavid_Corfield
• CommentTimeMay 25th 2016

Hmm, I wonder if (linear) HoTT could lend a hand with the topic of Mackey functors. There’s plenty of talk of induction and coinduction, fixed-points and quotients, as in Mackey functors, induction from restriction functors and coinduction from transfer functors:

Boltje’s plus constructions extend two well-known constructions on Mackey functors, the fixed-point functor and the fixed-quotient functor. In this paper, we show that the plus constructions are induction and coinduction functors of general module theory.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeMay 25th 2016

Certainly. As the entry Mackey functor mentions a bit, these are of similar nature as motives in the guise of “sheaves with transfer”, which in turn is a of the general kind of “pull-push integral transforms through correspondences”. There are some comments on the general similarity of these structures with structures naturally appearing in dependent linear homotopy theory in the very last section of “Quantization via linear homotopy types”.

• CommentRowNumber29.
• CommentAuthorDavid_Corfield
• CommentTimeMay 25th 2016

Pull-push transforms involve just the dependent sum and base change (linear polynomials functors), right? It sounds like Mackey functor people also think about coinduction/dependent product, and even more explicitly in work on (non-linear) polynomial functors/Tambara functors.

Anyway, perhaps there’s another opportunity for HoTT enthusiasts to do some calculations.

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeJan 28th 2019
• (edited Jan 28th 2019)

added pointer to Brauer induction theorem

While doing so I noticed that this entry is in bad shape. It’s too verbose without getting to the simple point. If nobody objects, I’ll eventually go and remove much of the old chatty material and bring in the actual abstract characterization and the explicit formula for induced representations right away

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeJan 28th 2019

added a section at the beginning (here) that just states the definition right away. Kept the previous material, but re-titled the subsection as “More exposition”.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeJan 28th 2019

added a section at the beginning (here) that just states the definition right away. Kept the previous material, but re-titled the subsection as “More exposition”.