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January 16, 2013

The Universal Property of Categories

Posted by Mike Shulman

Finally, the third paper I promised about generalized category theory is out:

  • Richard Garner and Mike Shulman, Enriched categories as a free cocompletion, arXiv

This paper has two parts. The first part is about the theory of enriched bicategories. For any monoidal bicategory 𝒱\mathcal{V}, we (or, rather, Richard, who did most of this part) define 𝒱\mathcal{V}-enriched bicategories, assemble them into a tricategory, define 𝒱\mathcal{V}-profunctors (modules) and weighted bilimits and bicolimits, and construct free cocompletions under such. It’s all a fairly straightforward categorification of ordinary enriched category theory, but although a number of people have defined and used enriched bicategories in the past, I think this is the most comprehensive development so far of their basic theory.

The second part is an application, which uses the theory of enriched bicategories to describe a universal property satisfied by the construction of enriched categories. I’ll explain a bit further below the fold, but the introduction of the paper gives an equivalently good (and more detailed) summary. You can also have a look at these slides from Octoberfest 2012.

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 SpanSpan to any familiar sort of morphism between categories. The best you can do is to identify functors with colax monad morphisms in SpanSpan 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)Mnd(K) of monads in a 2-category KK, the free cocompletion KL(K)KL(K) of KK under Kleisli objects for monads. It turns out that the morphisms in KL(K)KL(K) are again just colax monad morphisms in KK, but its 2-cells are somewhat laxer, in just such a way as to yield natural transformations when interpreted in SpanSpan.

However, we still have the problem that not every colax monad morphism in SpanSpan is a functor. If you know about proarrow equipments, you may notice that there’s clearly something equipment-y going on. That is, SpanSpan 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 SetSpanSet \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 SpanSpan. 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 SetSpanSet\hookrightarrow Span under an appropriate sort of Kleisli object in this world, we get the proarrow equipment CatProfCat\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).

Posted at January 16, 2013 7:54 PM UTC

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Re: The Universal Property of Categories

Are there any surprisingly familiar examples of enriched bicategories, to rival Lawvere’s metric spaces in the enriched category case?

Posted by: David Corfield on January 17, 2013 10:04 AM | Permalink | Reply to this

Re: The Universal Property of Categories

Well, Lawvere’s metric spaces are really a kind of enriched poset, so you’re asking me to categorify (or something) twice in one hop. (-: I don’t know any examples of enriched categories or bicategories that are as surprising as metric spaces: I would guess that once you start to impose associativity conditions, then it’s hard to get anything out that didn’t already ‘look like a category’ to begin with.

Posted by: Mike Shulman on January 17, 2013 12:29 PM | Permalink | Reply to this

example

I just described the example of up-to-homotopy homomorphisms, surprising or not, but I failed to make it a reply so it appears as a comment below this thread.

Posted by: stefan on January 18, 2013 8:30 PM | Permalink | Reply to this

Re: The Universal Property of Categories

My favorite example is probably not as surprising as you’d like, but it did surprise me. (It’s been a while since I thought this through so I apologize if I get a subscript wrong or worse!)

For a topological group XX and an A 4A_4 - space YY (where the latter satisfies the pentagon identity exactly), consider an up-to-homotopy homomorphism ff from XX to YY.

Considering YY as a bicategory with one object, we get an enriched bicategory AA whose objects are the points of XX and with A(a,b)=f(b 1a).A(a,b) = f(b^{-1}a). The surprising part is that an up-to-homotopy homomorphism from an associative HH-space to one that’s only associative up to homotopy obeys the same axiom as is required for enriched bicategories. The shape of that commuting diagram of 2-cells is cool too: it’s not quite the same as the associahedron since it has one more facet and one more vertex than the associahedron.

Posted by: stefan on January 18, 2013 8:25 PM | Permalink | Reply to this

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