Linearity for monoidal structures

This talk consists of two parts. In the first part, we explore partial linearity of categories in the following sense. We consider categories $ℂ$ equipped with a monoidal sum structure $(\oplus ,0)$ in the sense of [3] and the dual, a monoidal product structure $(\otimes ,1)$. In this setting, morphisms $X_1\oplus \mathrm{\cdots }\oplus X_n\to Y_1\oplus \mathrm{\cdots }\oplus Y_m$ are uniquely determined by matrices, as in linear categories in the sense of [4]. We further equip $ℂ$ with a natural transformation $i:X\oplus Y\to X\otimes Y$. Our first remark is that the mere existence of such $i$ immediately forces pointedness of $ℂ$. We call $i$ a prelineariser if it is compatible with the unitors of both monoidal structures in an appropriate sense. We show that $i$ is a prelineariser if and only if the matrix corresponding to $i$ is the identity matrix. We moreover show that a precursory coherence theorem holds: given any $\oplus$-word $w$ and any $\otimes$-word $v$ of the same length in the sense of [5], there exists a unique canonical morphism (with $i$ included in the definition of canonical) from $w$ to $v$, and that this morphism has matrix presentation the identity matrix. We call a category equipped with the above-mentioned structure, where $i$ is a prelineariser, a prelinear category. When $i$ is invertible, we call it a lineariser and we then speak of a partially linear category. We establish the full coherence theorem for partially linear categories.

When the sum structure is given by coproduct and the product structure is given by the Cartesian product, partially linear categories become linear categories. When both the sum structure and the product structure are given by the Cartesian product, partially linear categories become weakly unital categories [6], which include all unital categories [1]. In the second part of the talk we extend results on centrality in unital categories to our general context. Extending the concept of a central morphism introduced in [1], we say that a morphism $f:X\to Y$ in a prelinear category is central if the morphism exists. We show that in the partially linear context central morphisms admit “enrichment” in $\mathrm{𝐌𝐨𝐧}$ as in the case of unital categories. A preprint detailing the results of this talk can be found at [2].