ChapterZero

What is category theory good for, practically speaking? From what I understand of it, it allows you to make overarching statements about mathematical structures that are in a sense just semigroups: sets with associative binary relations. Probably there are some other technical requirements, but that’s the essence, right? So you can now treat Top, the category of topological spaces with homeomorphisms as the binary relation, as a category, and Grp, the category of groups with group morphism as the binary relation, as a category– look at them as different instantiations of the same type.

Why do you care? It seems like there’s a slight chance you could find some categorical theorem that could be translated into a nontrivial result in multiple categories, but I’ve never heard of it happening; I suspect there are two reasons for that: 1) recognizing a useful category theoretical result is probably hard, given the abstractness of the concepts involved, and 2) useful results seem (in my naivete, perhaps) to depend on the features that are abstracted away by category theory. Not that I’m saying such a useful result hasn’t been found, just I haven’t heard of it, which leads me to think if such results exist, they are rather technical and nonprofound.

Consider the first line from the introduction to Saunders Mac Lane’s ‘Categories for the Working Mathematician’:

Category theory starts from the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows.

Is that supposed to motivate further study? … because I’m not feeling it.

Possibly relevant posts:

• The Brave New World of Functional Analysis (3/8/2006)
• Beyond Undergraduate Mathematics (5/26/2005)
• speculations on the origins of matrix theory (10/12/2005)