Categories of Br\`egman operations and epistemic (co)monads

TL;DR

This paper develops a categorical framework for nonlinear quantum inference using Br extbackslash egman operations, linking convex analysis, resource theories, and logical inference in a unified mathematical setting.

Contribution

It introduces a novel categorical approach to nonlinear quantum inference based on Br extbackslash egman quasi-nonexpansive mappings, extending geometric and resource-theoretic concepts.

Findings

Constructs a categorical framework for nonlinear postquantum inference.

Provides concrete examples for semi-finite JBW-algebras and W*-algebras.

Establishes resource theories with resource monotones and functorial properties.

Abstract

We construct a categorical framework for nonlinear postquantum inference, with embeddings of convex closed sets of suitable reflexive Banach spaces as objects and pullbacks of Br\`egman quasi-nonexpansive mappings (in particular, constrained maximisations of Br\`egman relative entropies) as morphisms. It provides a nonlinear convex analytic analogue of Chencov's programme of geometric study of categories of linear positive maps between spaces of states, a working model of Mielnik's nonlinear transmitters, and a setting for nonlinear resource theories (with monoids of Br\`egman quasi-nonexpansive maps as free operations, their asymptotic fixed point sets as free sets, and Br\`egman relative entropies as resource monotones). We construct a range of concrete examples for semi-finite JBW-algebras and any W*-algebras. Due to relative entropy's asymmetry, all constructions have left and right versions, with Legendre duality inducing categorical equivalence between their well-defined restrictions. Inner groupoids of these categories implement the notion of statistical equivalence. The hom-sets of a subcategory of morphisms given by entropic projections have the structure of partially ordered commutative monoids (so, they are resource theories in Fritz's sense). Further restriction of objects to affine sets turns Br\`egman relative entropy into a functor. Finally, following Lawvere's adjointness paradigm for deductive logic, but with a semantic twist representing Jaynes' and Chencov's views on statistical inference, we introduce a category-theoretic multi-(co)agent setting for inductive inference theories, implemented by families of monads and comonads. We show that the br\`egmanian approach provides some special cases of this setting.