Double Orthogonal Factorization Systems II

This talk continues the presentation by Matthew Kukla on double orthogonal factorization systems (DOFS) by focusing on extensions of ordinary orthogonal factorization systems (OFS) to the double categorical setting [1, Section 5]. Given a double category whose category of objects and arrows carries an orthogonal factorization system, one may ask when this structure extends to a DOFS.

First, when the category of proarrows and double cells also carries an OFS, we present conditions under which chosen factorizations can be adjusted so that the source, target, and unit functors become strict morphisms of OFS, yielding a genuine double orthogonal factorization system. Using the notions of restrictions (cartesian cells) and extensions (opcartesian cells) in a double category [2] [3], we then give properties that ensure the existence of at least one DOFS extending a given OFS on the arrow category, and show that such a DOFS compares to any other DOFS sharing the same underlying OFS via a unique morphism of DOFS. As a particular case, in any fibrant double category [4], there exist a canonical initial and a canonical terminal DOFS over any given OFS on the arrows, bracketing all others. We illustrate these constructions with two concrete examples: the double category of relations enriched over a quantale, and the double category of spans.