The n-Category Café

One of the neat observations in the formal theory of monads is that a monad in the bicategory of spans is simply a category. If you’ve never seen this before, I encourage you to sit down and work it out.

Unfortunately, this fact is not fully satisfying to a category theorist (at least, not to this one), because it doesn’t behave as you might expect with respect to morphisms. The abstract notions of lax, colax, and strong monad morphisms do not specialize in $Span$ to any familiar sort of morphism between categories. The best you can do is to identify functors with colax monad morphisms in $Span$ whose underlying span is (induced by) a function. The situation with 2-cells is even worse: monad 2-cells between such monad morphisms are just equalities, there are no natural transformations in sight.

In “The formal theory of monads, II”, Lack and Street resolved this latter problem by considering, in place of the 2-category $Mnd(K)$ of monads in a 2-category $K$, the free cocompletion $KL(K)$ of $K$ under Kleisli objects for monads. It turns out that the morphisms in $KL(K)$ are again just colax monad morphisms in $K$, but its 2-cells are somewhat laxer, in just such a way as to yield natural transformations when interpreted in $Span$.

However, we still have the problem that not every colax monad morphism in $Span$ is a functor. If you know about proarrow equipments, you may notice that there’s clearly something equipment-y going on. That is, $Span$ is not just a bicategory, but an equipment — it has two different classes of morphisms between its objects, one “looser” than the other — and the condition for a colax monad morphism to be a functor is that its underlying morphism should be “tight” (i.e. a function rather than merely a span). So you might think, maybe we should take the free cocompletion “in the world of equipments”. One way to make sense of this is to regard an equipment as a (weak) F-category, and use an $\mathcal{F}$-enriched free cocompletion. This is why we needed the theory of free cocompletions of enriched bicategories.

Now if we start from the $\mathcal{F}$-bicategory $Set \hookrightarrow Span$ and take its free cocompletion under an $\mathcal{F}$-enriched sort of Kleisli object, we get an $\mathcal{F}$-bicategory of categories whose tight morphisms are indeed functors, but whose loose morphisms are general colax monad morphisms in $Span$. This is better, but still not ideal; it’d be much nicer if we got out a completely familiar structure, rather than one only part of which is familiar. And there’s an obvious thing we might hope to get: the equipment of categories, functors, and profunctors (which was, in fact, the motivating example for the definition of equipments).

It turns out that we can get this if we improve the enrichment further. Instead of mere equipments, we need to work in the world of equipments whose bicategory of loose morphisms (spans or profunctors) has some local colimits (colimits in hom-categories preserved by composition). Having local colimits is a sort of categorification of being an Ab-enriched category, and like that, it can be expressed by enrichment over a suitable base. It also has similar consequences, e.g. it makes products and coproducts coincide, and it also makes Kleisli objects and Eilenberg-Moore objects coincide. And if we perform the free cocompletion of $Set\hookrightarrow Span$ under an appropriate sort of Kleisli object in this world, we get the proarrow equipment $Cat\hookrightarrow Prof$. This is what I meant by the “universal property of categories”. It also generalizes to lots of other sorts of categories, but for that you’ll have to read the paper (or at least the introduction).