The n-Category Café

Sorry, no slides, since I decided a blackboard talk will convey to these philosophers the spirit of how mathematicians do category theory better than slides would: it involves drawing lots of diagrams in real time. And I don’t think they videotape the colloquia, either. But you’re not missing much: I’m sure you know everything I’ll say.

presumably they doesn’t distinguish it from homotopy type theory

Actually, they do. Both FOLDS and HoTT have a special treatment of equality, a notion of equivalence, and a relationship to higher-categorical structures, but all three of them are different!

FOLDS is not really a foundational system for mathematics. It’s more like a metalanguage in which to talk about general higher-categorical structures.

I was especially impressed by Olivia Caramello’s talk on topos theory and Brice Halimi’s talk on sketches: I learned some new math from these (and from reading some of Caramello’s papers). I unfortunately missed the very last talk, Hans Halvorson and Dimitris Tsementzis, speaking on a language that “forbids evil thoughts”:

Abstract: As Paul Benacerraf famously pointed out, set-theoretic foundations provide too much information for the working mathematician. Our talk is driven by the question: is it possible or desirable to develop a foundational framework that omits the irrelevant and distracting information? We consider one proposal motivated by the slogan that, “all mathematically relevant properties are structural.” Unlike many previous structuralist proposals, this proposal is linguistic (syntactic) in nature. Namely, we present a language L together with an intended semantics, such that the following hold: (1) L is rich enough to be considered foundational; (2) L satisfies the appropriate adequacy conditions with respect to its intended semantics; and crucially (3) well-formed sentences in L are necessarily invariant under isomorphisms of the relevant type of structure. This proposal is strongly influenced by Michael Makkai’s “structuralist foundation of abstract mathematics,” and his language FOLDS (first-order logic with dependent sorts).

If Mike Shulman attended this talk, perhaps he can say the most interesting thing that wasn’t in the abstract.

I posted this very nontechnical comment on Caramello’s talk on Google+, just to get nonexperts interested in topos theory, and we had a good discussion over there:

At the Category-Theoretic Foundations of Mathematics Workshop at U.C. Irvine this weekend, I met Olivia Caramello, who is a powerful advocate of topos theory as an approach to unifying mathematics. She was a student of one of the best topos theorists in the world, now she’s a research fellow at Cambridge University, and she lives and breathes math. I always enjoy meeting brilliant young mathematicians who are eager to explore mind-blowing new realms and confident in their power to do it. They can be scary, but they’re fun to see — sort of like a lion or tiger.

I’m not going to tell you what a topos is; I’ll just say some stuff about them. They were invented by Grothendieck in the 1960’s as part of his quest to prove some conjectures in number theory. The solutions of a bunch of polynomial equations give, indirectly, a topos, and he thought of this topos as a generalization of a space, hence the name ‘topos’. Later he and other realized that a topos is like a mathematical universe, since the world of set theory is one topos, but there are many more. So, you can think of topos theory as a grand generalization of set theory.

But then people realized that a topos is also like a theory. For many familiar mathematical gadgets - groups, rings, etcetera - there is a topos called the classifying topos of that gadget, which acts like an axiomatic theory of that gadget. In particular, any particular instance of that gadget in the world of sets - for example, any particular group - gives a map from the topos of sets to the classifying topos of that gadget, and conversely.

Olivia Caramello is deeply focused on classifying topoi and how they can be used as ‘bridges’ to relate seemingly different fields of math. She has an easy-to-read introduction to this idea here:

• Unifying theory.

My friend Colin McLarty, who also attended this conference, has written a nice history of topos theory that gives a good flavor of what it’s about, without revealing the — somewhat terrifying, at first — definition of a topos:

• The uses and abuses of the history of topos theory.

It’s taken me a long time to learn what little I know about topos theory — in part because I’m interested in lots of other things, and in part because I found it a bit intimidating for the first few decades. A long time ago, I wrote this introduction:

• Topos theory in a nutshell.

whose main virtue is that it points you to 2 free books on the subject: the easy one by Goldblatt, and the vastly more profound book by Barr and Wells. But now I’ve reached the point where topos theory makes a lot more sense! So, I should probably expand this page a bit someday.

If you know topos theory and/or a fair amount of logic, you can get a good intro to Caramello’s work here:

• Olivia Caramello, The unification of mathematics via topos theory.

However, this is not for the faint-hearted! Here’s a sample, just so you can decide if this is for you:

Now, the fundamental idea is the following: if we are able to express a property of a given geometric theory as a property of its classifying topos then we can attempt to express this property in terms of any of the other theories having the same classifying topos, so to obtain a relation between the original property and a new property of a different theory which is Morita-equivalent to it. The classifying topos thus acts as a sort of ‘bridge’ connecting different mathematical theories that are Morita-equivalent to each other, which can be used to transfer information and results from one theory to another. The purpose of the present paper is to show that this idea of toposes as unifying spaces is technically very feasible; the great amount and variety of results in the Ph.D. thesis [2] give clear evidence for the fruitfulness of this point of view and, as we shall argue in the course of the paper, a huge number of new insights into any field of Mathematics can be obtained as a result of the application of these techniques.

(A ‘geometric theory’ is not one about geometry; it’s a technical term for an axiom system obeying certain properties,which Caramello explains. As I hinted, many different kinds of mathematical gadgets can be described using geometric theories. She also explains the concept of “Morita equivalence”.)

In the subsequent discussion Timothy Gowers asked for some concrete, easily understood examples of the insights Caramello was getting from topos theory. We wound up discussing 1) the relation between Gödel’s completeness theorem and Deligne’s theorem on coherent topoi, and 2) Caramello’s new approach to Galois theory based on Fra¯ssé limits. It would be great if some experts on topos theory could straighten out any of the confusions we had, and provide some more examples.

Hans’ talk turned out to be essentially an advertisement for univalent foundations (homotopy type theory). I got the impression that that may have been a change since the writing of the abstract: that they used to believe FOLDS was the way to go, but are now convinced that UF is better.