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arXiv · 2601.14237

Partial Linearity in Categories

Roy Ferguson·Zurab Janelidze
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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.

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Roy Ferguson, Zurab Janelidze. 2026-01-20. Partial Linearity in Categories. https://arxiv.org/abs/2601.14237

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