Partial Linearity in Categories

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.