Homotopies as 2-Morphisms
Last time, while talking about homotopies as morphisms I said that I didn’t want to get too deeply into the reparameterization thing because it could get too complicated. But since when would I, of all people, shy away from 2-categories? In case it wasn’t obvious then, it’s because we’re actually going to extend in the other direction.
Given any two topological spaces and , we now don’t just have a set of continuous maps , we have a whole category consisting of those maps and homotopies between them. And I say that composition isn’t just a function that takes two (composable) maps and gives another one, it’s actually a functor.
So let’s say that we have maps , maps , and homotopies and . From this we can build a homotopy . The procedure is obvious: for any and , we just define
That is, the time- frame of the composed homotopy is the composition of the time- frames of the original homotopies. It should be straightforward to verify that this composition is (strictly) associative, and that the identity map — along with its identity homotopy — acts as an (also strict) identity.
What we need to show is that this composition is actually functorial. That is, we add maps and , change and to and , and add homotopies and . Then we have to check that
That is, if we stack onto and onto , and then compose them as defined above, we get the same result as if we compose with and with , and then stack the one onto the other.
This is pretty straightforward from a bird’s-eye view, but let’s check it in detail. On the left we have
Meanwhile, on the right we have
And so we do indeed have a 2-category with topological spaces as objects, continuous maps as 1-morphisms, and continuous homotopies as 2-morphisms. Of course, if we’re in a differential topological context we get a 2-category with differentiable manifolds as objects, smooth maps as 1-morphisms, and smooth homotopies as 2-morphisms.