Speaking of applying category theory to computational complexity, I've been thinking of NP-complete problems as terminal objects in the category of problems in NP with arrows \$$a \to b\$$ when \$$a\$$ can be reduced to \$$b\$$. I'm sure that's a trivial observation, but it was surprisingly handy when I was helping a friend understand how NP-completeness is special and why NP-hardness is something we'd ever think to consider.