## December 7, 2011

### Spans in 2-Categories: A Monoidal Tricategory

#### Posted by Alexander Hoffnung

A while back I posted a draft of a paper on the construction of a monoidal tricategory of spans. Since then the paper has changed quite a bit, and a new version appeared on the arXiv on Monday.

The following theorem is the main result of the paper:

Given a strict $2$-category $\mathcal{C}$ with finite pseudolimits, there is a monoidal tricategory $Span(\mathcal{C})$, consisting of:

• the same objects as $\mathcal{C}$,
• spans in $\mathcal{C}$ as $1$-morphisms,
• maps of spans in $\mathcal{C}$ as $2$-morphisms,
• maps of maps of spans in $\mathcal{C}$ as $3$-morphisms.

I usually think of spans as categorified linear operators, for example, the spans of groupoids in groupoidification. Also, spans, or more specifically, correspondences, are central in convolution operations in geometric representation theory. Here on the blog we have seen this in Ben-Zvi’s notes on geometric function theories. Of course, spans generalize relations between sets, so they show up in many other contexts as well.

This theorem has not changed from the previous version, although some changes to the structure of $Span(\mathcal{C})$ have been made. I want to comment on some of the changes here. In particular, I want to highlight the inclusion of Todd Trimble’s definition of a tetracategory.

The main changes are to the monoidal structure on the tricategory $Span(\mathcal{C})$. The structure in the previous version was too strict, so the new version corrects for this by specifying additional structure. Of course, this additional structure should satisfy coherence axioms as part of the monoidal structure. The result is a big change from the previous version in which the axioms were trivially satisfied.

In the paper, I define a monoidal tricategory as a one-object tetracategory. The coherence axioms to be satisfied are then Todd’s tetracategory axioms. It took me some time to understand the perturbations and other top-dimensional cells in the axioms well enough to write down the components of the span construction.

Todd was nice enough and patient enough to explain his method for writing down the tetracategory axioms to me. I have tried to pass along some of what Todd explained to me in this paper. Along with these “CliffsNotes” (apparently no longer “Cliff’s Notes”) for tetracategories, I have included the tetracategory definition itself, hopefully making it more accessible. If you have forgotten or have not seen it before, the handwritten definition, which currently lives on John Baez’s website, is 56 pages long!

Comments are very welcome.

Posted at December 7, 2011 2:14 AM UTC

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### Re: Spans in 2-Categories: A monoidal tricategory

I’d like to say publicly how grateful I am to Alex for all the hard work he poured into this. Over the course of a few weeks, he asked me a bunch of really good questions about that document on JB’s website, and the result is, I think, a much more readable definition now that it’s passed through Alex’s hands. It’s the most in-depth conversation I’ve had about tetracategories with anyone.

I’m sure we’ll be having more discussions. I sure hope so!

Posted by: Todd Trimble on December 7, 2011 12:31 PM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

So I have question. I know it is beyond the scope of what is in this paper, but perhaps Alex or Todd knows the answer. I should ask while this is fresh in your minds.

The homotopy hypothesis says that an n-groupoid is the same as an n-type. And so if you have a monoidal n-groupoid, where the objects are tensor-invertible, this should be the same as an (n+1)-type which is connected.

Similarly, we should have that a symmetric monoidal n-groupoid, with tensor invertible objects, should be the same as a stable n-type.

Now here is an interesting fact about stable homotopy types: they have an action of the sphere spectrum.

Now for low n, I know how to see this happening explicitly.

For example $\pi_1 \mathbb{S} = \mathbb{Z}/2$, and the action of this element means that whenever you have a symmetric monoidal groupoid (with invertible objects) you have a map which associates to each object an order 2 automorphism of the unit object.

You can see this explicitly. It comes from the self-braiding of that object.

$\pi_2 \mathbb{S} = \mathbb{Z}/2$, as well. This generator is the square of the element in degree one, and so the action there is basically derived from previous one. It is more-or-less the self-braiding of the self-braiding. =)

But for symmetric monoidal 3-groupoids something new is supposed to happen! $\pi_3 \mathbb{S} = \mathbb{Z}/24$. So for each object in a symmetric monoidal 3-groupoid (with invertible objects) we should be able to take that object and construct an automorphism of the identity of the identity of the unit object which is order 24!

I don’t know how to see this explicitly, and I was hoping that maybe, with your explicit hands-on knowledge, you might be able to see this phenomenon in action?

The most interesting part of this, I think, is the 3-torsion. I would be thrilled to even see an order 3 automorphism popping up somehow!

Posted by: Chris Schommer-Pries on December 9, 2011 3:26 PM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

Chris,

to compare to monoidal $n$-groupoids you need to consider pointed connected homotopy types. I don’t know for sure if, when you are considering symmetric monoidal $n$-groupoids, you need pointed stable homotopy types as well, but this seems to me to be necessary given the Cheng-Gurski results.

I don’t know what effect this would have on the action of the sphere spectrum, and whether this cuts down on the automorphisms available.

Posted by: David Roberts on December 11, 2011 11:44 PM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

All stable homotopy types, in the sense of the term “stable” used in algebraic topology, are pointed.

Posted by: Mike Shulman on December 12, 2011 5:25 AM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

Well, that blows that idea out of the water :)

Posted by: David Roberts on December 12, 2011 6:09 AM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

Chris wrote:

But for symmetric monoidal 3-groupoids something new is supposed to happen! $\pi_3 \mathbb{S} = \mathbb{Z}/24$. So for each object in a symmetric monoidal 3-groupoid (with invertible objects) we should be able to take that object and construct an automorphism of the identity of the identity of the unit object which is order 24!

Indeed, this is one of my never-attained holy grails.

I love all the appearances of the numbers 12 and 24 in the theory of modular forms, bosonic string theory, $\zeta(-1)$ and so on, and how they’re all interconnected. And as you probably know, they’re also connected to

$\pi_3 (\mathbb{S}) = \mathbb{Z}/24$

I explained this back in week102.

However, Alex and Todd haven’t got symmetric monoidal tricategories up and running yet, so we can’t work stably. The first hint of 24-ness in the homotopy groups of spheres is

$\pi_6 (S^2) = \mathbb{Z}/12$

So, it would be nice to understand the homotopy 6-type of the 2-sphere in a category-theoretic way. This should be the free 6-groupoid on a 2-morphism $f: 1_\ast \Rightarrow 1_\ast$. In other words, it should be the free braided monoidal 4-groupoid on one object.

So, it would be nice to understand braided monoidal tetracategories!

By the way, we also have

$\pi_6 (S^3) = \mathbb{Z}/12$

and this is more clearly connected to

$\pi_3 (\mathbb{S}) = \mathbb{Z}/24$

since 6 is 3 more than 3. However, I bet this $\mathbb{Z}/12$ is related to

$\pi_6 (S^2) = \mathbb{Z}/12$

somehow. I don’t know how.

Posted by: John Baez on December 16, 2011 6:59 AM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

The homotopy groups of $S^2$ and $S^3$ are the same because of the Hopf fibration, with fibre having homotopy only in dimension 1, hence the long exact sequence consists of isomorphisms.

Posted by: David Corfield on December 16, 2011 10:46 AM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

Oh, right!

Posted by: John Baez on December 17, 2011 8:05 AM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

I love this stuff too. I was so excited when I first realized how to see the Hopf element in the homotopy groups of spheres as coming from the braiding in a monoidal category. (You can see the unstable $\pi_3(S^2)=\mathbb{Z}$ in this way as well, of course, since in that case the braiding is not self-inverse.)

At the risk of sounding like a broken record, I’d like to point out that we don’t really need symmetric monoidal tricategories or tetracategories to study this question. What we need is just some “algebraic” viewpoint on $\infty$-groupoids which would enable us to describe things like braidings and syllepses, as well as to describe spheres as “freely generated” by certain cells.

Such a thing is exactly what we have in homotopy type theory. The types in homotopy type theory can be interpreted as homotopy types in Voevodsky’s univalent model (and, at least conjecturally, in other categories as well). But their “homotopy structure” is algebraic — paths and higher paths come with specified composition operations, braidings, and so on, all of which flow easily from the simple inductive definition of path types. (Formally, the structure gives rise to an $\omega$-groupoid in the sense of Batanin/Leinster.) Moreover, we also have a very nice way to define spheres as being freely generated by cells, using higher inductive types.

So far, I believe the state of the art in this direction are proofs that $\pi_1(S^1)=\mathbb{Z}$ (here) and that $\pi_n$ is abelian for $n\ge 2$ (here). But I also think the stage is set for rapid development. Also, merely exhibiting elements of homotopy groups of spheres should be easier than calculating the whole homotopy group, so I think it’s entirely within the realm of possibility to write down a 12-torsion element of $\pi_6(S^2)$ or $\pi_6(S^3)$. The way I would start is, of course, to go back to topology and see where these elements come from there, then try to interpret that algebraically.

Posted by: Mike Shulman on December 16, 2011 4:54 PM | Permalink | PGP Sig | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

Michael wrote:

At the risk of sounding like a broken record, I’d like to point out that we don’t really need symmetric monoidal tricategories or tetracategories to study this question. What we need is just some “algebraic” viewpoint on $\infty$-groupoids which would enable us to describe things like braidings and syllepses, as well as to describe spheres as “freely generated” by certain cells.

I’m eager for everyone to attack this kind of problem using whatever framework they happen to like.

People should also start thinking about what they can do with this astounding fact: for absolutely anything in the universe, any automorphism of the identity of the identity of that thing has $\mathbb{Z}/12$ acting as automorphisms of its identity of its identity!

For example: we already know that this astounding fact is related to the fact that the Picard group of the moduli stack of elliptic curves is $\mathbb{Z}/12$. But it would be nice to understand the connection as directly as possible!

Posted by: John Baez on December 17, 2011 8:17 AM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

According to the old program of categorification, there was the thought that $n Braid_k$ being the free $k$-tuply monoidal $n$-category, that the former could help you with diagrammatic notation for instances of the latter, and the latter could provide invariants for the former.

So would your monoidal tricategories give invariants for certain 3-manifolds in a 4-cube?

Posted by: David Corfield on December 9, 2011 6:05 PM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

Yes, Alex’s monoidal tricategories, or indeed any monoidal tricategories, should give invariants of 3-dimensional braids in a 4-dimensional cube. However, I don’t believe these are any more interesting than 1-dimensional braids in a 2-dimensional square, which are just parallel lines. I believe they will be just parallel 3-planes in the 4-cube.

In general, I conjectured that the free $k$-tuply monoidal $n$-category on one object describes $n$-braids in an $(n+k)$-dimensional cube. I think this only starts getting interesting when the codimension $k$ reaches 2. Since there’s no duality here, these $n$-braids can’t ‘doubling back’ or ‘zig-zag’ in any way. All we have is the monoidal structure, braiding, syllepsis and so on on. When $k = 1$, all we have the monoidal structure, which is what allows us to stack a bunch of parallel hyperplanes.

However, the monoidal tricategory $Span(\mathcal{C})$ should have duals for objects. So, it should give us invariants for somewhat more interesting submanifolds (with corners) sitting in a 4-dimensional cube.

In fact, Mike Stay has taken Alex’s monoidal tricategory $Span(\mathcal{C})$ and decategorified it, obtaining a monoidal bicategory. He’s then rigorously proved that this monoidal bicategory has duals for objects. In other words, it’s ‘compact’. This result will appear in a paper he’s writing, which will also become part of his thesis.

It seems that in general Alex’s monoidal tricategory $Span(\mathcal{C})$ does not have duals for morphisms. This is because the morphisms are maps of spans. There’s no way, in general, to turn such maps around. But I bet if the morphisms in $\mathcal{C}$ have duals, so do the morphisms in $Span(\mathcal{C})$.

I now like the idea of a monoidal tricategory with:

• the same objects as $\mathcal{C}$,
• spans in $\mathcal{C}$ as $1$-morphisms,
• spans between spans in $\mathcal{C}$ as $2$-morphisms,
• maps of spans of spans in $\mathcal{C}$ as $3$-morphisms.

This should have duals for morphisms as well as objects.

Jeffrey Morton and Jamie Vicary have done some truly amazing things with the monoidal bicategory obtained by decategorifying this sort of monoidal tricategory, in the special case where $\mathcal{C} = Gpd$. Unfortunately I’m not at liberty to reveal them!

Posted by: John Baez on December 16, 2011 6:28 AM | Permalink | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

…spans between spans in 𝒞 as 2-morphisms, maps of spans of spans in 𝒞 as 3-morphisms.

If $\mathcal{C}$ is a 2-category, though, won’t you also need 2-morphisms between maps of spans of spans as 4-morphisms?

I guess this is a categorification of Batanin’s monoidal globular category of spans. And also a concrete low-dimensional version of the $(\infty,n)$-category with duals $Fam_n$ that Lurie sketches in section 3.2 of On the Classification of Topological Field Theories (he says “correspondences” to mean what we call “spans”)?

Continuing my broken-record streak (-: let me suggest that a good way to approach these things might be through categories with cubical and other non-globular shapes. The bicategory of spans in a 1-category underlies a “fibrant” pseudo double category of spans, i.e. a pseudo internal 1-category in $1Cat$. Its monoidal structure is easy to get at in this way, since pseudo double categories form merely a 2-category (rather than a 3-category, like bicategories do) and a pseudomonoidal structure on a fibrant pseudo double category, as an object of this 2-category, gives rise to a monoidal structure on its underlying bicategory.

Similarly, the tricategory of spans and spans-of-spans in a 1-category underlies a fibrant pseudo triple category, constructed for instance here. They don’t consider fibrancy and lifting of monoidal structure, but it ought to work in the same way.

If we start from a 2-category, then we could expect to have to work 2-categorically all the way, but no more. Thus, the tricategory of spans in a 2-category should underlie an internal category in $2Cat$, while the tetracategory of spans and spans-of-spans in a 2-category should underlie a double-category-object in $2Cat$. (I would call these respectively a $(1\times 2)$-category and a $(1\times 1\times 2)$-category — an $(n\times k)$-category is an internal $n$-category object in $k Cat$ and so on.) This way the construction of a symmetric monoidal tricategory or tetracategory can be “factored” into a first construction which exists at no higher a category level than the category you started from, followed by a second construction which is abstract and general and completely determined by unique liftings.

Posted by: Mike Shulman on December 16, 2011 5:20 PM | Permalink | PGP Sig | Reply to this

### Re: Spans in 2-Categories: A monoidal tricategory

Michael wrote:

If $\mathcal{C}$ is a 2-category, though, won’t you also need 2-morphisms between maps of spans of spans as 4-morphisms?

Yeah, I guess so.

But in fact, all the good stuff Jeffrey Morton and Jamie Vicary are doing seems to require only the mere bicategory obtained by decategorifying this tetracategory twice. So for now, I’d happy with whatever is the quickest way to get ones hands on that.

Continuing my broken-record streak (-: let me suggest that a good way to approach these things might be through categories with cubical and other non-globular shapes.

Indeed! As you may know, Jeffrey likes cubical ways of thinking about spans-of-spans.

Old age arrives not when you start repeating yourself, but when you stop caring.

Posted by: John Baez on December 17, 2011 7:56 AM | Permalink | Reply to this
Read the post Compact Closed Bicategories
Weblog: The n-Category Café
Excerpt: Mike Stay has a new paper on compact closed bicategories and their many applications.
Tracked: June 6, 2012 2:08 AM

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