The n-Category Café

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.

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

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.

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.

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!

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!

…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.