If I remember my functional analysis correctly (hint: I probably don't), when considering Banach spaces we might want to work with up to 3 distinct choices of topology at a time: strong, weak, and weak-*. (Some of these coincide, I think, when the space is reflexive or Hilbert.)
At least naively, it would seem like the category of topological spaces is insufficient to describe such objects, since each object carries insufficient "data" (i.e. only one choice of topology instead of 3).
Is there a category-theoretic way to think about Banach spaces which is agnostic to which of the three topologies, strong, weak, weak-*, we might want to work with?
My gut inclination is that the answer is no, because I can't think of appropriate morphisms for such a category -- they would have to be continuous with respect to all 3 different topologies, whereas I think one would be interested in a function which is continuous with respect to any one of them.
That would lead to three different classes of morphisms, which would lead to an object related to a category (in particular which has categories as a special case) but nevertheless different.
Note: I would have expected these questions (1)(2) to contain the answer, but they don't for whatever reason. Also nLab's page does seem to indicate that category theory applied to Banach spaces is less than simple (something about short versus bounded linear maps and unit balls) but doesn't seem to address my question about choice of topology (the phrase "strong operator topology" is found only once on the page, and the word "weak" has no hits at all).
You mean the strong and weak; the weak-* topology is defined on duals. Anyway, one thing you can do is to consider the category of Banach spaces, together with not one but several forgetful functors to topological spaces, in this case one for the norm topology and one for the weak topology.