The category of “locally compact Hausdorff abelian topological groups”, denoted by LCA, relates many areas in mathematics, including harmonic analysis, topology, and number theory. 

A surprising fact about this category is that the algebraic K-theory spectrum of this category (as an exact category in the sense of Quillen), is the cofiber of K(Z) \to K(R). I’ll first state some fundamental results about LCA, including the celebrated “Pontryagin Duality”, and then give a sketch proof of the above theorem.