# What examples, exercises, topics, etc should I treat when teaching category theory to non-math people?

A couple friends of mine have become very enthusiastic about learning category theory (which I know at the level of category theory for the working mathematician), so I'm giving them lessons. One is a linguist and therefore quasi-mathy, the other studies ancient arabic poetry and has never previously been mathy, just became curious about set theory and category theory because they kept cropping up in philosophical discussions. The first session went really well, but there are a few points I need advice on

1) I know virtually nothing about language and very little about logic or philosophy, so my stock of examples is meager. I can only handwavily talk about categories of statements with arrows corresponding to implication so many times. I need lots of examples!

2) It's hard to know what they'll be excited about. I tried introducing groups (as one object categories) and showed them the integers and the symmetries of the square as such, in hopes of being able to use Grp for future examples. That would have lifted the top of my head off as a math newbie. They thought the idea was alright but weren't enthused. On the other hand they lost their heads when I explained what a commutative diagram is. One pointed at my example and said "that. is. sexy." Things I will not even consider include completeness criteria, theory of monads and the set valued yoneda lemma. I think limits and colimits and adjointness will be really appealing to them. I am doubtful about representability. It would be nice to be able to show them things which are surprising to them, which might be difficult since they have so few mathematical preconceptions to ruin.

3) They wanted exercises. It seems unlikely to me that they'll actually do them, but on the off chance that they do, I'm not sure how to come up with ones that align with their interests when I haven't got any understanding of their interests.

• Off the top of my head, maybe topos theory and its connections to logic might be interesting. Not sure how to introduce that to non-mathy people, though, especially if it's also not CS people where you could transition into it by connecting cartesian closed categories and simply-typed lambda calculus. – Daniel Schepler May 23 '17 at 19:54
• $\textbf{Set}$ might be a good example to use before doing groups. – Kaj Hansen May 23 '17 at 19:56
• On the other hand, maybe the free cartesian closed categories, with objects being "formal products of atomics" and arrows being "simply typed lambda expressions up to $\beta\eta$-equivalence" might be an appealing example of a category from a linguistic perspective. – Daniel Schepler May 23 '17 at 19:56
• I completely agree Kaj. Just need a good second category to send into Set. Daniel, I love the suggestions, although I unfortunately think they are too difficult to put into practice. – Sarah Griffith May 23 '17 at 20:29
• @user193072 Completely agree about "lack of mathematical maturity" being an obstacle, but, unfortunately, that's a big obstacle that I don't see guidance as dramatically mitigating that. Again, it doesn't block anything, it just slows everything down. I actually find it a bit unfortunate that representability is what you'd drop as it's equivalent to the notion of a universal property, arguably the core idea of category theory, and one with a good amount of weight from a philosophical perspective. It's also a notion that's comparable or simpler than (co)limits and adjunctions. – Derek Elkins May 25 '17 at 23:51