Generalizations of Bilinear Maps -- Technical Report

TL;DR

This paper extends the concept of bimorphisms in category theory, demonstrating their naturality and broad applicability, and recovers classical theorems using this generalized framework.

Contribution

It generalizes bimorphisms beyond traditional settings, clarifies necessary assumptions, and connects these notions to categorical axioms and classical results.

Findings

Many axioms in category theory can be expressed as bimorphisms.

The theory of bimorphisms extends to greater generality with minimal assumptions.

A simple proof of a classical theorem is recovered using the bimorphism perspective.

Abstract

Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in categorical approaches to finite model theory, we generalize the notion of bimorphism much further. To illustrate these maps are mathematically natural notions, we show that many common axioms in category theory can be phrased as certain morphisms being bimorphisms. We also show that much of the established theory of bimorphisms goes through in much greater generality. Our results carefully identify which assumptions are needed for the different components of the theory, including when good properties hold globally, or can at least be established locally. We include a brief string diagrammatic account of the bimorphism axiom, and conclude by recovering a simple proof of a classical theorem, emphasizing the efficacy of the bimorphism perspective.