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August 22, 2006

Lectures on n-Categories and Cohomology

Posted by John Baez

Here’s a paper that you n-category fanatics might enjoy - especially since it mentions a centaur, a faun-like thing, and a kid falling out of the back of a bus:

Lectures on n-Categories and Cohomology
John Baez and Michael Shulman

The goal of these talks was to explain how cohomology and other tools of algebraic topology are seen through the lens of n-category theory. Special topics include nonabelian cohomology, Postnikov towers, the theory of “n-stuff”, and n-categories for n = -1 and -2.

These talks were extremely informal, glossing over the difficulties involved in making certain things precise, just trying to sketch the big picture in an elementary way. It seemed useful to keep this informal tone in the notes. I cover a lot of material that seems hard to find spelled out anywhere, but nothing new here is due to me: anything not already known by experts was invented by James Dolan, Toby Bartels or Mike Shulman (who took notes, fixed lots of mistakes, and wrote the Appendix).

The talks were very informal, and so are these notes. A lengthy appendix clarifies certain puzzles and ventures into deeper waters such as higher topos theory. For readers who want more details, we include an annotated bibliography.

Posted at August 22, 2006 8:05 AM UTC

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Re: Lectures on n-Categories and Cohomology

I worked a lot today and feel tired now. In order to relax a bit I opened the above lecture notes and learned about (-1)-categories and (-2)-categories. (I had wanted to do this a while ago already.)

An nn-category is a Hom\mathrm{Hom}-thing of an (n+1)(n+1)-category.

Hence a (-1)-category is a Hom\mathrm{Hom}-thing of a 0-category.

And a (-2)-category is a Hom\mathrm{Hom}-thing of a -1-category.

We are being asked to figure out what a monoidal (-1)-categories is.

That’s not hard. At least not if you feel sufficiently relaxed about the meaning of the word “is”.

For me, the really hard part is to figure out what the difference between a monoidal (-1)-category and a (-2)-category is.

They shouldn’t be exactly the same, should they?

Hey, this blog is also about philosophy. So I am still on topic!!

As John discusses, calling the two possible (-1)-categories “true” and “false” or “equal” and “not equal” both seem to be reasonable choices.

Had I been asked for a name, though, I would probably have suggested “discrete category on 1-object set” and “discrete category on empty set”, instead.

After all, if we regard a 0-category SS as an \infty-category with all pp-morphisms identities for all pp, then Hom S(x,x)\mathrm{Hom}_S(x,x) is the \infty-category with a single object (namey the identity 1-morphism on xx) and all pp-morphisms identites. So it’s still an infinity-category!

Similarly Hom S(x,y)\mathrm{Hom}_S(x,y) for xyx \neq y. This is the \infty-category with no object.

That’s what my answer would have been.

However, with that answer I would have found that the unique (-2)-category is also an \infty-category with a single object and all pp-morphisms identities.

Same for monoidal (-1)-category. This is a 0-category with a single object, hence once again an \infty-category with a single object and all morphisms identies.

So it seems I am in trouble. I cannot distinguish between the unique monoidal 0-category, the unique (-2)-category and one of the two (-1)-categories.

However, from p.14 of the lecture notes mentioned above one finds that the experts distinguish the non-empty (-1)-category from the unique (-2)-category by using the distinction between TRUE and NECESSARILY TRUE.

Heh. It’s true that e πi=1e^{\pi i} = -1.

Or so I thought.

Maybe instead it is necessarily true??

How can I tell?

Is there a (-1)-category of solutions to the equation e πi=1e^{\pi i} = -1, or a (-2)-category. Or a monoidal (-1)-category?

OK, I go to bed now. Seems about time.

But before quitting, one more serious comment.

We use 1-functors to describe charged particles.

We use 2-functors to describe charged strings.

We use 3-functors to describe charged membranes.

We use 0-functors to describe charged instantons, aka (1)(-1)-branes with a 0-dimensional worldvolume.

Really, we do. At least some people do (\to).

Do we need (-1)-functors for anything?

Let’s see. By my way of looking at this situation, a (1)(-1)-functor is the same as a 0-functor on a monoidal 0-category.

OK, I can interpret that. That’s a charged instanton which is constrained to appear at a predescribed point in spacetime.

Not any point. A fixed point. The point being the single object in our monoidal 0-category.

All right, now it’s official. A D(2)D(-2)-brane is a D-instanton with predescribed point of appearance.

Maybe I should say it’s a D-instanton living in a 0-dimensional target space.

Yeah, that sounds right.

Posted by: urs on August 22, 2006 8:31 PM | Permalink | Reply to this

Re: Lectures on n-Categories and Cohomology

You sound a bit tired, sort of joking around, and I’m not sure how much you want serious replies to your questions. But, I feel the need to explain this stuff to everyone else.

Everything you say sounds right, but you may not have gotten a feeling for how (-1)-categories and (-2)-categories connect to logic. So, I want to say a tiny bit about that.

To review for the nonexperts… let’s figure out what (-1)-categories and (-2)-categories are. We’ll start with a couple of principles:

  • An n-category is a special sort of (n+1)-category.
  • A 0-category is a set.

So, whatever a (-2)-category is, it’s a special sort of (-1)-category, which in turn is a special sort of set.

From this viewpoint, (-2)-categories and (-1)-categories can’t be anything new or weird. They’re just certain special sets. As we’ll see, they’re certain incredibly famous sets. But, it will turn out to be handy to think of them other ways too.

To see what these sets are, we need another handy principle:

  • Given two objects x,y in an n-category, hom(x,y) is an (n-1)-category. Moreover, any (n-1)-category can arise this way.

So, given two objects x,y of a 0-category - that is, elements of a set! - hom(x,y) is a (-1)-category. Moreoever, any (-1)-category can arise this way.

What can hom(x,y) be like in this case? We need another principle:

  • If j > n, all j-morphisms in an n-category are identity morphisms.

So, if x and y are elements of a set, all the 1-morphisms in hom(x,y) are identity morphisms.

So, hom(x,y) has one morphism in it if x = y, and none in it otherwise.

So, viewed as a set, the (-1)-category hom(x,y) either has one element or none. Moreover, all (-1)-categories arise this way.

So, viewed as a set, a (-1)-category is just a set with 1 or 0 elements.

But, what’s nice is that we see Boolean logic showing up in n-category theory - tucked in at the bottom, down in the (-1)-categories. This “1” or “0” binary choice is really showing up because the elements x and y are either equal or not equal. So, we can - and should! - also think of the two (-1)-categories as being called true and false.

This may seem like a minor point, but Mike develops it into quite a grand one in the Appendix, where he considers the world of sets as a special case of the world of objects in a general topos - where instead of just two truth values, we have a whole Heyting algebra of truth values, as usual in intuitionistic logic.

The point is that just as topos theory lets us drastically generalize our notion of 0-categories (sets), it does the same for (-1)-categories (truth values). It also lets us generalize n-categories for higher values of n, which is even more exciting - but it’s fun, and important, to see how the simple roots of this tree of n-topos theory grow naturally into the fancy intricate branches.

Anyway, returning from topos theory to our classical logic of set theory, there are just two (-1)-categories in the universe: the empty set and the 1-element set. What about (-2)-categories?

Well, by one of our principles, we get these by choosing two objects x,y in a (-1)-category and looking at hom(x,y). We can’t choose two objects if our (-1)-category is the empty set, so let’s take the 1-element set. Then x = y and hom(x,y) has just the identity morphism in it… so hom(x,y) is again the 1-element set.

Moreover, all (-2)-categories arise this way.

So, there’s just one (-2)-category: the 1-element set.

We could call this equal or true. So you’re right, Urs: it’s just the same as the (-1)-category called “true” that we already saw. But it’s playing a somewhat different role now, because it expresses the fact that everything is necessarily equal to itself - this is necessarily true. We don’t have a binary choice at this level - we just have one choice. Instead of a bit of information, a (-2)-category gives no information at all.

As a little puzzle, I’ll ask: how many (-3)-categories are there, and what are they like?

You also seemed to feel funny about the fact that there’s just one monoidal (-1)-category, also named true. This might help you feel better:

  • For a 0-category to be monoidal (for a set to be a monoid) is extra structure - there’s a set of ways to make a set into a monoid.
  • For a 1-category to be monoidal is extra stuff - there’s a category of a ways to make a category into a monoidal category.
  • For a 2-category to be monoidal is extra 2-stuff - there’s a whole 2-category of ways to make a 2-category into a monoidal 2-category.

And so on… here we are using the yoga of properties, structure and stuff, with its extension to n-stuff.

So, it’s actually very nice that for a (-1)-category to be monoidal is a mere property - it either is or isn’t! In other words, there’s just a truth value of ways to (-1)-category monoidal!

Again, a small puzzle: what about monoidal (-2)-categories?

By the way, I never liked the use of the term p-brane to mean something whose extent in space is p-dimensional. It would be nicer to use p for the dimension of the entity in spacetime. Of course it’s too late to change it now, but anyway, then we’d have p-branes coupling to p-forms. 2-branes would be strings, 1-branes would be particles, and 0-branes would be… instantons!


Posted by: John Baez on August 23, 2006 8:29 AM | Permalink | Reply to this

Re: Lectures on n-Categories and Cohomology

I’m not sure how much you want serious replies to your questions

Sorry. I was in a silly mood, having worked too much.

But I am seriously interested in this.

Mike develops it into quite a grand one in the Appendix, where he considers the world of sets as a special case of the world of objects in a general topos - where instead of just two truth values, we have a whole Heyting algebra of truth values, as usual in intuitionistic logic.

Ah, I should have a look at the appendix then. This sounds very interesting.

While internal to the topos Set\mathrm{Set} (|n|-|n|)-categories are on the verge of being trivial, pretty much because sets are so “trivial”, this may change drastically as we look at (-n) categories in another topos, where sets are replaced with more fancy things.

As a little puzzle, I’ll ask: how many (-3)-categories are there, and what are they like?

The periodic table here stabilizes in the horizontal direction.

A (-3)-category (internal to Set\mathrm{Set}) is a Hom\mathrm{Hom}-thing of a (-2)-category. The latter is just the 1-element set {necTrue}\{\mathrm{necTrue}\}.

So the unique (-3)-category is

(1)End {necTrue}(necTrue)={Id necTrue}. \mathrm{End}_{\{\mathrm{necTrue}\}}(\mathrm{necTrue}) = \{ \mathrm{Id}_{\mathrm{necTrue}} \} \,.

Hence it is again a 1-element set.

Actually, I could write

(2)necTrue=Id true=Id Id x \mathrm{necTrue} = \mathrm{Id}_{\mathrm{true}} = \mathrm{Id}_{\mathrm{Id}_x}

And it continues this way. A (-4)-category is the 1-element set

(3){Id Id Id Id x}. \left\{ \mathrm{Id}_{\mathrm{Id}_{\mathrm{Id}_{\mathrm{Id}_{x}}}} \right\} \,.

As one can see, one use of (4)(-4)-categories internal to Set\mathrm{Set} is as a test for the MathML capabilities of your browser. ;-)

I never liked the use of the term pp-brane to mean something whose extent in space is pp-dimensional.

True. But it harmonizes well with another awkward counting system - namely that of gerbes.

Since what they call a 1-gerbe is already a 2-structure, they run into the problem that bundles are 00-gerbes, while functions are (-1)-gerbes.

Whenever you find yourself starting counting at (-1), you know you have chosen wrong conventions at some point.

But if you do it consistently, at least the indices match again.

So a 1-brane (string) couples to a 1-gerbe.

A 0-brane (point) couples to a 0-gerbe (bundle).

A (-1)-brane (instanton) couples to a (-1)-gerbe (function).

We should hence invent an alternative terminology for brane which fits into

nn-bundles : (n1n-1)-gerbes :: ?? : (n1)(n-1)-branes.

I suggest nn-particle.

nn-bundles : (n1n-1)-gerbes :: nn-particles : (n1)(n-1)-branes.

So a particle is a 1-particle is a point.

A string is a 2-particle.

A (-1)-brane is a 0-particle.

nn-particles couple to nn-bundles with connection. \Leftrightarrow (n1)(n-1)-branes couple to (n1)(n-1)-gerbes with connection.

Posted by: urs on August 23, 2006 9:55 AM | Permalink | Reply to this
Read the post Connes on Spectral Geometry of the Standard Model, I
Weblog: The n-Category Café
Excerpt: Connes is coming closer to the spectral triple encoding the standard model coupled to gravity. Part I: some background material.
Tracked: September 6, 2006 12:20 PM

Re: Lectures on n-Categories and Cohomology

I was looking again at the Lectures on nn-Categories and Cohomology to see if could find a hint towards the answer of the following question:

What’s an action nn-groupoid and how is it characterized by morphisms to an nn-group?

In week 249 John mentions that

Any groupoid with a faithful functor to GG is equivalent to the action groupoid X//GX//G for some action of GG on some set XX.

I like to think of it this way: I write BG\mathbf{B} G for the group GG regarded as a one-object groupoid and then notice that every action groupoid sits in a sequence of groupoids

XX//GBG, X \to X//G \to \mathbf{B} G \,,

where the X:=Disc(X)X := Disc(X) on the left is the discrete groupoid over XX (XX as objects, no nontrivial morphisms).

In particular, if X=GX=G and we have the right action of GG on itself we get

GG//GBG, G \to G//G \to \mathbf{B} G \,,

which is the groupoid version of the universal GG-bundle, realizing EG:=G//G\mathbf{E}G := G//G as the action groupoid of GG action on itself.

What I want to understand is: How does this generalize to higher nn?

Given an nn-group GG and writing BG\mathbf{B} G for its one-object nn-groupoid incarnation, which sequences of nn-groupoids

XX//GBG X \to X//G \to \mathbf{B}G

I am entitled to address as coming from action nn-groupoids?

I was looking at the lecture notes by John and Mike since they talk about the generalization of “faithful” to higher nn (e.g. p. 18).

There the observation of course is that for 1-functors, faithful means surjective on 2-morphisms (identities).

So I suppose it would make sense to call an nn-functor faithful if it is (n+1)(n+1)-surjective, i.e. surjective on equations between nn-morphisms. But does that lead to the right characterization of action nn-groupoids?

It looks intuitively all right, but one problem is that one would hope there to be a notion of \infty-action groupoid, but what’s (+1)(\infty+1)-surjective?

In principle it should be straightforward to work it out:

for GG an nn-group and CC an nn-category with a forgetful (hm…) nn-functor to (n1)Grpd(n-1)Grpd, we should look at actions of GG on objects in CC ρ:BGC \rho : \mathbf{B} G \to C then send this forward to nGrpdn Grpd

ρ^:BGρC(n1)GrpdnGrpd \hat \rho : \mathbf{B} G \stackrel{\rho}{\to} C \to (n-1)Grpd \hookrightarrow n Grpd

and then take a weak colimit there:

X//G:=colim BGρ. X//G := colim_{\mathbf{B} G}\rho \,.

That’s how it works for ordinary action 1-groupoids.

By the universal property of the colimit we canonically get the morphism

X//GBG X//G \to \mathbf{B} G

and one should check what characteizes the morphisms obtained this way. Such that one could say that for every nn-functor

VBG V \to \mathbf{B} G

with these and those properties, VV is equivalent to a colimit of the above kind.

Does anyone know what these properties are? It depends on what we take a “weak nn-colimit” to be.

I’d be most happy if we could do this in ωCat\omega Cat (strict \infty-categories). But whatever works, I’d be glad to see it.

Posted by: Urs Schreiber on March 25, 2008 11:48 PM | Permalink | Reply to this

Re: Lectures on n-Categories and Cohomology

If the ordinary action groupoid can be derived as a pullback of the forgetful functor from Pointed set to Set, might we not be able to think of an action nn-groupoid similarly?

I don’t think we ever took the step of properly working out action 2-groupoids, despite having in our hands the classifier, (Pointedcat) +Cat(Pointed cat)^+ \to Cat.

I tried to make a start here in a).

Posted by: David Corfield on March 26, 2008 10:05 AM | Permalink | Reply to this

Re: Lectures on n-Categories and Cohomology

the ordinary action groupoid can be derived as a pullback of the forgetful functor from Pointed set to Set

Thanks, David, I had forgotten about that. Didn’t properly follow that discussion back then anyway. I’ll go back to that now.

I tried to make a start here in a).

Great, thanks for reminding me.

When we form the pullback X//G (PointednCat) BG ρ nCat \array{ X//G &\to& (Pointed n-Cat) \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& n-Cat }

is it sufficient to talk about strict pullbacks? (Don’t ask me what i mean by “sufficient”.)

Do we know what (PointedωCat) (Pointed \omega Cat) might mean?

Notice I am not after computing these higher action groupoids right this moment. I want to figure out their best description first.

Posted by: Urs Schreiber on March 26, 2008 4:48 PM | Permalink | Reply to this

Re: Lectures on n-Categories and Cohomology

Well, given my comment in another thread I suppose I should answer this last question myself:

assuming all required pullbacks exist in (ωCat, Gray)Cat(\omega Cat,\otimes_{Gray})-Cat, I should form there the pullback diagram

T ptωCat Hom(2,ωCat) dom pt ωCat \array{ T_{pt}\omega Cat &\to& Hom(2,\omega Cat) \\ \downarrow && \downarrow^{dom} \\ pt &\to& \omega Cat }

as discussed in Tangent categories.

Then T ptωCat ωCat codom Hom(2,ωCat) \array{ T_{pt}\omega Cat &\to& \omega Cat \\ \searrow && \nearrow_{codom} \\ & Hom(2,\omega Cat) }

gives the projection down to ωCat\omega Cat and the “kernel” s 1pts^{-1}pt that hopefully exists. Then

s 1pt T ptωCat ωCat \array{ s^{-1} pt \\ \downarrow \\ T_{pt}\omega Cat \\ \downarrow \\ \omega Cat }

would be the universal ωCat\omega Cat-ω\omega-bundle.

Does that exist the way I am imagining here?

(Darn, I don’t even know if (ωCat, Gray)Cat(\omega Cat, \otimes_{Gray})-Cat) has all pullbacks.)

Posted by: Urs Schreiber on March 26, 2008 6:44 PM | Permalink | Reply to this

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