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July 4, 2007

Supercategories

Posted by Urs Schreiber

Motivated by general questions in supersymmetric QFT, I would like to better understand some of the “arrow-theory” behind supersymmetry, finding a formulation which gives a systematic way to internalize the concept into various contexts. For instance, people have a pretty good purely arrow-theoretic understanding of finite-group QFT, such as Dijkgraaf-Witten theory. Can we understand how to superize this systematically, in a context where many of the standard tools one finds in the literature are simply not applicable?

In order to both motivate and further introduce the problem, I might, in a followup entry, start looking into the following

Exercise (The Willertonesque super 2-particle). Simon Willerton has demonstrated (see The Baby Version of Freed-Hopkins-Teleman) that the “quantum theory of the 2-particle propagating on a finite group” has a beautiful arrow-theoretic formulation:

Take the parameter space of the 2-particle to be the fundamental groupoid of the circle par:=Σ. \mathrm{par} := \Sigma \mathbb{Z} \,. Take its target space to be a finite group tar:=ΣG. \mathrm{tar} := \Sigma G \,. Then configuration space is the groupoid conf:=Funct(Σ,ΣG):=ΛG, \mathrm{conf} := \mathrm{Funct}(\Sigma \mathbb{Z}, \Sigma G) := \Lambda G \,, which plays the role of the loop group of the finite group GG. The fact that this 2-particle is charged gives rise to a 2-vector bundle on this configuration space, and quantum states of the 2-particle are sections of this. In the simplest case (see the above entry for the more general case), this simply means that a state here is a representation ψ:ΛGVect \psi : \Lambda G \to \mathrm{Vect} of the configuration space groupoid on vector spaces.

There is more structure here, but for the moment concentrate on this basic data. The point of this is that everything is purely combinatorial, well defined, and exhibits just the bare structure of the QFT here, stripped of all distracting technicalities.

The exercise is then: do the analogous discussion for the super 2-particle. Figure out what the super-parameter space of the super 2-particle in the above sense is, what its super-configuration space supergroupoid is and what its super-representations on supervector space are like.

Clearly the first step to make any progress at all here is to get a reasonable good understanding what supersymmetry really is, such as to apply it to this situation. So, this entry here is just about this question: What is the arrow theory of supersymmetry?

As an attempt to approach this exercise, I’ll introduce the concept of a supercategory, which is supposed to be to that of a supergroup like categories are to groups. I feel that this concept helps extracting some of essence of what is going on.

It turns out that this is closely related to another structure which has appeared in 2-dimensional QFT, that of G-equivariant categories.

It’s an exercise. I can’t be sure that I am on the right track. But I would like to share the following, as I proceed.

In the pdf

Supercategories

I give the following definition, and try to indicate why I think it is useful.

Definition. A super category is a diagram Id C C s \array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow& C \\ & \searrow \nearrow_{s} } in Cat\mathrm{Cat} such that Id Id C C C s s= Id C Id C Id. \array{ & \nearrow \searrow^{\mathrm{Id}} & & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow & C &\Downarrow & C \\ & \searrow \nearrow_{s} & & \searrow \nearrow_{s} } \;\; = \;\; \array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow \mathrm{Id}& C \\ & \searrow \nearrow_{\mathrm{Id}} } \,.

Personally, I find it helpful to think of Id C C s \array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow & C \\ & \searrow \nearrow_{s} } as the “flow of a (odd) vector field” on CC, using the philosophy of What is a Lie derivative really?

Definition. Given a supercategory CC, a grading and a grading operator on CC is a subcategory C 0CC_0 \hookrightarrow C preserved by ss, together with a nontrivial transformation Id C 0 b C 0 Id \array{ & \nearrow \searrow^{\mathrm{Id}} \\ C_0 &\Downarrow b& C_0 \\ & \searrow \nearrow_{\mathrm{Id}} } the “fermion number operator” (think (1) fermionnumber (-1)^{\mathrm{fermion}\, \mathrm{number}} ) such that Id Id C 0 C 0 s C 0 C 0 s C 0 Id Id \array{ & \nearrow \searrow^{\mathrm{Id}} & & & & \nearrow \searrow^{\mathrm{Id}} & & \\ C_0 &\Downarrow & C_0 &\stackrel{s}{\to}& C_0 &\Downarrow & C_0 &\stackrel{s}{\to}& C_0 \\ & \searrow \nearrow_{\mathrm{Id}} & & & & \searrow \nearrow_{\mathrm{Id}} & & } is central and involutive with respect to the horizontal composition of natural transformations.

Here C 0=C evenC oddC_0 = C_{\mathrm{even}} \oplus C_{\mathrm{odd}} is to be thought of as the subcategory of CC which contains exactly the grading-preserving morphisms.

In the usual QFT applications, we expect to see monoidal supercategories, which are braided for QFTs in dimensions two or higher and symmetrically braided in dimension three or higher. The notion of braided monoidal supercategories naturally leads over to the GG-equivariant fusion categories considered by Turaev and Kirillov.

In

GG-equivariant monoidal categories

there is an attempt to understand the structure here as a generalization of the concept of kk-tuply stablized nn-categories:

an ordinary braided monoidal category is the same as a one-object, one-morphism 3-category. What, in this sense, is a braided monoidal supercategory?

It seems to be the answer is this:

like an ordinary braided monoidal catgeory is a 3-category which in lowest degrees looks like the trivial 2-group, a braided monoidal supercategory is a 3-category which in lowest degree looks like the strict 2-group that comes from the crossed module G (2)=( 2Id 2). G_{(2)} = (\mathbb{Z}_2 \stackrel{\mathrm{Id}}{\to} \mathbb{Z}_2) \,. I am calling this generalization of stabilization of nn-categories currently G (2)G_{(2)}-stabilization. So the claim would be that braided monoidal supercategories come from G (2)G_{(2)}-stabilized 3-categories, with G (2)G_{(2)} the above strict 2-group.

Indepently of its appearance here in the context of supersymmetry, I would very much enjoy to hear any comments on this generalized notion of stabilization. Has it been considered anywhere?

Posted at July 4, 2007 9:37 PM UTC

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29 Comments & 7 Trackbacks

Re: Supercategories

What confuses me here is that you envisage supercategories as generalizing supergroups. I don’t know what a supergroup is pic, but it doesn’t seem that a supercategory generalizes one. A supercategory is “a thing”, and not “a collection of things”… this is weird.

Posted by: Bruce Bartlett on July 5, 2007 11:03 AM | Permalink | Reply to this

Re: Supercategories

Ok, I think I get it now! A supercategory is a category CC, equipped with a functor s:CCs : C \rightarrow C and some natural transformation. Right. Is a one object supergroupoid (i.e. where the underlying category CC is a groupoid) a supergroup?

Posted by: Bruce Bartlett on July 5, 2007 11:08 AM | Permalink | Reply to this

Re: Supercategories

Is a one object supergroupoid (i.e. where the underlying category C is a groupoid) a supergroup?

Yes, exactly. That’s the idea.

Supergroups are usually defined in terms of subgroups of automorphism groups of objects in these graded vector spaces Mod A\mathrm{Mod}_A.

So, here I would say that a supergroup is a one-object supergroupoid.

Posted by: urs on July 5, 2007 11:12 AM | Permalink | Reply to this

Re: Supercategories

I haven’t talked about morphism of supercategories yet.

In fact, I think one should go back one step and first consider the concept of a category with a GG-flow on it. Then morphisms of that. Then supercategories as categories with a 2\mathbb{Z}_2-flow on them (an “odd vector field”) and one additional condition.

I am thinking here along the lines of the disucssion in What is a Lie derivative, really?

So:

Definition. For GG any group, a GG-flow on a category CC is an action R:ΣGAut(C) R: \Sigma G \to \mathrm{Aut}(C) together with a collection of natural transformations Id C C R g \array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow& C \\ & \searrow \nearrow_{R_g} } for all gGg \in G, respecting that action.

A morphism of two categories CC and CC' with a GG-flow on them is a functor F:CC F : C \to C' such that Id C C R g F C=C F Id C C R g. \array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow& C \\ & \searrow \nearrow_{R_g} & \downarrow \\ && \downarrow^{F} \\ && C' } \;\;\;\; = \;\;\;\; \array{ C \\ \downarrow^F \\ \downarrow & \nearrow \searrow^{\mathrm{Id}} \\ C' &\Downarrow& C' \\ & \searrow \nearrow_{R'_g} } \,.

The point is that a GG-flow is a GG-action such that everything is connected to the identity by natural transformations. The components of these natural transformations are the “flow lines” of the flow. Manifestly so, if we consider flows on the path groupoid, as described in Isham on Arrow Fields. Here we have a generalization of this concept of a flow of a vector field, which also allows us to consider flows of such weird things like “odd” vector fields.

I was claiming that this is one nice way to understand the Cartan condition on a connection on the total space of a GG-bundle

Posted by: urs on July 5, 2007 11:28 AM | Permalink | Reply to this

Re: Supercategories

Consider the algebra of functions which live on the “super loop groupoid” of a group, under convolution. It seems that they have an anti-involution on them, satisfying some conditions. Does this relate to Jandl algebras?

Posted by: Bruce Bartlett on July 5, 2007 1:13 PM | Permalink | Reply to this

Re: Supercategories

Consider the algebra of functions which live on the “super loop groupoid”

Good, that’s the next step. Are you just thinking of any supergroupoid right now or do you really have a concrete realization of the super loop groupoid in mind?

(By the way, I am having trouble deciding in every case when to write “supersomething” or “super something”.)

I am asking, because there are two decisions to be made, it seems, when considering the super loop groupoid:

a) how do we superize parameter space?

b) how do we superize target space?

(Since no super structure is also a super structure, that case is of course included.)

Even more elemantary: let’s look at just the point particle, modeld by the trivial one-object category. Its configuration space of maps to ΣG\Sigma G is just ΣG\Sigma G itself.

What is the superpoint?

It seems there are precisely two choices:

i) the superpoint is pt/ 2\mathrm{pt}/\mathbb{Z}_2, i.e. the category Σ 2\Sigma \mathbb{Z}_2

ii) the superpoint is the action groupoid of 2\mathbb{Z}_2 acting on itself 2// 2\mathbb{Z}_2//\mathbb{Z}_2.

In the first case the functor ss would be the identity, in the second case it would switch the two copies.

I am not completely sure which one to use.

Posted by: urs on July 5, 2007 1:25 PM | Permalink | Reply to this

Re: Supercategories

You’ve got an excited audience watching from the sidelines. Neat stuff! *applause*

One quick question though, if it is “categorify” rather than “categorize”, shouldn’t it be “superify” instead of “superize”? :)

Posted by: Eric on July 5, 2007 4:31 PM | Permalink | Reply to this

Re: Supercategories

if it is “categorify” rather than “categorize”, shouldn’t it be “superify” instead of “superize”?

Oh, that’s right, I hadn’t thought about this. Yes, I guess it should be “superify”.

Posted by: urs on July 5, 2007 4:35 PM | Permalink | Reply to this

Re: Supercategories

As for me, how about “supersize”!? That’s when you know you are doing some serious maths, when you can supersize it.

Posted by: Bruce Bartlett on July 5, 2007 6:12 PM | Permalink | Reply to this

Re: Supercategories

Does this relate to Jandl algebras?

Right now I cannot give you a detailed answer. But in fact I am running into this question all over the place here: on the abstract level we are talking about it can get a little hard to decide if some 2\mathbb{Z}_2 grading one runs across is due to a super structure or due to the fact that one is working in a context of unoriented 2-particles, described by Jandl structures.

Posted by: urs on July 5, 2007 1:35 PM | Permalink | Reply to this

Re: Supercategories

one should go back one step and first consider the concept of a category with a GG-flow on it. Then morphisms of that. Then supercategories as categories with a 2\mathbb{Z}_2-flow on them (an “odd vector field”)

I have now updated Supercategories accordingly.

There is now first a section which recalls the concept category with a GG-flow on it, which also lists the basic examples that show that this definition is a reasonable definition.

Then a supercategory is simply a category with a 2\mathbb{Z}_2-flow on it.

Posted by: urs on July 5, 2007 2:09 PM | Permalink | Reply to this

Re: Supercategories

Back when I did my MSc, several of my algebra classes handled GG-graded algebras in various guises and for at least simple kinds of GG - which is to say \mathbb{Z} and 2\mathbb{Z}_2. I encountered, repeatedly, the comment that “super-X” just meant 2\mathbb{Z}_2-graded X, and couldn’t figure out why the super-X crowd wouldn’t just say that.

Does this impression hold true here as well? That the central recognition is the presence of a 2\mathbb{Z}_2-grading somewhere in the structure?

Posted by: Mikael Johansson on July 5, 2007 7:56 PM | Permalink | Reply to this

Re: Supercategories

“super-X” just meant 2\mathbb{Z}_2-graded XX, and couldn’t figure out why the super-XX crowd wouldn’t just say that.

For instance, there is

a) the category of 2\mathbb{Z}_2-graded vector spaces

and

a) the category of super vector spaces.

These are different! They are both equal as monoidal categories with duals. But they differ as braided monoidal categories.

The braiding on the former is the ordinary braiding VWWV V \otimes W \stackrel{\sim}{\to} W \otimes V of vector spaces. The braiding on the latter is the one which introduces a minus sign whenever an odd-graded vector space passes by another odd graded vector space. This is different.

Similarly, there are

a) 2\mathbb{Z}_2-graded Lie algebras

and

b) super Lie algebras.

And these are not the same. The bracket of the former is skew-symmetric, the bracket of the latter is graded skew symmetric.

This is a result of the fact that the former are Lie algebras internal to 2\mathbb{Z}_2-graded vector spaces, while the latter are Lie algebras internal to super vector spaces. And for the definition of a Lie algebra, one needs to make use of the braiding.

This pattern continues. Many structures appear in a 2\mathbb{Z}_2-version and in a super version. These differ by whether or not there are minus signs introduced whenever the order of two odd objects is interchanged.

Given any abstract braided monoidal category with a 2\mathbb{Z}_2-grading. How do we determine if it is super in the above sense?

We do this by checking what happens to an odd object as we transport it along a little looping. This operation is the combination of a unit, a counit and one braiding operation. In the super case this braiding picks up a sign.

This way, carrying objects through a little looping is the same as acting on them the with a “fermion number operator” (1) fermionnumber. (-1)^{\mathrm{fermion}\; \mathrm{number}} \,.

In a merely 2\mathbb{Z}_2-graded category, everything has fermion number 0 under this looping operation. In a true supercategory, some of the objects have odd fermion number. And the 2\mathbb{Z}_2-grading exactly interchanges even fermion number with odd fermion number.

Posted by: urs on July 5, 2007 9:49 PM | Permalink | Reply to this

Re: Supercategories

I think the point I was somewhere making - which most probably ends up being at odds with at least the standard terminology on the physical side, thus motivating this terminology - is that whenever I’ve seen a graded algebraic structure introduced, the assumption is that all operations are graded in the commutativity-like operations they have. So the category of 2\mathbb{Z}_2-graded Lie algebras is one that I would have defined to be graded skew symmetric, whereas the category you describe under the same name is “merely” a category of … hmmm … possibly Lie 2\mathbb{Z}_2-graded algebras.

Though admittedly, this usage ends up being clumsy in the extreme. I simply never saw the point of including a grading but not have it do anything with the algebraic operations it coexists with.

Posted by: Mikael Johansson on July 5, 2007 10:49 PM | Permalink | Reply to this

Re: Supercategories

I simply never saw the point of including a grading but not have it do anything with the algebraic operations it coexists with.

If you look at some of the references in TWF 254 (by Kac, Leites, Shchepochkina and perhaps yours truly), you’ll find the term “consistent grading”, which means a Z-grading consistent with Grassmann parity.

Whenever you have a Z-grading, you can cook up a Z_2-grading by considering the degree mod 2; it is consistent iff if all odd degree generators are fermionic and all even degree generators bosonic. E.g., every Z-grading of a proper Lie algebra gives you a Z_2-grading, clearly inconsistent. IIRC, GSW makes a great fuss about this for E_8.

Posted by: Thomas Larsson on July 17, 2007 6:01 PM | Permalink | Reply to this

Re: Supercategories

So, by the way, this is one point why I am going through all this here:

I thought it was worth wasting bandwidth with an entry on “supercategories” because it seemed to me that something was missing:

there is the concept of a supergroupoid as for instance in John’s paper on the categorified Gelfand-Naimark theorem. That is defined to be a groupoid with an involutive endomorphism of the identity functor on it.

The idea is that this transformation multiplies each object with (1) fermionnumber(-1)^{\mathrm{fermion}\; \mathrm{number}}.

But that alone, I think, doesn’t quite make a groupoid a “supergroupoid” in what should be the more or less standard meaning of super:

I might have a category with three simple objects, two off them have fermion number 0 the third one fermion number 1 under such an involutive endomorphism. Clearly, there is no symmetry between odd and even fermion number here. So this should not count as a supercategory.

So, there is actually a second condition: the presence of a “parity changing operator” which is compatible with the involutive endomorphism of the identity. This “parity changing operator”, I think, is best thought of as a “ 2\mathbb{Z}_2-flow on the category” #.

So a “supercategory” should be, I think,

a category with

a) a 2\mathbb{Z}_2-flow on it

and

b) a compatible “grading operator” on it

the way I have defined it in the above definition.

This doesn’t mention a braiding yet, and on purpose so. But if we do assume there is also a braiding around, there is a nice, way, it seems, to put this into this picture. That’s what the last part of supercategories is about, which makes the relation to G-equivariant monoidal categories.

I should polish that description a little. But I don’t have time to do so right now. I’ll be on vacation until July 20!

Posted by: urs on July 6, 2007 6:57 AM | Permalink | Reply to this

Re: Supercategories

One tiny comment:

I’ll refine the above definition of morphisms of categories with GG-flow:

we should allow arbitrary morphisms. Conjugating these with the flows for gGg \in G on the domain and with that for g 1g^{-1} on the codomain induces a GG-flow on the Hom-space itself, thus making the Hom-categories of categories with GG-flow itself a category with GG-flow. That’s what we want. What I had above is then just the Id\mathrm{Id}-graded part of the Hom-space.

Posted by: Urs Schreiber on July 16, 2007 5:37 PM | Permalink | Reply to this

Re: Supercategories

Noah Snyder once explained to me how the category of super vector spaces should be defined. As a monoidal abelian category, it is just the category of Z/2 graded vector spaces with the grading not imposing any condition on the morphisms. However, super vector spaces was equipped with an unusual commutator. This commutator had the magical effect that if you wrote down the diagrammatic definition of a commutative algebra object or a lie algebra object in the category of super vector spaces, you got skew commutative algebras and super Lie Algebras.

I’ll shoot Noah an e-mail and see if he’ll post an explanation or a reference.

Posted by: David Speyer on July 5, 2007 11:24 PM | Permalink | Reply to this

Re: Supercategories

However, super vector spaces was equipped with an unusual commutator. This commutator had the magical effect that if you wrote down the diagrammatic definition of a commutative algebra object or a lie algebra object in the category of super vector spaces, you got skew commutative algebras and super Lie Algebras.

Just in case it wasn’t clear: this is exactly what I was talking about in the above comment. The “commutator” here is the braiding, which is nontrivial for super vector spaces, but trivial for 2\mathbb{Z}_2-graded vector spaces.

Posted by: urs on July 6, 2007 6:41 AM | Permalink | Reply to this

Re: Supercategories

Hmmmm. I’m not sure how universal these conventions are. If you’d have asked me then I’d have said that “ 2{\mathbb{Z}}_2-graded” and “super” were synonymous. Certainly to topologists “graded” (i.e., \mathbb{Z}-graded) means “with the signs”.

Urs, do you have different names for \mathbb{Z}-graded things with and without the signs? Are there many 2\mathbb{Z}_2-graded things that you meet in everyday life that aren’t super?

Posted by: Simon Willerton on July 6, 2007 10:36 AM | Permalink | Reply to this

Re: Supercategories

[Urs is on holiday and claims that as an excuse for not posting at the Café. He emailed me the following and asked me to post it for him. Simon.]

There is a category Vect[G]\text{Vect}[G] of GG-graded vector spaces for every group GG, and this is usually taken to come equipped with the obvious braiding coming from Vect\text{Vect}. From this general point of view, there is no reason to treat G=Z 2G=\mathbf{Z}_2 differently.

To give an example from the literature: at Polyvector Super-Poincaré Algebras I mention a paper which talks about classification of Super-Lie algebras. In this paper, everything is done both for the Z 2\mathbf{Z}_2-graded as well as for the super case. The difference is only in the braiding, namely in the choice of whether or not the bracket is skew or graded skew.

Posted by: urs on July 10, 2007 9:43 PM | Permalink | Reply to this

Re: Supercategories

Simon wrote:

Are there many 2\mathbb{Z}_2-graded things that you meet in everyday life that aren’t super?

Not me. But, I was surprised and pleased to meet one recently: it turns out that a homogeneous space G/HG/H deserves to be called a symmetric space if the group GG is equipped with an involution σ:GG, \sigma : G \to G, σ 2=1 \sigma^2 = 1 whose fixed points are precisely the subgroup HH. This makes the Lie algebra of GG into a /2\mathbb{Z}/2-graded Lie algebra for which the Lie algebra of HH is the even part. So, a major step in classifying symmetric space is taking Lie algebras (simple ones, say) and finding ways to make them /2\mathbb{Z}/2-graded.

I can’t help but wonder if this has something to do with bosons and fermions.

PS - As I get more interested in number theory I become more aware that 2\mathbb{Z}_2 is used for the 2-adic integers, so I now use /2\mathbb{Z}/2 for the integers mod 2. Besides, it often makes sense to pronounce // as ‘mod’. But, I’m not stuffy enough to use /2\mathbb{Z}/2\mathbb{Z}.

PPS - As a fun puzzle for math fiends: given any group GG, give a cute classification of the ways you can take the usual monoidal category of GG-graded vector spaces and make it symmetric. (As Urs notes, for G=/2G = \mathbb{Z}/2 there are two — the vanilla way, and the super way.)

Posted by: John Baez on July 17, 2007 7:40 PM | Permalink | Reply to this

Re: Supercategories

As a fun puzzle for math fiends: given any group G, give a cute classification of the ways you can take the usual monoidal category of G-graded vector spaces and make it symmetric.

I’ve seen it written that for abelian groups, the braidings on Vect[G]Vect[G] are classified by H ab 3(G, ×)H^3_ab (G, \mathbb{C}^\times), the third Eilenberg-Maclane abelian cohomology group. This group is apparently in bijection with the quadratic forms on GG.

I’m not sure how requiring the braiding to be symmetric influences any of this.

I don’t know about this Eilenberg-Maclane cohomology, so I’d be interested in any explanations you might have.

Posted by: Bruce Bartlett on July 17, 2007 8:53 PM | Permalink | Reply to this

Re: Supercategories

This stuff is explained pretty nicely in Joyal and Street’s paper on braided monoidal categories — probably best in this version that was never published:

I’ll just summarize. Suppose you have a skeletal category CC with the group GG as its set of objects and the abelian group AA as the group of endomorphisms of any object.

Suppose you want to make CC into a monoidal category where the tensor product of objects is given by multiplication in GG. Then the ways to do it are classified by the cohomology group

H 3(K(G,1),A) H^3(K(G,1), A)

where K(G,1)K(G,1) is the 1st Eilenberg–Mac Lane space of GG: the space whose only nonvanishing homotopy group is π 1=G\pi_1 = G.

Next, suppose you want to make CC into a braided monoidal category, where again the tensor product of objects is given by multiplication in GG. Now the ways to do it are classifed by

H 4(K(G,2),A) H^4(K(G,2), A)

where K(G,2)K(G,2) is the 2nd Eilenberg–Mac Lane space of GG: the space whose only nonvanishing homotopy group is π 2=G\pi_2 = G. Note: now GG needs to be abelian for any of this stuff to work!

If you have the tao of categorification in your bones, you can guess what’s next:

Next, suppose you want to make CC into a symmetirc monoidal category, where again the tensor product of objects is given by multiplication in GG. Now the ways to do it are classifed by

H 5(K(G,3),A) H^5(K(G,3), A)

where K(G,3)K(G,3) is the 3rd Eilenberg–Mac Lane space of GG: the space whose only nonvanishing homotopy group is π 3=G\pi_3 = G. Again, GG needs to be abelian.

At this point, it stabilizes:

H n+2(K(G,n),A)H n+3(K(G,n+1),A) H^{n+2}(K(G,n), A) \cong H^{n+3}(K(G,n+1), A)

when nn hits 3. This follows from the Freudenthal Suspension Theorem, and you can get a feel for it if you ponder the periodic table of nn-categories.

The first cohomology group I mentioned,

H 3(K(G,1),A) H^3(K(G,1), A)

is just another name for H 3(G,A)H^3(G,A) — you can compute it purely algebraically, without knowing about Eilenberg–Mac Lane spaces.

The higher ones I mentioned can also be computed in a purely algebraic way — and this is in fact what Eilenberg and Mac Lane figured out how to do in their famous series of papers on the bar construction!

However, you can compute them yourself if you know a bit about braided and symmetric monoidal categories, using the stuff I just said. Or cheat and read Joyal–Street.

As you might guess, all this is part of a much bigger pattern. Some of it is explained in this paper:

Posted by: John Baez on July 18, 2007 4:26 PM | Permalink | Reply to this

Re: Supercategories

Thanks for the explanation!

I would like to see the next simplest examples spelled out explicitly:

what are all the symmetric braided monoidal structures on Vect[ n]\mathrm{Vect}[\mathbb{Z}_n] for low nn?

I understand that you gave all the information to look up and/or work out the answer myself. Maybe I will, as soon as I find the time. But if somebody has some quick hints, I’d greatly enjoy it.

Posted by: Urs Schreiber on July 20, 2007 9:28 PM | Permalink | Reply to this

Re: Supercategories

given any group GG, give a cute classification of the ways you can take the usual monoidal category of GG-graded vector spaces and make it [braided and] symmetric.

I was wondering about this question while thinking about this stuff here. When trying to understand what things really are like, it often helps to lift all degeneracies and consider something more complicated than what one is really interested in. Sometimes simplicity of structures makes things harder to understand.

So, here I have the feeling it would be easier to understand the super-world if we could do everything not just for 2\mathbb{Z}_2 but for general (finite, I guess) groups.

Another aspect which I am struggling with is the role vector spaces play in the super-world: are they essential or do they just hide a more fundamental structure at work in the background?

This is akin to the question in your Tale of Gropoidification.

I feel this urge to rid my world of vector spaces. Even though I know this doesn’t make my life easier…

Posted by: Urs Schreiber on July 18, 2007 12:13 PM | Permalink | Reply to this

Re: Supercategories

While hanging around in Conil de la Frontera (from where I sent my last telegram to the nn-café), I did think a little (just a little) more about supercategories and all that. Here are some comments.

One major guiding light in this business is that we need to reproduce the following fact:

Morphisms from the superpoint to any ordinary space form the (odd) tangent bundle of that space.

I had made some comments on this here before (in other threads), but now it feels like I am better understanding what’s going on.

First: what is the tangent bundle, really?

Let CC be any category. It’s tangent space at any object xObj(C)x \in \mathrm{Obj}(C), which I’ll write T x(C) T_x(C) ought to be the category whose objects are morphisms in CC starting at xx and whose morphisms are commuting triangles.

The entire “tangent bundle” of CC is then the disjoint union of all these categories TC:= xObj(C)T xC. T C := \oplus_{x \in \mathrm{Obj}(C)} T_x C \,.

We want to find a notion of supercategories such that with CC regarded as a supercategory in the trivial way, and with pt\mathbf{pt} the superpoint, we have TC=Hom(pt,C). T C = \mathrm{Hom}(\mathbf{pt},C) \,.

And we want this to be such that it generalizes seamlessly to nn-categories.

I think it works like this:

Let pt:={} \mathrm{pt} := \{\bullet\} be the ordinary point, i.e. the category with a single object and a single morphism.

Then let pt:={} \mathbf{pt} := \{ \bullet \to \circ\} be the the category with two objects and one nontrivial morphism going between these. Think of \bullet as the body and of \circ as the soul of the point.

(I don’t like this terminology, nor much of the rest of the “super”-terminology, but it is standard and I am mentioning it in order to help follow the modelling process.)

(Later I’ll require this morphism to be an isomorphism. This will make pt\mathbf{pt} a category with nontrivial 2\mathbb{Z}_2-flow.)

The canonical inclusion {} {} \array{ \{\bullet\} \\ \downarrow^\subset \\ \{\bullet \to \circ\} } is crucial for everything to follow.

Thinking of the ordinary category CC as a supercategory in the trivial way means looking at the obvious inclusion C = C \array{ C \\ \downarrow^= \\ C }

(You know, in the silly standard terminology we’d say that CC has no soul. Poor CC.)

So, a morphism from the superpoint to CC is a morphism ptC \mathbf{pt} \to C which covers a morphism ptC \mathrm{pt} \to C in that pt C pt C. \array{ \mathrm{pt} &\to& C \\ \downarrow^\subset && \downarrow^\subset \\ \mathbf{pt} &\to& C } \,. Moreover, a 2-morphism between such morphisms is a 2-morphism pt C \array{ & \nearrow \searrow \\ \mathbf{pt} &\Downarrow& C \\ & \searrow \nearrow } which vanishes when pulled back to pt\mathrm{pt}: pt pt C =ptC. \array{ &&& \nearrow \searrow \\ \mathrm{pt} &\to& \mathbf{pt} &\Downarrow& C \\ &&& \searrow \nearrow } \;\; = \;\; \mathrm{pt} \to C \,.

This way we indeed get that TC=Hom(pt,C). T C = \mathrm{Hom}(\mathbf{pt},C) \,.

And it is obvious how to generalize this to CC an arbitrary nn-category.

Here is how we reproduce the ordinary tangent bundle of an ordinary manifold XX this way:

we need to model XX as a category. Take it to be the 2-groupoid P 2(X) P_2(X) whose objects are the points of XX, whose morphisms are piecewise smooth parameterized paths in XX and whose 2-morphisms are thin homotopy classes of homotopies between all those paths whose tangent vectors at source and target coincide.

Then (TP 2(X)) =TX, (T P_2(X))_\sim = T X \,, where () (\cdot)_\sim here denotes dividing out 2-isomorphisms.

That’s also the motivating idea behind the concept of tangent space here: we take all morphisms emanating at a given object and then let the 2-morphisms between them identify all those which “have the same tangent” at that object.

All this happens to tie in nicely with the stuff about INN(G)\mathrm{INN}(G), see The Inner Automorphism 3-Group of a Strict 2-Group, and in fact answers at least in part the question I was posing in a comment to that:

Let GG be an ordinary group. Then the “tangent space” of the category ΣG\Sigma G at the single object \bullet is T (ΣG)=T Id ΣG(Hom(Cat))=INN(G). T_\bullet (\Sigma G) = T_{\mathrm{Id}_{\Sigma G}} (\mathrm{Hom}(\mathrm{Cat})) = \mathrm{INN}(G) \,.

The structure of INN(G)\mathrm{INN}(G) as a groupoid is manifest from its realization as T (ΣG)T_\bullet (\Sigma G), while the monoidal structure on it (the 2-group structure) is manifest from its realization as T Id ΣG(Hom(Cat))T_{\mathrm{Id}_{\Sigma G}} (\mathrm{Hom}(\mathrm{Cat})).

For G 2G_2 any strict 2-group, we have T (ΣG 2)=INN(G 2)T Id ΣG 2(Hom(2Cat)). T_\bullet (\Sigma G_2) = \mathrm{INN}(G_2) \subset T_{\mathrm{Id}_{\Sigma G_2}} (\mathrm{Hom}(2\mathrm{Cat})) \,.

The idenitification INN(G 2)=T (ΣG 2) \mathrm{INN}(G_2) = T_\bullet (\Sigma G_2) is a quick way to encode all the pictures that David Roberts and I have in that paper. The inclusion INN(G 2)T Id ΣG 2(Hom(2Cat)) \mathrm{INN}(G_2) \subset T_{\mathrm{Id}_{\Sigma G_2}} (\mathrm{Hom}(2\mathrm{Cat})) is a quick way to understand the monoidal structure on INN(G 2)\mathrm{INN}(G_2).

Posted by: Urs Schreiber on July 16, 2007 6:23 PM | Permalink | Reply to this

Re: Supercategories

I realize a stupid typo in my comment above: where it has Hom(nCat) \mathrm{Hom}(n\mathrm{Cat}) I of course mean Mor(nCat) \mathrm{Mor}(n\mathrm{Cat}) the category whose objects are nn-functors, whose morphisms are natural transformations etc.

And I should have maybe emphasized more how those “tangent categories at identity functors” reproduce the categorical flows of vector fields, that I was talking about in What is a Lie derivative, really?

Posted by: Urs Schreiber on July 18, 2007 12:22 PM | Permalink | Reply to this

Re: Supercategories

I’m interested in the question of whether the tangent bundle of a category defined in this manner has the same homotopy type as the category.


I must admit this interpretation came as a pleasant surprise to me, because I was trying to generalise the notion of universal bundle for a (many object) groupoid to a universal bundle for a (many object) 2-groupoid and the diagrammatic calculus in Urs’ and my paper was a way to remove any reliance on the 2-groupoid in question being a 2-group (i.e. having one object). This full result is unfortunately still running around loose inside my head and I need to capture it ;)


Posted by: David Roberts on July 20, 2007 9:14 PM | Permalink | Reply to this
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