Partial Linearity in Categories
Abstract
In this paper we generalise the notion of linearity (in the sense of Lawvere) to a category C equipped with a compatible sum structure and product structure. In this context, any morphism f from an n-fold sum to an n-fold product has a unique n by m matrix presentation, but a morphism for a given matrix does not necessarily exist. We define the sum and product to be compatible if there exists a natural transformation i from sum to product with matrix presentation the identity and define C to be partially linear if such an i is invertible. We establish a coherence theorem for partially linear categories. We generalise the notion of a central morphism to this setting, and show that the central morphisms of a partially linear category admit enrichment over monoids.
Cite
@article{arxiv.2601.14237,
title = {Partial Linearity in Categories},
author = {Roy Ferguson and Zurab Janelidze},
journal= {arXiv preprint arXiv:2601.14237},
year = {2026}
}
Comments
Added proofs and corrected claims: 1. That centrality in terms of matrix presentation corresponds to centrality with respect to both sum and product. We have the forward implication, but the converse is not known. 2. That partially linear corresponds to central morphisms enriched over monoids. The forward implication holds, but the converse implication is false