The most recent book of Nick Katz [see https://web.math.princeton.edu/~nmk/mellin398.pdf ] proves extremely concrete equidistribution theorems for certain families of exponential sums. Categories enter in three essential ways (at least): (1) all the work going to Deligne's Weil II version of the Riemann Hypothesis over finite fields; (2) the theory of perverse sheaves; (3) the Tannakian formalism to recover a group from a category. In this, the new contribution of Katz in this book is (3): essentially, the equidistribution is proved using the Weyl equidistribution criterion, and all analytic estimates follow from (1). But if one doesn't know that there is a group underlying the families of sums (or rather the unitarized Frobenius automorphisms which give rise to these sums), one doesn't know what these estimates are really proving.
For more traditional families of sums, one uses instead Deligne's Equidistribution Theorem, where the group is given concretely as monodromy group of a lisse sheaf, but Katz's family are not parameterized by an algebraic variety, and the Tannakian category arises by looking at a category of perverse sheaves equiped with a suitable form of multiplicative convolution.
This is, I think, completely amazing...
