Dare I demand that you spend more of your time, when I have been so fortunate as to have had months of free -- free! -- tuition in my favourite area of study by one of the world's foremost experts? The temerity! - but here I am. I can only apologise for being so inactive in the comment threads - like others, I will make time my excuse. The course, for me, has been on that razor edge of productive learning between the trivial and the impossible that maths seems to skate, and I'm enjoying it immensely.
- My intuition around constructing "free" things has really improved
- The intuition about adjunctions being a "best guess" where there's no exact answer - I've had the technical definition memorised for years, but it took the course for the intuition to drop
- Enriched categories as glorified posets! They are much easier to understand this way
- String diagrams have clicked for me - the monoidy ones I'd seen previously had seemed abstruse, but the lemon meringue diagram just nailed it
- The way the monoidal structure interplays with posets - I could have taken a stab to guess the laws, but working it out with real-ish examples really makes it click
- Using free categories & functors to model database migrations was enlightening, and is enormously relevant to my work.
- I already had an intuition for profunctors as "pipeline shaped things with input & output" from programming, but working through constructing the enriched composition & categorical structure has really fleshed out the concept.
- So here's a fun, and slightly embarrassing one. I think seeing the connection between feasibility relation composition & matrices was supposed to help me understand profunctors, but it actually helped me get..... *ahem* matrices. So I never looked at matrices after high school, and never really understood why they were composed end-on-end with these specific rote-learned multiplication rules. The feasibility matrices got me there.
This is from a combination of the book & the course, but I never would have gotten this far into the book in this level of detail without the course.
I fervently hope that you will continue the course, but will be nothing but grateful if you understandably decide to wrap it up here. Thank you John. That said, operads are my favourite thing in the world, and I might cry if we miss out.