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Factorization Systems for Restriction Categories

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Robin Cockett, G.S.H. Cruttwell, Jonathan Gallagher, and Dorette Pronk

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.

Keywords:
Restriction categories, Factorization Systems

2020 MSC:
18B10, 18A32

*Theory and Applications of Categories,*
Vol. 42, 2024,
No. 7, pp 145-171.

Published 2024-07-22.

http://www.tac.mta.ca/tac/volumes/42/7/42-07.pdf

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