Representation conjecture for M*-categories via Hilbert M*-modules

Prove that for any M*-category C with a separating object A, the hom-functor C(A, −): C → Set factors through the forgetful functor Hilb_{C(A,A)} → Set via an equivalence of M*-categories; equivalently, establish that C is equivalent, as an M*-category, to the category Hilb_{R} of Hilbert M*-modules over the endomorphism M*-ring R = C(A,A).

Background

The paper introduces M*-categories as R*-categories endowed with codirected ℓ²-limits of contractions, yielding analytic completeness properties such as monotone completeness and orthogonal completeness. These categories abstract and unify key features of Hilbert spaces, Hilbert W*-modules, and unitary representations.

In outlining future directions, the authors propose the framework of M*-rings (symmetric ordered -rings that are both monotone and orthogonally complete) and Hilbert M-modules (orthogonally complete inner product modules over M*-rings). They suggest that categories of Hilbert M*-modules over a given M*-ring R, denoted Hilb_R, form M*-categories. The conjecture aims to characterize all M*-categories (under the presence of a separating object) as such Hilb_R categories, paralleling the Gabriel–Popescu theorem for abelian categories.