I skimmed through books 1, 4, and 5 of my new batch and am currently skimming through 3; it seems I don’t have the mathematical prerequisites to get much out of 2. It will take me a long time to digest all of the interesting things I’ve learned, but I thought I’d discuss an interesting idea coming from Lawvere and Schanuel.

An important idea in mathematics is to reduce an object into its “connected components.” This has various meanings depending on context; it is perhaps clearest in the categories and , and also has a sensible meaning in, for example, for a group . Lawvere and Schanuel suggest the following way to understand several of the examples that occur in practice:

Let be a concrete category with a forgetful functor . If it exists, let be the left adjoint to . Then describes the “discrete” (i.e. “totally disconnected”) objects of , and, if it exists, the left adjoint to is a functor describing the “connected components” of an object in .

I think this is a nice illustration of a construction that is illuminated by the abstract approach, so I’ll briefly describe how this works for a few of my favorite categories.