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[Oct. 4th, 2012|08:11 am]
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Here's some very vague category theory questions.

Say I'm just a humble mathematician, not having already drunk the category theory kool-aid. I want to convince myself that the definition of category is "sensible" or "good" or whatever. I could try to look to the fact that it appears to characterize how sets and functions behave. The reason, I tell myself, that associativity is required, is because composition of functions between sets really is associative. The reason that we require identities is that, for any set, there does inevitably exist an identity function on it.

Now, the Yoneda lemma is a crucial formal reflection of this intuition: every category can in fact be seen as a concrete category made up of sets and functions.

Question 1: is there a kind of converse to the Yoneda lemma that feels like a model-theoretic completeness theorem? Does the "theory of a category" (i.e. "there are some objects and some arrows and identities and composites and the composites are associative") capture everything that's true, in an appropriate sense, of sets and functions? Or am I thinking of this backwards and the Yoneda lemma actually is in the completeness direction? I'm not sure.

Question 2: What do I replace "sets and functions" with to get bicategories? What's the simplest ("free-est") concrete mathematical data structure that I want to use in place of function, that has the property that it's associative up to isomorphism, but not on the nose?

The first things that come to mind for Q2 --- the examples of non-strict bicategories most familiar to me --- are topological-spaces/continuous functions/homotopies and sets/spans of sets/maps-of-spans. But the homotopy bicategory seems needlessly complicated --- surely there's some simpler, more plain-set-theoretic example that works? And spans aren't "free" enough, I think. It is a true fact that for any span there's a converse span, but this dualization isn't available in a general bicategory.

Oh, wait, um, the homotopy bicategory also admits time-reversal of homotopies, doesn't it. So scratch that, too.

I tried reading about the Yoneda lemma for bicategories hoping for clarification, but instead was met with the usual incomprehensible zoo of notions of subtly different weakened thingies and weakened equivalences. And I wish I had more concrete litmus tests and diagnostic tools for understanding such things, which is why I ask these questions in the first place.


Hm, the more I think about it, the more I think Q1 is answered by "yes, the Yoneda lemma is a kind of completeness theorem". If there were some property that "all models satisfied" (i.e. all collections of sets and functions between them satisfied) but was not captured by the axiomatization of categories that we know, then converting some category that failed to have that property to a concrete category should fail.

I'm still curious about Q2 though.


Wait, no, I still hunger for some more satisfying answer to Q1 after all. I could just as truthfully say "every category can be faithfully embedded into the category of sets and relations" rather than sets and functions, but the axiomatic theory-of-a-category doesn't let me form the converse of an arrow, which is nonetheless a semantically legitimate thing to do with relations.

[User Picture]From: bhudson
2012-10-04 06:21 pm (UTC)
If, in my soon to be copious free time, I wanted to read an intro to category theory, what is the category-theory-for-dummies material you'd recommend? I'm starting from linear algebra and geometry are cool, but sets are scary.
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[User Picture]From: jcreed
2012-10-04 10:12 pm (UTC)
I remember Barr and Wells and Crole getting mentioned a lot as accessible introductions for CS-minded people. MacLane is a bit dry, but not too bad.

Linear algebra actually gives you a pretty good leg up. If you go back and read the original paper from 1940-something,("General Theory of Natural Equivalences") the motivating example is trying to formalize the intuition that, although a (finite-dimensional) linear space V and its dual space V* are isomorphic, it's not a "natural" isomorphism (for instance, it depends on a choice of basis) --- but the isomorphism between V and V** is! And so to define "natural isomorphisms" they found they needed to define "functors" (e.g. the the dualization operator that gives me V* from V, and which also gives me f* : W* → V* from f: V → W), and to define functors they needed to define "categories" (e.g., the ambient setting of linear algebra, where the dramatis personae are linear spaces and linear maps). So, bam, category theory was invented.

Edited at 2012-10-04 10:13 pm (UTC)
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