April 6, 2010

New Paper on the Hecke Bicategory

Posted by Alexander Hoffnung

Hi everyone!

I have just finished a draft of an expository style paper called The Hecke Bicategory on groupoidified Hecke algebras and permutation representations. This paper is a companion to the latest papers in the Higher Dimensional Algebra series – HDA 7: Groupoidification and HDA 8: The Hecke Bicategory (in progress).

I would be happy to hear comments and corrections (no matter how small).

The paper is available on my website and the arXiv.

The goal of this paper is to recall the fundamental theorem of Hecke operators, which was first sketched in the seminar on geometric representation theory by John Baez and James Dolan in fall of 2007.

There has been a bunch of discussion on this theorem and closely related topics in the blog entries associated to each lecture. This paper represents my take on the fundamental theorem of Hecke operators, including a very simple categorification of the category of permutation representations of a finite group $G$.

Sections 2, 3, and 4, discuss the relationships between bicategories of spans and matrices. The focus is on the bicategory of spans of finite $G$-sets.

Sections 5 and 6 recall the definitions of Hecke algebras and degroupoidification, respectively.

Section 7 and 8 reinterpret the hom-categories of the bicategory of spans of $G$-sets as ‘nice topoi’, hinting at the implicit structure as a category enriched over the monoidal bicategory of nice topoi, cocontinuous functors, and natural transformations.

Section 9 introduces the definition of an enriched bicategory and the change of base theorem, which we use to prove the fundamental theorem of Hecke operators.

Section 10 illustrates some parts of the equivalence between the monoidal bicategory of nice topoi and the monoidal bicategory of spans of groupoids, which, along with the change of base theorem of the previous section, allows us to apply degroupoidification to the bicategory of spans of finite $G$-sets.

Section 11 gives the precise statement and a sketch of the proof of the fundamental theorem of Hecke operators.

Finally, in Section 12, we take a closer look at the categorified Hecke algebras arising from the construction in the previous section, and hint at some applications to categorified knot invariants. There is a lot more work to do in this direction!

Enjoy!

Posted at April 6, 2010 7:44 PM UTC

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Re: New Paper on the Hecke Bicategory

Very nice! A couple of trivialities:

p. 7 “…we are claiming that THEIR is a way…” typo.

p. 12 “One answer is that while the morphisms between categories are functors, the morphisms between topoi require specified extra structure and must satisfy extra properties. Such a morphism is called a geometric morphism”. That makes it sound as though geometric morphisms are the only choice when dealing with topoi, when in fact there are also logical functors.

Posted by: David Corfield on April 7, 2010 3:35 PM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Thanks David!

Your point about including logical functors is a good one. I will rewrite this paragraph.

I was trying to make a distinction between the morphisms between categories and those between topoi, saying that geometric morphisms are functors that require extra structure, say a left adjoint, and must satisfy extra properties.

I probably was not being careful enough. I think I should really only make a distinction for arbitrary topoi, but not for Grothendieck topoi, which I am focusing on (more or less).

The maps between Grothendieck topoi are functors with extra properties, but not really any extra structure. This is just because for Grothendieck topoi, we do not have to define a geometric morphism as an adjunction, but rather, just as a functor which preserves all small colimits and finite limits. Then we are guaranteed the right adjoint.

Posted by: Alex Hoffnung on April 7, 2010 9:58 PM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

saying that geometric morphisms are functors that require extra structure, say a left adjoint

I usually regard “having a right adjoint” as a property of a functor, rather than structure on it, since adjoints are unique up to unique specified isomorphism when they exist. For categories not satisfying the adjoint functor theorem, the property of “having a right adjoint” may not be expressible in more explicit terms as being equivalent to colimit-preservation, but that doesn’t seem to me to make it any more of a “structure” in that case.

Posted by: Mike Shulman on April 8, 2010 6:30 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Yes, it’s best to say ‘having a right adjoint’ is an extra property of a functor, not extra structure.

The reason is that if a functor has two right adjoints, these are naturally isomorphic — in a god-given way!

So, we are allowed to speak of ‘the’ right adjoint of a functor, when it exists. There aren’t truly different choices of right adjoint: it either exists or not. So, having a right adjoint is just a property, not extra structure. We speak of extra ‘structure’ when it’s possible to make different choices of that structure.

Whether I’m bald or not — that’s a property. If I decide to put on a hat, the hat is extra structure. But if all hats were isomorphic in a god-given way, we could say that wearing a hat is just a property.

Just a little message from the Department of Properties, Structure and Stuff.

Posted by: John Baez on April 8, 2010 5:40 PM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

That makes a lot of sense. Thanks guys. I was hoping one or both of you would chime in about this.

Posted by: Alex Hoffnung on April 8, 2010 5:43 PM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

I think the paper is very nice. Here are some of my thoughts and suggestions.

Section 3 definition 1: I think it would be clearer to say “…with a chosen basis such that the action of G maps basis vectors to basis vectors.” Since any representation of G can be given a basis as a vector space.

Section 4: I feel like it would be helpful, when defining a bicategory, to mention that morphisms can be composed. Likewise in the examples.

middle of p6: Have you defined what it means for a span to be “irreducible”?

Sections 5 and 12 are very different in character from the rest of the paper, and I found them next-to-impossible to understand without a lot of background at my fingertips that I don’t have. I’m sure you don’t want to spend the time to give people the necessary background, but you might want to at least warn people that the necessary background is different in different sections.

Section 6, first paragraph: I don’t think that functor was defined in section 5. Again on p9 there is an I think misplaced reference to section 5.

Section 6, Definition 3: I would put a comma after “where $g\cdot x = x'$”.

Bottom of p9: “arbitrary spans of groupoids form a bicategory (after taking isomorphism classes)” is not really clear to me; what are you taking isomorphism classes of? Perhaps it has something to do with the following paragraph? If so, maybe a rearrangement would help.

And my only comment with mathematical content: I don’t think that jointly faithful spans are closed under composition. Let $G$ be any groupoid and consider the spans $1 \leftarrow G \to G$ and $G\leftarrow G \to 1$. They are each jointly faithful, but their composite is $1\leftarrow G \to 1$, which is not. I think you need to perform some sort of “faithful reflection” in order to define this bicategory (also see below).

Middle of p10: I would leave out the comma after “whereas.”

Bottom of p10: I don’t think “note that” is quite appropriate here; it seems to me that really you are defining what a “categorified vector” is.

The notion of “essential inverse image” is confusing me again; I remember talking about this with John and maybe deciding that the terminology was off? At issue is the point that the morphisms between objects of $p^{-1}(x)$ don’t need to respect the isomorphisms $p(v) \cong x$ in $\mathcal{G}$, whereas in the “essential inverse image” I would expect them to have to. Maybe something like “full inverse image”?

Thinking about this some more, it occurs to me that maybe it would be clearer to phrase the whole theory in terms of connected components of a groupoid, rather than isomorphism classes of objects. Of course, the two are in canonical bijection, but for instance, the full inverse image of an object looks kind of weird to me (since the morphisms don’t respect the isomorphisms as above), but as the ordinary inverse image of a connected component it makes much more sense.

I felt a little confused at the end of Section 6, and reading back I think it was because the action groupoid $X\sslash G$ was introduced at the beginning of that section as motivation for groupoids, but it never came back in that section. Maybe some more signposting would help. I think I was also expecting some comments to the effect that the operation $S\mathcal{V}$ is a linear operator, and that this in fact defines a functor.

Bottom of p12: I would leave out the comma after “Any functor $f\colon G\to H$”.

Top of p13: Maybe the words “Kan extension” should be said?

Section 7 Lemma 9: This looks like a special case of a result that $G Set / Z \simeq \hat{Z\sslash G}$ for any $G$-set $Z$.

How committed are you to the term “nice topos”? I don’t find it especially evocative myself. I also wonder whether there is really any point to observing that these categories are topoi? The notion of geometric morphism is mentioned early on, but doesn’t really seem to enter the picture at all—the morphisms you use between these categories are just the cocontinuous functors. I think it might be clearer not to even mention that they are topoi, and just give them a more precise name (something like “presheaf categories of groupoids” but maybe shorter—“groupoidified vector space”?).

Top of p15: I like the point of view that a groupoid is a categorified “basis” and its presheaf category is the corresponding categorified “vector space.” Here is something else that might be helpful to say here: a groupoid can be recovered up to equivalence from its presheaf category, just as a basis can be recovered up to isomorphism from its vector space, but in each case the equivalence/isomorphism is non-canonical.

Section 10: I was a little confused here. It seems to me that in Section 8 we already defined a functor in the other direction $Span\to Nice$. One presumes that the functor being defined here is the inverse of that one, but maybe this connection should be made explicitly.

Also, it might be helpful to remind the reader here that the unadorned name “$Span$” is being used to denote the bicategory of groupoids and jointly faithful spans.

I guess the notions of topos theory do enter briefly with the idea of “points” of a topos. However, I think there’s a more direct way to recover a groupoid $G$, up to equivalence, from its presheaf category $\hat{G}$: it consists of the objects of $\hat{G}$ which are “small-projective” in that $hom_{\hat{G}}(X,-)$ preserves all colimits. And once you have that, then the “free cocompletion” yoga gives you the rest of the functor, as on p20 (although I think there is also a hidden use here of the fact that any groupoid is isomorphic to its opposite). Perhaps you have some other reason to want to use topos-theoretic language that isn’t apparent from this expository paper, though.

It might also be helpful to observe that a cocontinuous functor between presheaf categories is the same thing as a profunctor, so that your bicategory $Nice$ is equivalent to the bicategory of groupoids and profunctors between them. Then the category-of-elements construction becomes the usual interpretation of a profunctor as a two-sided discrete fibration, and the “faithful reflection” used to compose spans of groupoids is a special case of the composition of such two-sided fibrations. (Noting that every functor between groupoids is a Grothendieck fibration in the loose sense of Street.)

Finally, end of Section 11, Claim 17: I think it would be helpful to spell out a bit more explicitly what that equivalence means. I.e. it means that given two permutation representations $X$ and $Y$, the vector space of intertwining operators between them can be constructed as the degroupoidification of the groupoid whose presheaf category is the category of spans of G-sets from $X$ to $Y$ — which is just the groupoid $(X\times Y)\sslash G$. Am I parsing that correctly?

This is neat stuff!

Posted by: Mike Shulman on April 9, 2010 6:52 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Hi Mike -

I think the paper is very nice. Here are some of my thoughts and suggestions.

Thanks! And, thank you for the close read and all of the comments. Sorry for the slow reply. Simon was nice enough to make a bunch of comments as well, so I have been busy making the changes that he suggested. I have posted a new draft of the paper keeping the same link from the original posting. I have kept the old draft online.

Section 3 definition 1: I think it would be clearer to say “…with a chosen basis such that the action of G maps basis vectors to basis vectors.” Since any representation of G can be given a basis as a vector space.

Right! This was a mistake. I am glad you brought it to my attention. I have fixed this throughout the paper. Now $G$-sets and permutation representations of $G$ are in bijective correspondence.

Section 4: I feel like it would be helpful, when defining a bicategory, to mention that morphisms can be composed. Likewise in the examples.

I agree. I have added more to the definition of bicategory. I will probably add a bit more later. I will try to flesh out the examples a bit too.

Sections 5 and 12 are very different in character from the rest of the paper, and I found them next-to-impossible to understand without a lot of background at my fingertips that I don’t have. I’m sure you don’t want to spend the time to give people the necessary background, but you might want to at least warn people that the necessary background is different in different sections.

I am not sure how much of the background I want to explain yet, but nonetheless, this is a good point. I am imagining this as a paper aimed at representation theorists who have some amount of category theory on hand. Maybe I need to balance it a little more and keep in mind category theorists who have some amount of representation theory on hand.

Section 6, first paragraph: I don’t think that functor was defined in section 5. Again on p9 there is an I think misplaced reference to section 5.

I think this was a confusion between section references and citations to another paper, HDA7 in this case.

And my only comment with mathematical content: I don’t think that jointly faithful spans are closed under composition.

Wow! Oops! That was bad. I am trying to figure out how I want to fix this now. I will correct the problem in the next draft.

The notion of “essential inverse image” is confusing me again; I remember talking about this with John and maybe deciding that the terminology was off? At issue is the point that the morphisms between objects of $p^{-1}(x)$ don’t need to respect the isomorphisms $p(v) \cong x$ in $\mathcal{G}$, whereas in the “essential inverse image” I would expect them to have to. Maybe something like “full inverse image”?

I will get back to this issue in a bit as well.

I felt a little confused at the end of Section 6, and reading back I think it was because the action groupoid $X\sslash G$ was introduced at the beginning of that section as motivation for groupoids, but it never came back in that section.

I split this into two sections. I think this makes the ideas a bit more clear. I will see how it reads on the next go-around.

How committed are you to the term “nice topos”? … I think it might be clearer not to even mention that they are topoi,…

I am not committed at all to the name. I think the fact that it is a topos will be important in papers in progress, where we do use a good amount of topos theory. How about groupoid-sets, somewhat of an analogy with $G$-sets?

I have gone through and removed most appearances of the word topos, and I have redefined various notions avoiding the use of geometric morphisms. I will probably put a little of the topos theory language back in as I want to say something about why the map from $GrpdSet$ to $Span$ is an equivalence. This involves linearizing spans of groupoids using the left adjoints which show up in the induced essential geometric morphisms. Of course, I can avoid the term geometric morphism and jut describe these as pairs of adjoint functors with certain properties.

Here is something else that might be helpful to say here: a groupoid can be recovered up to equivalence from its presheaf category, just as a basis can be recovered up to isomorphism from its vector space, but in each case the equivalence/isomorphism is non-canonical.

Thanks. I will go through and make sure that the groupoid/basis – presheaf category/vector space analogy is clear.

I guess the notions of topos theory do enter briefly with the idea of “points” of a topos.

I have redefined points as left-exact cocontinuous functors from $Set$ to $\hat{G}$ for now. It seems to me at the moment that this requires the least overhead, but I am sure I will change my mind a number of times.

It might also be helpful to observe that a cocontinuous functor between presheaf categories is the same thing as a profunctor, so that your bicategory $Nice$ is equivalent to the bicategory of groupoids and profunctors between them.

Right, I will come back to this on the next time around.

On a related note, thanks for pointing out the nLab page for Grothendieck fibrations. I am sitting in on a seminar on stacks, and was really bothered by the strictness in the definition of a category fibered in groupoids. It is hard to get people to answer questions like “Why is this definition so strict?” I was happy to see the nLab page answers this in part.

Finally, end of Section 11, Claim 17: I think it would be helpful to spell out a bit more explicitly what that equivalence means. I.e. it means that given two permutation representations $X$ and $Y$, the vector space of intertwining operators between them can be constructed as the degroupoidification of the groupoid whose presheaf category is the category of spans of G-sets from $X$ to $Y$ — which is just the groupoid $(X\times Y)\sslash G$. Am I parsing that correctly?

Exactly! I have included this explanation.

More later.

Posted by: Alex Hoffnung on April 12, 2010 10:09 PM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

How about groupoid-sets, somewhat of an analogy with G-sets?

That’s a possibility, I guess. But my first guess at the meaning of “groupoid-set” would be a single presheaf on a groupoid, rather than the category of such; since for a group G a “G-set” is a single presheaf on $\mathbf{B} G$.

If the question were just about what to call the bicategory, I think you probably couldn’t do much better than $GpdProf$, i.e. the bicategory of groupoids and profunctors. But that doesn’t help if you want a word other than “groupoid” with which to refer to the objects of that bicategory, one which emphasizes that you’re viewing them via their corresponding presheaf topoi.

Here’s a slightly facetious suggestion: orbiset. Since an “orbifold” or “orbispace” is a manifold or space with extra groupoid structure, especially viewed as a generalization of the action groupoid of some G-manifold or G-space, it makes sense for an “orbiset” to be a set with extra groupoid structure — that is to say, a groupoid — especially viewed as a generalization of the action groupoid of some G-set, which seems to be your point of view. Moreover, the word “orbiset” has the connotation of thinking of the groupoid as some sort of space whose points have automorphisms, which is also more or less the effect of passing to its topos of presheaves (if we view topoi as generalized spaces).

I’m not sure whether I think that is actually a good idea, but I thought I would throw it out there.

Posted by: Mike Shulman on April 13, 2010 3:21 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Alex said

Now $G$-sets and permutation representations of $G$ are in bijective correspondence.

Do you mean that the isomorphism classes of these things are in bijective correspondence?

Posted by: Simon Willerton on April 13, 2010 12:05 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Actually, I am a bit confused now.

After fixing a field, I thought I defined G-sets and permutation representations of G to be exactly the same thing.

What mistake am I making here?

Posted by: Alex Hoffnung on April 13, 2010 12:33 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Well, according to the definition in the current version, a permutation representation consists of a $G$-set $X$, together with a vector space of which $X$ is a basis and on which $G$ acts linearly extending its action on $X$. So given the $G$-set $X$, the rest of the data is determined uniquely up to unique isomorphism, but no more than that; a given set can be the basis of more than one vector space. So I think there is an equivalence between the groupoid of $G$-sets and $G$-equivariant isomorphisms on the one hand, and the groupoid of permutation representations and chosen-basis-preserving linear isomorphisms on the other hand.

Posted by: Mike Shulman on April 13, 2010 2:53 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

You were essentially making an evil assertion. You were basically saying (I think) that the category of permutation representations and the category of $G$-sets are isomorphic. As Mike says, the obvious functors between them will only give you an equivalence.

Posted by: Simon Willerton on April 13, 2010 3:20 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

I took a little nap and cleared out some of the cobwebs. Reading the comments, I see the problem I was having.

I was not claiming that the categories should be isomorphic, because this is not even close to true unless you do something drastic, like restrict to basis preserving linear maps. Then you still get just an equivalence as Mike pointed out above.

I was not even considering the categories at all. I was just looking at the function between the set of finite $G$-sets and the set of permutation reps of $G$.

The mistake was convincing myself that given a $G$-set, there could only be one permutation rep with that basis. Of course, as Mike said, the right thing to say is that the rep is unique up to unique isomorphism.

Posted by: Alex Hoffnung on April 13, 2010 4:40 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Hi Mike,

Can you explain what you mean by “faithful reflection”? I do not know this terminology.

Posted by: Alex Hoffnung on April 15, 2010 3:43 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Sure. Whenever X is a property that objects of a category $C$ can have, and I talk about the “X reflection” of an object $c\in C$, I mean that the full subcategory of $C$ consisting of the objects satisfying X is reflective and I want to apply the left adjoint reflector to $c$.

Thus, for example, we have the “posetal reflection,” “groupoidal reflection,” and “discrete reflection” of a category, the last of which is also known as its set of connected components. Similarly, the “subterminal reflection” of an object of a topos is otherwise known as its “support.”

More generally, we can apply this idea in slice categories. Thus, for instance, the “monic reflection” of a function $f\colon X\to Y$ is its image, using the fact that monics are reflective in $Set/Y$ because (epi,mono) is a factorization system. Likewise, in $Cat$ we have a factorization system (bijective-on-objects,full+faithful) which produces the “fully faithful reflection” of a functor, i.e. its full image. But we also have another factorization system (b.o.+full, faithful) which produces the “faithful reflection” of any functor. Explicitly, the faithful reflection of $f\colon A\to B$ is obtained from $A$ by identifying all pairs of parallel arrows in $A$ which get identified by $f$.

When constructing a bicategory of relations from a regular category, we perform composition by pullback as in the bicategory of spans, but then we apply the “monic reflection” to get back to a relation (= a jointly monic span). For this to work, we need the dual class of (strong) epics to be stable under pullback. Likewise, when constructing a bicategory of jointly-faithful spans of groupoids, we can perform composition by pullback as in the bicategory of spans of groupoids, but then we apply the faithful reflection, which works because b.o.+full maps of groupoids are stable under pullback.

Posted by: Mike Shulman on April 15, 2010 7:33 AM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

the full subcategory of C consisting of the objects satisfying X is reflective

An existing link is reflective subcategory

Posted by: Urs Schreiber on April 15, 2010 8:18 PM | Permalink | Reply to this

Re: New Paper on the Hecke Bicategory

Yes, thanks, sorry; I was in a hurry and didn’t test my links. I’ve fixed it.

Posted by: Mike Shulman on April 15, 2010 9:33 PM | Permalink | Reply to this

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