### Category Theory in Barcelona

#### Posted by Tom Leinster

I’m excited to be in Barcelona to help Joachim
Kock teach an introductory course on category
theory. (That’s a link to
bgsmath.cat — categorical activities in
Catalonia have the added charm of a *.cat* web address.) We have a wide audience
of PhD and masters students, specializing in subjects from topology to
operator algebras to number theory, and representing three Barcelona
universities.

We’re taking it at a brisk pace. First of all we’re working through my textbook, at a rate of one chapter a day, for six days spread over two weeks. Then we’re going to spend a week on more advanced topics. Today Joachim did Chapter 1 (categories, functors and natural transformations), and tomorrow I’ll do Chapter 2 (adjunctions).

I’d like to use this post for two things: to invite questions and participation from the audience, and to collect slogans. Let me explain…

Joachim pointed out today that category theory is full of slogans. Here’s the first one:

It’s more important how things interact than what they “are”.

As he observed, the question of what things “are” is slippery. Let me quote a bit from my book:

In his excellent book

Mathematics: A Very Short Introduction, Timothy Gowers considers the question: “What is the black king in chess?”. He swiftly points out that this question is rather peculiar. It is not important that the black kingisa small piece of wood, painted a certain colour and carved into a certain shape. We could equally well use a scrap of paper with “BK” written on it. What matters is what the black kingdoes: it can move in certain ways but not others, according to the rules of chess.

In a categorical context, what an object “does” means how it interacts with the world around it — the category in which it lives.

Tomorrow I’ll proclaim some more slogans — I have some in mind. But I’d
like to hear from you too. **What are the most important slogans
in category theory?** And what do they mean to you?

I’d also like to try an experiment. The classes move rather quickly, so there’s not a huge amount of time in them for discussion or questions. But I’d like to invite students in the class to ask questions here. You can post anonymously — no one will know it’s you — and with any luck, you’ll get interesting answers from multiple points of view. So please, don’t be inhibited: ask whatever’s on your mind. You can even include LaTeX, in more or less the usual way: just put stuff between dollar signs. No tinguis por!

## Re: Category Theory in Barcelona

Today’s first slogan:

This requires a bit of refinement, as objects of a category can also be objects of a 2-category… but it’s a starting point for the definition of equivalence, which I’ll be doing in a couple of hours.