Post #2057060

I see this pattern everywhere as I explore higher end math as a #math casual: a category C\mathcal{C} C (modules, bundles, representations, sheaves, projections in an algebra) decomposes into atoms (simples, irreducibles, points, primes, factors), and there's a trace-like map from C\mathcal{C} C to a more linear object (cohomology, class functions, cyclic homology, the modular spectrum) that turns the categorical structure into something computable, and that — in good cases — lets you reconstruct C\mathcal{C} C from the atoms plus the trace data. The atoms live on a space (Spec, classifying space, moduli space), and that space is the classifying space in a sense: not a thing but a parameter space for the irreducible building blocks.