I will introduce the p-adic local Langlands correspondence for GL(2,Qp), in the forms established by Colmez and Paskunas, and then give an interpretation of it as an embedding of categories (a form of "localization"). Time permitting, I will also discuss local-global formulas for singular cohomology of modular curves that you can get from this framework. This is joint work with James Newton and Carl Wang-Erickson.