So this paper of mine was recently accepted by the Journal of Homotopy and Related Structures (the version that was accepted should be reflected on the arXiv by tomorrow – i.e. July 10 – I’m not sure about the journal ). It’s been a while since I sent out the earliest version, and most of the changes have involved figuring out who the audience is, and consequently what could be left out. I guess that’s a side-effect of taking an excerpt from my thesis, which was much longer. In any case, it now seems to have reached a final point. Some of what was in it – the section about cobordisms – is now in a paper (in progress) about TQFT. I don’t see anywhere else to include the other missing bit, however, which has to do with Lawvere theories, and since I just wrote a bunch about MakkaiFest, I thought I might include some of that here.

The paper came about because I was trying to write my thesis, which describes an extended TQFT as a 2-functor (and considers how it could produce a version of 3D quantum gravity). The 2-functor

$Z_G : nCob_2 \rightarrow 2Vect$

(or into $2Hilb$) is an ETQFT. The construction of the 2-functor uses the fact that you can get spans of groupoids out of cospans of manifolds – and in particular, out of cobordisms. One problem is how to describe $nCob_2$ so that this works. It’s actually most naturally a cubical 2-category of some kind. The strict version of this concept is a double category – which has (in principle separate) categories of horizontal and vertical of morphisms, as well as square 2-cells. Ideally, one would like a “weak” version, where composition of squares and morphisms can be only weakly associative (and have weak unit laws). A “pseudocategory” implements this where the only higher-dimensional morphisms are the squares, but it turns out to be strict in one direction, and weak in the other. As it happens, it’s a big pain to use only squares for the 2-morphisms.

Initially it seemed I would have to define a whole new structure to get weak composition in both directions, because in both directions, composition represents gluing bits of manifolds together along boundaries – using a diffeomorphism (or a smooth homeomorphism, depending on which kind of manifolds we’re dealing with). I called it a “double bicategory” and started trying to define it along the same lines as a double category. It then turned out that Dominic Verity had already defined a “double bicategory” – you can read the paper where I talk about how the notions are related. Here I want to talk about a few aspects which I cut out of the paper along the way.

The idea is that there are two ways of “categorifying”: internalization, and enrichment. A bicategory is a category enriched in $Cat$, the category of categories – for any two elements, there’s a whole hom-category of morphisms (and 2-morphisms). A double category is a category internal to $Cat$. This means you can think of it as a category of objects and a category of morphisms, equipped with functors satisfying all the usual properties for the maps in the definition of a category: composition functors, unit functors, and so forth. This definition turns out to be equivalent to the usual one. So I thought: why not do the same with bicategories?

Thus, the way I defined double bicategory was: “A bicategory internal to $Bicat$“. In the paper as it stands, that’s all I say. What I cut out was a sort of dangling loose end pointing toward Lawvere theories – or rather, a variant thereof – finite limit theories (for something more detailed, see this recent paper by Lack and Rosicky). As I mentioned in the previous post, a Lawvere theory is an approach to universal algebra – it formally defines a kind of object (e.g. group, ring, abelian group, etc.) as a functor from a category $T$ which is the “theory” of such objects, while the functor is a “model” of the theory.

What makes it “universal” algebra is that it can involve definitions with many sorts of objects, many operations, given as arrows, of different arities (number of inputs and outputs). This last makes sense in the monoidal context, and in particular Cartesian. Making decisions like this – what class of categories and functors we’re dealing with – specifies which doctrine the theory lives in. In the case of bicategories, this is the doctrine of categories with finite limits. In a Lawvere theory in the original sense, the doctrine is categories with finite products – so if there’s an object $G$, there are also objects $G^n$ for all $n$. Then there are things like multiplication maps $m : G^2 \rightarrow G$ and so on. For a category or bicategory, multiplication might be partial – so we need finite limits. A model of a theory in this doctrine is a limit-preserving functor.

So what does the theory of bicategories look like? It’s easy enough to see if you think that a (small) bicategory is a “bicategory in $Sets$“, and reproduce the usual definition, omitting reference to sets. It has objects $Ob$, $Mor$, and $2Mor$. (This fact already means this is a “multi-sorted” theory, which goes beyond what can be done with another approach to universal algebra based on monads). Funthermore, there are maps between these objects, interpreted as source, target, and identity maps of various sorts. These form diagrams, and since we’re in a finite limit theory, there must be various objects like $Pairs = Mor \times_{Ob} Mor$ which for sets would have the interpretation “pairs of composable morphisms”. Then there’s a composition map $\circ : Pairs \rightarrow Mor$… and so on. In short, in describing the axioms for a bicategory in a “nice” way (i.e. in terms of arrows, commuting diagrams, etc.), we’re giving a presentation of a certain category, $Th(Bicat)$, in generators and relations. Then a model of the theory is a functor $Th(Bicat) \rightarrow \mathcal{C}$ – picking out a “bicategory in $\mathcal{C}$“.

Now, a bicategory in $Sets$ is a bicategory. But a bicategory in $Bicat$ is another matter. First of all, I should say there’s something kind of odd here, since $Bicat$ is most naturally regarded as a tricategory. However, we can regard it as a category by disregarding higher morphisms and taking 2-functors only up to equivalence to make $Bicat$ into an honest category with associative composition. Thus, if we have a functor $F : Th(Bicat) \rightarrow Bicat$, we have:

• Bicategories $F(Ob)$, latex $F(Mor)$, and $F(2Mor)$
• 2-Functors $F(s)$, $F(\circ)$ and so on
• satisfying conditions implied by the bicategory axioms

But each of those bicategories (in $Sets$!) has sets of objects, morphisms, and 2-morphisms, and one can break all the functors apart into three collections of maps acting on each of these three levels. They’ll satisfy all the conditions from the axioms – in fact, they make three new bicategories. So, for example, the object-sets of the bicategories $F(Ob)$, $F(Mor)$ and $F(2Mor)$ form a bicategory using the object maps of the 2-functors $F(s)$ and so on.

So if we say the original bicategories $F(Ob)$ and so on are “horizontal”, and these new ones are “vertical”, we have something resembling a double category, but weak (since bicategories are weak) in both directions. The result is most naturally a four-dimensional structure (the 2-morphisms in $2Mor$ are most conveniently drawn as 4d, which is shown in Table 2 of the paper).

Now, the paper as it is describes all this structure without explicitly mentioning the theory $Th(Bicat)$ except in passing – one can define “internal bicategory” without it. This is why this is a “loose end” of this paper: a major benefit of using Lawvere-style theories is the availability of morphisms of theories, which don’t come up here.

In any case, with this 4D structure in hand, what I do in the paper is (a) get some conditions that allow one to decategorify it down to Verity’s version of “double bicategory” (and even down to a bicategory); and (b) show that couble cospans are an example (double spans would do equally well, but the application is to cobordisms, which are cospans). My own reason for wanting to get down to a 2D structure is the application to extended TQFT, which means we want a 2-category of cobordisms, thought of in terms of (co)spans.

Maybe in a subsequent post I’ll talk about the example itself, but one point about internalization does occur to me. Double cospans give an example of a double bicategory in the sense above – a strict model of $Th(Bicat)$ in $Bicat$. In fact, they consist of “(co)spans of (co)spans” in a way that Marco Grandis formalized in terms of powers $\Lambda^n$, where $\Lambda$ is the diagram (i.e. category) $\bullet \leftarrow \bullet \rightarrow \bullet$. One can actually think of this in terms of internalization: these are spans in a category whose objects are spans in $\mathcal{C}$, and whose morphisms are triples of maps in $C$ linking two spans (likewise for the span-map 2-morphisms). Yet it’s manifestly edge-symmetric: both the horizontal and vertical bicategories are the same.

As I mentioned in the previous post, there are lots of nice examples of double categories which are not edge-symmetric – sets, functions, and relations; or rings, homomorphisms, and bimodules, say. In fact, the second is only a pseudocategory – weak in one direction (composition of bimodules by tensor product is really only defined up to isomorphism). This is a significant thing about non-edge-symmetric examples. There’s much less motive for assuming both directions are equally strict. It’s also more natural in some ways: a pseudocategory is a weak model of $Th(Cat)$ in $Cat$ – equations in the theory are represented by (coherent) isomorphisms. This is the most general situation, and a strict model is a special case.

In the bicategory world, as I said, $Bicat$ is a tricategory, so weaker models than the one I’ve given are possible – though they’re not symmetric, and so while one direction has composition and units as weak as a bicategory, the other direction will be weaker still. Robert Paré, in a conversation at MakkaiFest, suggested that a nice definition for a cubical n-category might have each direction being one step weaker than the previous one – a natural generalization of pseudocategories. Maybe there’s a way to make this seem natural in terms of internalization? One can iterate internalizing: having defined double bicategories, collect them together and find models of $Th(Bicat)$ in $DblBicat$, and so forth. Maybe doing this as weakly as possible would give this tower of increasing weakness.

Now, I don’t have a great punchline to sum all this up, except that internalization seems to be an interesting lens with which to look at cubical n-categories.