Factorization Systems for Restriction Categories

Robin Cockett ; G.S.H. Cruttwell ; Jonathan Gallagher ; Dorette Pronk

Theory and applications of categories, Hofstra Festschrift, Tome 42 (2024), pp. 145-171

Résumé

This paper describes the notion of a (latent) factorization system for a restriction category. Analogous to factorization systems for ordinary categories, a description in terms of a Galois connection based on an orthogonality relation is given. The orthogonality relation involving a unique cross-map (or lifting), however, is between a pair, consisting of a map together with a restriction idempotent, and a map. This gives a significantly different character to the theory.

A restriction category in which the idempotents split is precisely a partial map category whose partiality is given by the system of monics which split the restriction idempotents. A restriction factorization on such a partial map category is completely characterized by a factorization on the total maps which is stable with respect to the system of monics; that is, pullbacks of both E and M-maps along these special monics are (respectively) E and M-maps.

Examples of latent factorization systems are discussed. In particular, one source of latent factorizations is provided by lifting latent factorizations from the base of a latent fibration into the total category: this is a generalization of the observation that one can lift a factorization from the base of an ordinary fibration into the total category.

Détail

Publié le : 2024-07-22

Classification : 18B10, 18A32
Keywords: Restriction categories, Factorization Systems