Factorisation systems and equipments

Keisuke Hoshino – 29 November 2023

An orthogonal factorization system (OFS) can be regarded as a cartesian fibration, while a cartesian equipment naturally gives rise to a cartesian fibration as well. We establish sufficient conditions for a cartesian equipment to induce an orthogonal factorisation system as a cartesian fibration. On the other hand, Shulman introduced a way to obtain an equipment from a monoidal bifibration, and we observe applying it to a stable OFS results in a cartesian equipment. This construction generalises the cartesian equipment of relations, and the outcome is termed the cartesian equipment of relations relative to a stable OFS. Lastly, we note that by appropriately strengthening the conditions for a cartesian equipment to induce an OFS, we precisely identify the class of cartesian equipments of relations relative to stable OFSs. (Joint work with Hayato Nasu.)