(Warning: I’m trying to talk about things I don’t really understand in this post, so feel free to correct me if you see a statement that’s obviously wrong.)
Why are continuous functions the “correct” notion of homomorphism between topological spaces?
The “obvious” way to define homomorphisms for a large class of objects involves thinking of them as “sets with extra structure,” and then homomorphisms are functions that preserve that extra structure. In category theory this is formalized by the notion of a concrete category, i.e. a category with a good notion of forgetful functor. For topological spaces this is the functor which gives the set of points.
However, a naive interpretation of “preserving additional structure” suggests that homomorphisms between topological spaces should be open maps, and this isn’t the case. So what gives?
I’m not sure. But rather than take the concrete perspective I’d like to talk about another general principle for defining a good notion of homomorphism.
Little insight: If you can realize a structure as a small category, homomorphisms should be functors.
