Last time, I mentioned that I would have changed up how I presented the initial information. A few people couldn’t make it to the first lecture. So I offered to show up an hour early this week and essentially give the first lecture again. As you might expect, the second time around was much better! This time I started off with the informal definition, then dove into examples: Set, Mat, Vec, FinVect… I also followed John’s advice from Zulip and did not even bother with a formal definition at this stage. Some added benefits from this is that I started drawing commutative diagrams a lot more, and with context. So now I don’t have to worry as much about how I’m going to introduce them and motivate them.
Another thing I wanted to mention about the first lecture is that some students were not picking up precisely what I meant when I was saying that a category has morphisms. It’s hard for me to convey precisely what the confusion was, but I think they weren’t sure if a category has a morphism vs has a set of morphisms vs has a function vs I don’t know what else. I wasn’t sure how to resolve this mid-lecture. The student just asked me after class and I think the confusion got resolved by me giving a bunch of examples, and explaining how you can make up subcategories by removing objects and/or morphisms.
In the second lecture I picked up a few dangling threads. I recounted the informal definition, again ignoring the formal one, and recalled briefly the category Set. The first new thing for this lecture was to define the path category of a graph. In retrospect, I think I was talking about diagrams and this path category too close together. At least one student got mixed up about it based on a question they asked. I also wish I had demonstrated a graph with some inequivalent parallel paths.
At this point I switched to talking about isomorphisms. I threw out the definition at the very end of the first lecture, but with only mid-air discussion, not much meat. So today I spelled out that there are a bunch of theorems about isomorphisms in math: Cantor–Schröder–Bernstein for the category Set, the very long list of equivalent conditions you learn in linear algebra for Mat and FinVect. I gave the example in my old blog post Homeomorphism is not just continuous bijection as a non-theorem in Top.
The last thing I did was talk about monoids. I wrote down the definition, I should not even assume they have encountered groups before. And then I asked if anybody could see some similarity to the definition of category. Someone said the operation must be composition. So I wrote down four blank lines for us to fill in: objects, morphisms, identity, composition, and wrote * in the last line. They were able to fill up the last three lines and I let the first one hang for a bit. This is the trickiest bit the first time you learn that monoids are one-object categories: what is the object??? It’s nothing. A student suggested we call it Fish. Theorem: Monoids are the same thing as one-object categories.
After class, a physics student started asking about groups, but couldn’t quite figure out what her question was. So I just gave away the second punchline that groups are monoids with inverse, and thus they are one-object categories with all morphisms invertible. Next time I think I’ll start by actually giving examples of monoids so that doesn’t hang in their brain as just an abstract useless thing, and give the group addendum. I should also get to functors so we can really start cooking.
Joe Moeller 
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