I have heard the requests to start posting lecture notes. I’ll get to it soon. These posts are my reflections on the experience of putting this course together and how the students respond.
I’ve committed one of the cardinal sins of teaching category theory. I never mentioned the fact that the naming convention for categories changes from category to category based on both historical and preferential reasons. So at least one student was very confused that Set(x,y) was the collection of functions, but Mat(n,m) was the collection of matrices. I knew I should have said this, but I just forgot.
In the third lecture, I finally got to the definition of functor. Before that, I wanted to finish explaining monoids and groups as one-object categories. A student asked what the benefit is of realizing that monoids are categories. This thoroughly nerd-sniped me. I was getting ready to talk about functors anyway, so I kept almost talking about actions and representations as functors, and then remembering I hadn’t defined that yet.
My first non-trivial example of a functor was the free vector space generated by a finite set. This was perhaps another misstep. I wanted an example that was accessible to the whole audience, and I am assuming that everyone has had at least linear algebra. But there are many levels of linear algebra courses, and this concept was not familiar to most of the audience it seemed. As a result, I kept trying to squeeze info out of them to help me build this functor, but mostly nobody seemed to follow. I’ve now realized the analogous functor into Mat is obviously way easier to define and probe for understanding the functoriality conditions.
I think I need to do a better job of highlighting where all this is going. It’s a nice idea to try to highlight the utility of each concept as it is being introduced, not just how this and that math concept exemplify it. Unfortunately, I don’t think a functor is a very useful concept in isolation. So if simply finding a new structure for something to fit into isn’t enticing, then it can’t believably be portrayed as useful until later. I’d love to hear people’s thoughts on this point.
Joe Moeller 
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