  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.
