Theorem: Finite-monodromy categorical equivalence

There is a complex-linear dagger equivalence

Its inverse sends $(H_k{)}_k$ to

The construction extends degreewise to bounded complexes and their mapping complexes.

Proof. Lemma lem:projectors decomposes each fiber into orthogonal monodromy eigenspaces. The projectors are parallel, so they decompose the entire local system.

A parallel map $A$ commutes with $U$, hence with every $P_k$. It is therefore precisely a family of maps $A_k:H_k\to {H^{\prime }}_k$. Conversely, any such family defines a parallel map of the direct sums. Composition, adjoints, and identities are blockwise.

The decomposition and recombination maps are inverse natural unitaries. For complexes, the differentials and all graded intertwiners commute with the projectors. The same decomposition therefore respects differentials in both objects and mapping complexes.