Now we’ve said a lot about the category of topological spaces
and continuous maps between them. In particular we’ve seen that
it’s complete and cocomplete — it has all limits and colimits.
But we’ve still yet to see any good examples of topological
spaces. That will change soon. First, though, I want to point
out something we can do with these limits: we can define
topological groups. Specifically, a topological group is a group
object in the category of topological spaces. That is, it’s a
topological space along with continuous functions , , and that
satisfy the usual commutative diagrams. A morphism of
topological groups is then just like a homomorphism of groups,
but by a continuous function between the underlying topological
spaces. Alternately we can think of it as a group to which we’ve
added a topology so that the group operations are continuous.
But as we’ve seen, a topological structure feels a bit floppier
than a group structure, so it’s not really as easy to think of a
“topology object” in a catego