The n-Category Café

Hi John,

As you know, the reason I’m interested in this stuff is firstly, of course, because I’m a big fan of the HDA series, but also because many of the patterns which crop up in your and Jim’s course on Geometric Representation Theory also crop up in 2-representations; one just has to sprinkle the words “$U(1)$-central extension” here and there.

But unless I am still confused it seems to me that, without that Doplicher-Roberts reconstruction stuff (which sounds very interesting), the current version of the Fundamental Theorem of Hecke Operators has no actual content!

Of course that’s a very strange thing to say, so let me explain myself. I’m going to break up the theorem into two parts.

(a) The “intertwiners and groupoids” part. John wrote:

So, for any pair of finite G-sets A and B, we have a hom-groupoid

$hom(A,B)=(A\times B)//G$

And, if we `degroupoidify’ this groupoid — take the free vector space on the set of isomorphism classes — we get the vector space of intertwiners from the permutation representation associated to $A$, to the permutation representation associated to $B$!

The important thing is: do you see why the last paragraph is true?

Yes… though I didn’t see it at first. Now that I think I understand it, wouldn’t most mathematicians consider this as sort of trivial though?

Here’s my (perhaps flawed) understanding of it. For any two representations $\sigma$ and $\rho$ of $G$, the space of intertwiners between them computes as the space of invariants of the tensor product (using the dual of the first one),

So for permutation representations, one computes: $$\begin{aligned} Hom(\mathbb{C}[A], \mathbb{C}[B]) &\cong Hom(1, \mathbb{C}[A]^\vee \otimes \mathbb{C}[B]) \\ &\cong Hom(1, \mathbb{C}[A] \otimes \mathbb{C}[B]) \\ &\cong Hom(1, \mathbb{C}[A \times B]) \\ &= linear combinations of orbits in \,\, (A \times B)//G, \end{aligned}$$ which is precisely what you are saying (though it seems to be an elementary computation).

(b) The “degroupoidification/topos” part. In your notes, the final theorem is stated as:

Then you say:

The point of this theorem is that the familiar category $Perm(G)$ is obtained from the Hecke bicategory $Hecke(G)$ by a ‘degroupoidification’ process, namely $\overline{D}$, after a slight change in viewpoint, namely $\overline{K}$.

For any others reading this, let me explain what is being stated here. $Hecke(G)$ is the bicategory with:

• objects are finite $G$-sets
• given $G$-sets $A$, $B$, the hom-category $hom(A,B)$ is the category of functors from $(A \times B)//G$ into $Set$.

The Fundamental Theorem of Hecke Algebras then states that if we apply “Process $\overline{D} \circ \overline{K}$” to $Hecke(G)$, then we gets $Perm(G)$. What are these processes?

Well the first one, “Process $\overline{K}$”, is the machine which takes $Hecke(G)$ and forms a new bicategory, which has

• the same objects, namely finite $G$-sets
• and hom-groupoids given by $Hom(A,B) = (A \times B)//G$.

But without any Doplicher-Roberts-style input, “Process $\overline{K}$” has no content! It simply transforms hom-categories into groupoids,

but it doesn’t do it intrinsically , it just does it by definition!

So the theorem doesn’t start with the cool bicategory $Hecke(G)$, perform some abstract intrinsic manipulations, and then output $Perm(G)$. Rather, it sort of cheats - it converts $Hecke(G)$ by hand into a bicategory where the hom-sets are $(A \times B)//G$… and then states that if we apply “Process $\overline{D}$” to this new bicategory - in other words, if we degroupoidify the hom-groupoids - we’ll get out $Perm(G)$. But this last step is basically trivial - it is precisely part (a) above!

In other words, in the way I am understanding it, “Process $\overline{K}$” has no content, while the degroupoidification “Process $\overline{D}$” is elementary, so the theorem looks empty. On the other hand, I know that there is definitely something to this theorem… so what is it?

Bruce wrote:

Now that I think I understand it, wouldn’t most mathematicians consider this as sort of trivial though?

Sure, if they understood it. Most mathematicians think anything they really understand is trivial. I think that’s why we usually explain things so badly: to keep other mathematicians — and even ourselves — from really understanding things, and then labelling them ‘trivial’.

If I’d just told you that there was a systematic way to take the category of permutation representations of a finite group and boost it up to a bicategory, you might have been impressed. Or maybe not — after all, there are some very trivial ways to do this, e.g. by throwing in identity 2-morphisms.

If I’d told you that no, I wasn’t doing anything silly like that, and by the way: this bicategory makes no mention of the complex numbers or any other ground field, you might have been impressed.

I could probably even have revealed more details without letting you understand what was going on. I could have said: I’m going to take any category $Perm(G)$ of finite-dimensional representations of a finite group, and obtain it from a bicategory $Hecke(G)$ consisting of

• finite $G$-sets
• spans of finite $G$-sets
• maps of spans of finite $G$-sets

by means of a systematic process that turns certain nice bicategories into $Vect$-enriched categories. That sounds sort of impressive.

But, since I explained it well enough for you to understand how it really works, you think it’s trivial.

Actually, you only discussed the easiest part in your comment: how the hom-groupoids

$$hom(A,B)=(A \times B)//G$$

turn into the hom-spaces

$$hom(\mathbb{C}[A], \mathbb{C}[B])$$

when we degroupoidify. We also have to check that composition gets preserved! We start with a composition like this

$$\circ : hom(A,B) \otimes hom(B,C) \to hom(A,C)$$

which really means

$$\array{ \circ : (A \times B)//G &\times& (B \times C)//G &\to &(A \times C)//G }$$

But notice, the arrow here is not a functor — it’s a span! Jim explained this span at the end of his lecture. I think it’s sort of cool.

And then, we need to check that this span gets sent to the usual composition of intertwining operators

$$\circ : hom(\mathbb{C}[A], \mathbb{C}[B]) \otimes hom(\mathbb{C}[B], \mathbb{C}[C]) \to hom(\mathbb{C}[A], \mathbb{C}[C])$$

For this, of course, we need to use the recipe for turning spans of groupoids into linear operators.

There’s also the part about getting an instrinsic characterization of ‘nice topoi’, which will allow the reconstruction of a finite groupoid $X$ from the nice topos $Set^X$. As you note, that should obscure things enough to make the result more impressive.

Okay — it’ll actually be interesting, too. But not as interesting as you seem to think. After all, this part is bound to work, if the outline I’ve sketched so far is correct. As soon as we know

$$\array{ J : [finite groupoids, spans, equivalences of spans] &\to& [topoi, functors, natural isomorphisms] \\ X &\mapsto & Set^X }$$

is ‘one-to-one’ (in the suitable bicategorical sense), then we can characterize its ‘image’

$$Nice = [nice topoi, nice functors, nice natural isomorphisms]$$

and get an equivalence

$$J : [finite groupoids, spans, equivalences of spans] \to Nice$$

which will then have an ‘inverse’

$$K : Nice \to [finite groupoids, spans, equivalences of spans]$$

which is our ‘reconstruction theorem’.

Indeed, all these ‘reconstruction theorems’ have a certain stage magic quality to them: you make a big deal of pulling a rabbit out of the hat, and it impresses everyone tremendously — but all you did was invert the process of putting the rabbit in. All these theorems do is characterize the image of a 1-1 function and then construct its inverse — or some categorified version thereof!

But let me explain why I actually like the ‘Fundamental Theorem of Hecke Operators’.

Most of all, it suggests how to take big wads of finite group representation theory and categorify it! For example, those ‘Hecke algebras’ people like to talk about — those $q$-deformed versions of the group algebras of symmetric groups — are really algebras naturally sitting inside $Perm(G)$ where $G = GL(n,F_q)$. And using the ‘Fundamental Theorem’, it turns out these Hecke algebras can be categorified! An algebra is a one-object category enriched over $Vect$; when we categorify the Hecke algebra we’ll get a one-object bicategory enriched over $Nice$ — some sort of monoidal category. And since the Hecke algebra is a quotient of the braid group algebra, closely related to the Jones polynomial, this means we’ll have categorified some aspects of ‘quantum topology’.

Jim already explained this back in lecture 13 — that’s what all those braid diagrams were about. The ‘Fundamental Theorem’ just puts this in its proper context.

Next quarter we’ll say more about this and a bunch of other applications…

Wow! This stuff sounds great! It’ll take me a while to absorb it, but let me start by explaining why I did something absurd:

Tom wrote:

Isn’t it clear anyway that $G Tor \simeq G$, for any group $G$?

Yes, it’s clear: for any group $G$, all left $G$-torsors are isomorphic, so a skeleton of $G Tor$ has one object, which might as well be $G$ itself, viewed as a $G$-torsor. The automorphism group of $G$ as a left $G$-torsor is just $G$ itself, acting via right translations. So, $G Tor \simeq G$.

So, once I got my paws on $G Tor$ I was done getting $G$. And I should have known this; I’m a big fan of this fact. It was absurd for me to do any further finagling to recover $G$. Why did I bother getting $G$ as automorphisms of the forgetful functor $G Tor \to Set$? It’s because I was in the mood for ‘Tannaka–Krein reconstruction’, where we recover an algebraic gadget from endomorphisms of a forgetful functor.

While it was absurd, it wasn’t wrong. Once we remember $G \simeq G Tor$, we see the automorphism group of the forgetful functor $G Tor \to Set$ is the same as the group of all transformations of $G$ that commute with left translations — and this is just $G$, acting via right translations!

John,

Here are some very preliminary thoughts on this issue you brought up in your notes, on characterizing ‘nice toposes’ in intrinsic terms (p. 4). I think it should be possible to state it all much more cleanly, but this is a first pass.

A ‘nice’ topos is the category of presheaves over some finite groupoid $C$. One nice fact is

Theorem. $C$ is a groupoid iff $Set^{C^{op}}$ is a Boolean topos.

(‘Boolean’ means that every subobject $i: c \to d$ has a complementary subobject: one whose union with $i$ is all of $d$, and whose intersection with $i$ is initial (empty). It’s easy to see that $G$-Set, for $G$ a groupoid, is Boolean.)

So it would be enough to give an intrinsic characterization of presheaf toposes, and add the word ‘Boolean’.

I’ll start with a simple characterization theorem for presheaf toposes. I’ve talked about this a little elsewhere, but looking at what I said there it seems brutally abstract, and it would be better to say it again a little more nicely.

By the way, Tom has of course already stated some pertinent facts on presheaf toposes in the language of toposes and geometric morphisms. But I want to put geometric morphisms a little to the side for the time being, and talk more in the language of presheaf categories and cocontinuous maps, because that’s where nice toposes and nice functors actually live.

Let $E$ be a small-cocomplete category. Here is the key definition (due to Lawvere?):

Definition. An object $e$ of $E$ is tiny if the hom-functor $E(e, -): E \to Set$ preserves small colimits.

(By the way: in email you were singling out the role of indecomposable = transitive $G$-sets. These are those presheaves $X$ on $G$ [or on $G^{op}$ if you wish] such that $hom(X, -)$ preserves small coproducts. Such objects are sometimes called connected objects. Here I’m strengthening that condition to connected and projective: $hom(X, -)$ also preserves coequalizers.)

Notice that when $E$ is a presheaf topos $Set^{C^{op}}$, the representable presheaves $hom(-, c)$ are tiny, since

$$Set^{C^{op}}(hom(-, c), F) \cong F(c) = eval_c(F)$$

by Yoneda, and evaluation at an object $c$ preserves colimits of presheaves since colimits of presheaves are computed object-wise.

In fact, the Cauchy completion of $C$ is equivalent to the full subcategory of tiny presheaves on $C$. This gives a reconstruction of a small category $C$ from $Set^{C^{op}}$, if $C$ is already Cauchy complete (as it will be in the case where $C$ is a groupoid, as Tom has already pointed out).

To characterize those cocomplete categories $E$ which are presheaf categories, it therefore makes sense to focus on the full subcategory of tiny objects of $E$, and ask what else we need. Well, in the presheaf case, every presheaf can be expressed as a canonical colimit of representables (= tiny presheaves). We say that the representables are dense in the category of presheaves. Thus it makes sense to focus on the condition that the tiny objects form a dense subcategory.

(Returning to your email, you had observed that every $G$-set is the coproduct of indecomposables, i.e., the coproduct of objects $X$ whose representables $hom(X, -)$ are coproduct-preserving. Here, we are pursuing an analogous statement, that every $G$-set is the colimit of objects $X$ whose representables $hom(X, -)$ are colimit-preserving.)

More precisely, a full subcategory $i: C \to E$ is dense if every object $e$ of $E$ is the colimit of the cone of $C$ over $e$, i.e., the colimit of the composite

$$(i \darr e) \stackrel{proj}{\to} C \stackrel{i}{\to} E.$$

In the language of weighted colimits, this says that the canonical map

$$E(i-, e) \otimes_C i = \int^c E(i c, e) \otimes i c \to e$$

is an isomorphism for every $e$ (the tensor inside the coend is a coproduct of copies of $i c$ indexed over the hom-set $E(i c, e)$). This arrow is actually the counit of an adjunction between $E$ and $Set^{C^{op}}$:

$$E(X \otimes_C i, e) \cong Set^{C^{op}}(X, E(i-, e));$$

this adjunction simply restates the universal property of weighted colimits. I will call this adjunction the Kan adjunction induced by the functor $i: C \to E$.

Here then is the characterization or recognition theorem for presheaf toposes. I think this theorem goes pretty far back in the day; I have some dim trace memory that Lawvere told me it’s in Marta Bunge’s thesis.

Theorem. Let $E$ be small-cocomplete. $E$ is equivalent to a presheaf category iff the full subcategory of tiny objects, $i: Tiny(E) \to E$, is essentially small and dense.

Let me just prove the “if” direction. The equivalence $E \simeq Set^{Tiny(E)^{op}}$ is given by the Kan adjunction of $i: C = Tiny(E) \to E$. As we have seen, density of $i$ is equivalent to the fact that the counit of this adjunction is an isomorphism. It remains to see that the unit of the Kan adjunction is also an isomorphism. But for every weight $X: C^{op} \to Set$, the component of the unit at $X$ is given by the composite

$$\array{ X(c) & \cong & \int^{c^'} X(c') \otimes C(c, c') & [by the Yoneda lemma] \\ & \cong & \int^{c'} X(c') \otimes E(i c, i c') & [the embedding i is full] \\ & \cong & E(i c, \int^{c'} X(c') \otimes i c') & [by the fact that c is tiny] \\ & = & E(i c, X \otimes_C i). & }$$

(To be continued; this is probably not yet the most illuminating theorem to apply toward the problem of intrinsically characterizing nice toposes.)

Tom wrote:

For a small category $C$,

$$HOM(Set, Set^C) \simeq Flat(C^{op}, Set)$$

where the right-hand side is the category of flat functors and natural transformations between them. I talked about this here. I didn’t explain what ‘flat’ meant, but for groupoids it just means ‘torsor’.

Cool!

In your previous comment about flat functors, you wrote:

OK… but what’s a flat functor? I won’t give the definition, but to a first approximation, a flat functor is a functor that preserves finite limits. This is actually true when $C$ has finite limits. It’s also true that flat functors preserve all finite limits that exist in $C$. It’s somehow a bit daft to think about finite-limit-preserving functors on a category that doesn’t have all finite limits; flatness is the righteous concept.

Somehow this scared me off, as if you were saying “I could tell you what flat functors actually are, but that would drive you stark raving mad, so I’ll spare you…”

So, this time I looked at Mac Lane and Moerdijk (which I’d already been perusing) and was mildly relieved to discover that this concept of ‘flat functor’ is closely akin to the concept of ‘flat module’ for a ring — which is something I know from homological algebra: namely, a module such that tensoring with it is left exact.

More precisely: for any category $C$, they define a way to ‘tensor’ a functor $F: C \to Set$ with a functor $G : C^{op} \to Set$ and get a set. Then, they say a functor $F : C \to Set$ is flat if tensoring with $F$ is left exact.

Hey — is this ‘tensoring’ just what some people call a ‘coend’? If so, I actually understand it.

Anyway, while I don’t feel I understand this, it’s not driving me stark raving mad.

The main problem with me understanding this topos stuff is not that any piece of it seems devilishly tricky: it’s that there’s so much of it… it seems locally trivial, but globally nontrivial.

Your argument that nice functors = cocontinuous functors is very, how shall we say: nice! And it seems you are also suggesting the additional nice fact that nice transformations are just transformations.

So in other words, the 2-category NICE is biequivalent to the bicategory $Bim_{gpd}$ of groupoids and bimodules (= profunctors = distributors) between them. The 2-category $Gpd$ of groupoids and functors sits inside of NICE, where a functor $f: G \to H$ induces a nice functor

$$Set^{G^{op}} \to Set^{H^{op}},$$

namely the Kan extension which is left adjoint to the pulling-back functor

$$Set^{f^{op}}: Set^{H^{op}} \to Set^{G^{op}}$$

which is itself a nice functor. Or in other words, that we have a homomorphism

$$i: Gpd \to Bim_{gpd}$$

which maps 1-cells to left adjoint 1-cells. Conversely, since groupoids are Cauchy complete, every left adjoint 1-cell in $Bim_{gpd}$ arises in this way.

$Bim_{gpd}$ is an example of what is called a cartesian bicategory, a notion meant to capture a ‘generalized calculus of relations’. This would include the calculus of ordinary relations in a suitably nice category (a regular category), but is intended to cover also the 2-categorical calculus of spans in a finitely complete category, and of internal categories and bimodules in a finitely complete category, and other examples of similar ‘relational’ character. In recent years cartesian bicategories have undergone a renaissance, due to efforts of people like Wood, Carboni, Kelly, and Walters to properly address the technical monoidal bicategory aspects (which hadn’t been worked out when Carboni and Walters first introduced the notion).

Anyway, we now have a tie-in between Groupoidification and the theory of cartesian bicategories which is already pretty well-developed, and at this point I think I can give a good answer to the question ‘why groupoids?’ (or perhaps better, ‘why not categories?’). We can get a hint of this by looking at what Denis-Charles said above: one can play a similar game with nice functors between more general presheaf categories,

$$F: Set^C \to Set^D,$$

where they basically amount to functors $f: C^{op} \times D \to Set$ (that is, bimodules), or to discrete fibrations of the form

$$El(f) \to C^{op} \times D$$

which gives a span in $Cat$ from $C^{op}$ to $D$. Notice the variance however! But in $Gpd$, we can straighten out the variance so as to consider spans from $G$ to $H$.

But more to the point, the real reason that spans of groupoids work so well is that in $Bim_{gpd}$, we have a Beck-Chevalley condition which allows us to compose spans – this Beck-Chevalley condition doesn’t hold in the bicategory of categories and bimodules! For example: when we want to compose a nice functor $g_{!}f^*$ given by a span

$$H \stackrel{f}{\leftarrow} G \stackrel{g}{\to} J$$

with another nice functor $k_{!}h^*$ given by a span

$$J \stackrel{h}{\leftarrow} K \stackrel{k}{\to} L,$$

we have to work $k_{!} h^{\star} g_{!} f^\star$ into the form $(-)_! (-)^*$, which we do by passing through the weak pullback

$$\array{ P & \stackrel{q}{\to} & K \\ p \darr & \swArrow & \darr h \\ G & \stackrel{g}{\to} & J }$$

and applying the Beck-Chevalley condition, which gives the invertibility of the canonical map $q_{!} p^\star \to h^\star g_{!}$. This means we can write

$$k_! h^\star g_! f^\star \cong (k q)_! (f p)^\star$$

and all is well. But, we need groupoids in order for Beck-Chevalley to work.

Over on the cartesian bicategories side, the Beck-Chevalley condition is closely tied to the fact that groupoids and bimodules form what is called a discrete cartesian bicategory (which is the special case of Beck-Chevalley where the weak pullback involved is the coassociativity square for the diagonal map $\delta: G \to G \times G$). If you write out the Beck-Chevalley condition in that special case, it says that $G$ is a Frobenius monoid in $Bim_{gpd}$, where the comultiplication is the bimodule $\delta_!: G \to G \times G$ and the multiplication is the bimodule $\delta^*: G \times G \to G$.

I commented earlier on Beck-Chevalley for bimodules, but I didn’t see clearly how it ties in with Groupoidification until after seeing Denis-Charles’s comment. Thanks, Denis-Charles!