A factorization system over a subcategory is a common generalization of an orthogonal factorization system and a strict factorization system, in which factorizations are only unique up to zigzags belonging to some specified subcategory.
Let be a category, and let , , and be wide subcategories of with and . Given a morphism in , let denote the non-full subcategory of the over-under-category (double comma category) :
We say that is a factorization system over if is connected (and thus, in particular, inhabited).
If consists of only the identities in , then a factorization system over is a strict factorization system.
If is the core of , then a factorization system over is an orthogonal factorization system
If is the canonical inclusion of (a skeleton of) into a Lawvere theory , then a factorization system over is a decomposition of into a distributive law of two other Lawvere theories.
Suppose given a category . Then to give a category equipped with an identity-on-objects functor and a factorization system over is the same as to give a distributive law between two monads on in the bicategory Prof. The two monads are the categories and , and their composite is .
Last revised on September 14, 2018 at 14:32:06. See the history of this page for a list of all contributions to it.