The n-Category Café

There’s a hell of a lot math going on in this blog that I don’t follow. Probably most of it. It’s a sad business sometimes…

More in a bit.

“Would this be part of “groupoidification”?

What I meant was the particular meaning that John has given the term: in my words:

Groupoidification is the study of group representations in terms of action groupoids and their generalization.

By some cosmic coincidence, when John began spreading vague hints about how things beautify once we pass from reps to the corresponding action groupoids, thereby lifting the categorical dimension a bit, I happened to be thinking day and night about another strange lift in categoricaal dimension, and suddenly it occurred to me that both might actually be the same kind of shifts, or at least closely related.

At the moment I am thinking that JohnJimTodd’s “groupoidification” is to obstruction theory like associated bundles are to principal bundles. (That’s not supposed to be intelligible without further details. If it was, we need to talk ;-)

But I must say, as I did early on when John mentioned this the very first time: I find the term “groupoidification” rather unsuggestive of what it actually refers to.

If I were in any position to do so, I would urge its inventors to change that term to something else. It’s not too late yet, so far “groupoidification” is mainly media hype for a product that you cannot yet buy.

My suggestion: let’s think about what the concept of “action groupoid” really means (let’s think about action 2-groupoids to lift some accidental degeneracies of low dimension).

When we have figured out that action groupoids really mean XYZ, then we rename “groupoidification” to “XYZified representation theory”.

Auf seinen Kopf, über seinen Kopf… prepositions are so confusing.

Yes. For instance in German “ein Kommentar über seinen Kopf” is both “a comment on his head” and “a comment over his head” – only that the second version doesn’t have the standard metaphorical meaning, in German, of “beyond his grasp”.

Poor Danny is now trying to learn this mess they call German grammar. I am glad I don’t have to, anymore. But maybe Danny can soon tell me the names of all these rules – since I don’t know them…

Danny and I had lots of fun when we had to get all this paperwork done the days after he arrived. The first words he learned were Einwohnermeldeamt, Bezirksamt and, watch out, Gebühreneinzugszentrale (the latter one is that institution which guarantees that one can here switch on the TV without being reduced to an intellectual ameba within seconds. Instead it still takes minutes…)

I often neglect your posts — but I always feel a tinge of guilt about this

Don’t worry, I wasn’t complaining. We are all best when we do what we feel like at the moment. I literally just meant to say that I know the feeling of waiting eagerly for technical comments on blog entries.

If I had five grad students at my disposal to delegate to them finishing the 3/4-done things I have, I’d be probably switching to other things. Personal interest probably gauges itself somewhere in between the poles of felt necessity and felt fascination

And of felt need for sleep. Right now I need to call it quits. But thanks a lot for the detailed comment on classifying topoi and orbi-simplices. I’ll get back to you on that tomorrow.

I wrote:

I want the students (the actual ones in the classroom!) to get a good solid grasp of some classic pieces of linear algebra — the representation theory of symmetric groups and $\mathrm{GL}(n,F_q)$’s — before I start groupoidifying them.

In other words, I want the kids to take a good look at some examples before I hit these examples with a powerful sledgehammer.

And why do I want them to see these examples? It’s because amazing things happen when we groupoidify them!

It’s the usual story: take the most beautiful mathematics, categorify it, and it gets even better.

I explained this starting on page 4 of the notes for week 8 of the Winter 2004 quantum gravity seminar, and continue in week 9.

Ah, thanks. See, I wasn’t aware of these particular notes on weak quotients.

For those not having read them: therein John talks about how to conceive the weak quotient of an action

$$\rho :\Sigma G\to \Sigma {\mathrm{Aut}}_C(V)$$

of a group $G$ on object $V$ in a category $C$ as the weak colimit

$$V//G:={\mathrm{colim}}_{\Sigma G}\rho$$

(or rather, he spells out what this means in order to save his student’s nerves).

What I had in mind was a related but different way to think of this: we might define $V//G$ by looking at the weak cokernel of $\rho$

$$\Sigma G\stackrel{\rho }{\to }\Sigma {\mathrm{Aut}}_C(V)\to \mathrm{wcoker}(\rho )$$

and then identify, somewhat more indirectly, $V//G$ as any object in a 2-category $C\prime$ such that

$$\mathrm{wcoker}(\rho )\simeq \Sigma {\mathrm{Aut}}_{(C\prime )}(V//G)$$

I have an application in mind where we don’t care so much about the objects but rather about their automorphism $n$-groups.

More precisely, I have an application here where to every rep of $G$ we also want a rep of $${\mathrm{INN}}_0(G):=\mathrm{wcoker}({\mathrm{Id}}_G)$$ “in a compatible way”.

So I am interested in the situation

$$\begin{array}{ccc}G& \to & \mathrm{wcoker}({\mathrm{Id}}_G)\\ {\downarrow }^{\rho }\\ {\mathrm{Aut}}_C(V)\end{array}.$$

With the above considerations I can complete this cone as

$$\begin{array}{ccc}G& \to & \mathrm{wcoker}({\mathrm{Id}}_G)\\ {\downarrow }^{\rho }& {\searrow }^0& \downarrow \\ {\mathrm{Aut}}_C(V)& \to & {\mathrm{Aut}}_{C\prime }(V//G)\end{array}$$

with tranformations filling the triangles.

Right. It’s what you probably were already guessing: $2^{(-)}$ is alternate notation for the contravariant functor

But an interesting question is why this is called the “generic Boolean algebra”, and this enters into what Tom was explaining.

As Tom said, whenever you have a finitary algebraic theory, like the theory of groups or the theory of Boolean algebras, it is very easy to build the corresponding classifying topos: it is the topos of presheaves of the form

where ${\mathrm{Alg}}_{\mathrm{fp}}$ is the category of finitely presentable algebras of the theory and their homomorphisms (i.e., algebras which arise from taking coequalizers of maps between finitely generated free algebras). Now, what we mean by the classifying topos is that it is universal in some sense among pairs

where $E$ is a Grothendieck topos and $M$ is an internal model of the theory as interpreted in $E$. In the case of the theory of Boolean algebras, the classifying topos is the topos of functors

equipped with a certain model, a universal model, which we are calling the “generic Boolean algebra”. In this case, the generic model is the underlying set functor

it means that $U$ comes equipped with internal Boolean algebra operations $\wedge :U\times U\to U$, $\neg :U\to U$, etc. The universal property says that given a topos-model pair $(E,M)$, there exists a left exact left adjoint

and an isomorphism $\varphi :{\chi }_M(U)\to M$, and this pair $({\chi }_M,\varphi )$ is unique up to natural isomorphism.

This recipe works the same way for any finitary algebraic theory: the classifying topos is $[{\mathrm{Alg}}_{\mathrm{fp}},\mathrm{Set}]$, and its generic model is the underlying-set functor $U:{\mathrm{Alg}}_{\mathrm{fp}}\to \mathrm{Set}$ (which carries a structure of internal model of the theory), and there is this universal property as above. (If you want to view the theory of Boolean algebras as a finitary algebraic theory, one is obliged to include the terminal or degenerate Boolean algebra as a model, as Tom was doing. If you wish to restrict to non-degenerate Boolean algebras, you can do that as Lawvere was doing, but the technical details are slightly less smooth and simple.)

Finally, Lawvere is invoking the equivalence

between the opposite of (nonempty) finite sets and (nondegenerate) Boolean algebras, where $P$ takes $S$ to the Boolean algebra $2^S$. Because of this equivalence, you can just as well say that $[{\mathrm{Fin}}^{\mathrm{op}},\mathrm{Set}]$, equipped with the functor $2^{(-)}$ as model, is the classifying topos for Boolean algebras, and that’s what Lawvere was saying.

Charlie writes:

Lawvere writes:

“generic Boolean algebra $2^{()}$ (in the usual 2-valued notation)”

Can you please figure out what Lawvere is saying here, e.g., I understand what a Boolean algebra with $2^n$ elements is but I don’t know what Lawvere means by his notation?

$X^Y$ is a standard notation for the set of functions from a set $Y$ to a set $X$. So, $2^X$ is the set of functions from the set $X$ to the set $2=\{\mathrm{0,1}\}$.

Since subsets of $X$ are described by their characteristic functions

$$\chi :X\to 2$$

we can also think of $2^X$ as the set of subsets of $X$. This is often called the ‘power set’ of $X$ — and now it should be clear why!

$2^X$ is a finite Boolean algebra: we can take unions, intersections and complements of subsets of $X$, and they satisfy the usual rules of classical logic.

Moreoever, all finite Boolean algebras are of this sort. In fact, we get a contravariant equivalence of categories

$$2^{(-)}:{\mathrm{FinSet}}^{\mathrm{op}}\to \mathrm{FinBool}$$

sending each finite set $X$ to the finite Boolean algebra $2^X$. It’s contravariant, since a function $f:X\to Y$ lets us take the inverse image of any subset of $Y$ and get a subset of $X$, giving a function going backwards from $2^Y$ to $2^X$.

Note: taking inverse images preserves all the Boolean algebra operations: union, intersection and complement. One can easily get confused here, since a function $f:X\to Y$ also lets us take the image of any subset of $X$ and get a subset of $Y$. However, the image of the complement of a subset isn’t always the complement of the image! Intersections aren’t preserved either. So, we don’t get a Boolean algebra homomorphism by taking images.

There’s much more to say but I hope this helps.

John wrote:

It takes a while to figure out what it means for an operation with a bunch of inputs and a bunch of outputs to commute with some other operation with a bunch of inputs and a bunch of outputs. But, if you draw these operations as black boxes with wires going in and coming out, you can do it.

Hey! I just bumped into a picture of this, over at the Secret Blogging Seminar:

It’s in an article by Noah Snyder on the field with one element. It’s no coincidence: Durov’s ‘generalized rings’ are algebraic theories where every operation commutes with every other in the above sense.

I want to sleep on it too, but that sounds right to me, and it’s actually a very nice observation. Perhaps more on this later…

I wouldn’t mind taking a stab at this. We have here at least three ways of considering groups as models of a theory:

• Models of a Lawvere or finite products theory. Here the theory may be reconstructed as the category opposite to the category of finitely generated free groups $F[n]$. This theory is a category with finite products. A group in a category with finite products $E$ is identified with a product-preserving functor $${\mathrm{FreeGrp}}_{\mathrm{fg}}^{\mathrm{op}}\to E.$$
• Models of a “left exact” theory. Here the theory may be reconstructed as the category opposite to the category of finitely presentable groups; this theory is a finitely complete category. A group in a finitely complete category $E$ is identified with a left exact or finite-limit preserving functor $${\mathrm{Grp}}_{\mathrm{fp}}^{\mathrm{op}}\to E.$$
• Models of a “small-complete” theory. Here the theory may be reconstructed as the category opposite to the category of all groups; this theory is a small-complete category. A group in a small-complete $E$ is identified with a continuous (small-limit preserving) functor $${\mathrm{Grp}}^{\mathrm{op}}\to E.$$

Let’s see how this is supposed to work. My point is that there is a uniform argument which specializes to each case, where the free group on one generator $F[1]$ plays the role of a “generic group” which generates the theory at hand. I’ll run through the argument in the case Robin touched on, the case of models of small-limit theories in small-complete categories $E$.

We begin by noticing the underlying-set functor

carries a tautological group structure. This means we have group operations $m:U\times U\to U$, $\mathrm{inv}:U\to U$, etc.: this boils down to the obvious fact that for any group $G$, the underlying set $U(G)$ carries a group structure (well, I did say it was tautological!).

This functor is represented by the free group on one generator $F[1]$, i.e., we have a natural isomorphism $U\cong \mathrm{Grp}(F[1],-)$. The group structure on $U$ translates to a cogroup structure on the representing object $F[1]$ as an object of $\mathrm{Grp}$, or to a group structure as an object of ${\mathrm{Grp}}^{\mathrm{op}}$. In conceptual terms, what is going on is that via the (small-limit preserving) Yoneda embedding

the object $F[1]$ inherits a group structure from the tautological group structure on $\mathrm{hom}(F[1],-)\cong U$.

Consequently, a small-limit preserving functor

will carry the group object $F[1]$ to a group object in $E$ (because small-limit preserving, indeed product-preserving functors carry groups to groups).

Conversely, given a group object $G$ in $E$, there is (up to isomorphism) a unique small-limit preserving functor

which takes the group $F[1]$ to the group $G$. In the case $E=\mathrm{Set}$, this should be obvious: the functor is just the representable $\mathrm{hom}(-,G)$. And actually, the general case can be reduced to the case of Set: a group $G$ in $E$ gives, for each object $X$ of $E$, a hom-group $E(X,G)$ in Set. Hence for each object $X$ we get a limit-preserving functor

or a limit-preserving functor

which factors through a limit-preserving functor (which I will call)

via the Yoneda embedding! (The uniqueness can be deduced from the fact that every group is a colimit of a diagram involving just copies of $F[1]$, whence every object in ${\mathrm{Grp}}^{\mathrm{op}}$ is a corresponding limit, which must then be preserved by a continuous functor.)

The argument we just gave carries over essentially verbatim to the other types of theories, where we restrict the class of limits considered. Actually, it also carries over essentially verbatim to any algebraic theory $T$ instead of the theory of groups.

Remark: each of the full inclusions

can be thought of as a “completion” of some kind to a larger class of limits, e.g., the first inclusion is the universal extension of a category with finite products to a finitely complete category.

Finally, to connect up with the amazing, mind-boggling fact Robin mentioned: for many small-complete categories $E$ which arise in practice, it is true that continuous functors ${\mathrm{Grp}}^{\mathrm{op}}\to E$ coincide with right adjoints. (That would be true of the case where $E$ is a Grothendieck topos, for instance, or where $E$ is a category of algebras.) By the yoga of adjoint functors, these would correspond in turn to left adjoints

and thence to right adjoints

as Robin was saying. I’d like to hold off on identifying this class of categories, or maybe someone like Robin would like to fill us in further, or offer up a different point of view.