Partial Linearity in Categories

TL;DR

This paper extends the concept of linearity in categories to partially linear categories with compatible sum and product structures, establishing a coherence theorem and exploring central morphisms.

Contribution

It introduces the notion of partial linearity in categories, generalizing Lawvere's linearity, and proves a coherence theorem for these categories.

Findings

Unique matrix presentation for morphisms from sums to products

Existence of a natural transformation linking sum and product

Central morphisms admit monoid enrichment

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.