Double categories of relations relative to factorisation systems

Abstract: The double category of relations and the double category of spans have historically been key examples in the study of double categories. More generally, given a class $\mathcal{M}$ of morphisms in a finitely complete category $\mathbf{C}$, one can define an $\mathcal{M}$-relation $A \nrightarrow B$ as a span that belongs to $\mathcal{M}$ as a morphism into $A\times B$. If $\mathcal{M}$ is the right class of a stable factorisation system, the $\mathcal{M}$-relations and morphisms in $\mathbf{C}$ form a double category.

In this talk, I will first give a characterisation of double categories that arise from stable factorisation systems in this manner.

I will then explore how different classes of stable factorisation systems can be characterised in terms of their corresponding double categories. This include the (regular epi, mono) factorisation system on a regular category, which has been studied by Carboni and Walters in terms of cartesian bicategories, and by Lambert in terms of cartesian double categories. By considering the (isomorphism, all) factorisation system, we also recover the double category of spans, a structure that has been examined bicategorically by Lack, Walters, and Wood, and double-categorically by Aleiferi. I will explain how these known results are related to our general theorem.

This talk is based on joint work with Hayato Nasu, and the results can be found in our paper available at arxiv.org/abs/2310.19428.

category theory

Audience: researchers in the topic

( video )

Second Virtual Workshop on Double Categories