Factorisation systems and (virtual) equipments

Keisuke Hoshino

RIMS Kyoto University



Abstract: This talk explores the relationships between orthogonal factorisation systems (OFS), Cartesian fibrations, and Cartesian equipments, focusing on the connections and generalisations they suggest. These ideas form the backbone of the results from the joint work with Hayato Nasu, which explores the characterisation of double categories of relations.

First, we introduce a way to view an OFS as a Cartesian fibration in terms of stable systems. Next, we delve into double categories and cartesian equipments, highlighting connections to Cartesian fibrations. Moreover, we introduce the result by Shulman that gives a sufficient condition for a Cartesian fibration to yield a Cartesian equipment, which gives rise to double categories of relations relative to stable OFSs.

One of our key results is the constructon of OFSs from Cartesian equipments. Notably, this construction restores the stable OFS when applied to a double category of relations, and provides the well-known comprehensive factorisation system when applied to Prof: the equipment of categories, functors, and profunctors.

Finally, we discuss an ongoing generalisation to virtual equipments. This framework aims to unify and extend existing results on comprehensive factorisations in various contexts, including the work of Tholen and Yeganeh on Burroni’s T-categories.