[Koshkin]

> most mathematicians

Agree, but only on the level on which they do not care about the abstract math (algebra, topology, etc.) in general. As soon as you step into the territory of the abstract math, especially where different disciplines blend, such as homology and cohomology, category theory (and its diagram language) helps a lot to clarify things. (Incidentally, a lot of this stuff is now part of the "applied math" as well, having found its way into theoretical physics, for example.)

[spekcular]

I'm not sure I'm comfortable characterizing the fields where category theory is useful as "abstract math." Modern PDE is plenty abstract, for example. Probably it's better to say that the usefulness of category theory is proportional to the problem's distance from algebraic topology and algebraic geometry.

I also am reluctant to characterize theoretical physics as "applied math." I haven't seen anyone who calls themselves an applied mathematician use category theory in a substantive way (where here I am thinking about numerical computing, mathematical biology, and so on).