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.