Category A is isomorphic to category Bop
in Very Basic Noncommutative Geometry (ahahahaha "very basic") is blowing my mind.
Okay first of all it gave me an answer to what noncommutative geometry is, which I never knew, though I'd heard about it a lot. There's a theorem, the Gelfand-Naimark theorem, that says that the category of every nice (i.e. locally compact hausdorff) topological space is equivalent to the dual of the category of some commutative algebraic gadgets (namely, C* algebras, which I don't really understand at all). But also there's lots of noncommutative C* algebras. What are the topological things that correspond to those? Welllll, not locally compact hausdorff topological spaces, that's for sure. They're all "commutative".
So noncommutative geometry seems to be just... the study of noncommutative C* algebras. But since Gelfand-Naimark is true, this is viewed as if it's the study of a "noncommutative" generalization of topological spaces.
Separately, a Lovecraftian shiver tickles my brain upon seeing
(aﬃne schemes) = (commutative rings)op
on page 19, because, like, commutative rings are so garden variety I think I have a half-dozen growing in my backyard amongst the weeds, but schemes are naked gibbering madness that drive mathematicians crazy for their hubris, or so I hear.