nLab
factorization system over a subcategory

Contents

Idea

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.

Definition

Let CC be a subcategory, and let JJ, EE, and MM be wide subcategories of CC with JEJ\subseteq E and JMJ\subseteq M. Given a morphism f:xyf\colon x\to y in CC, let Fact J E,M(f)Fact^{E,M}_J(f) denote the non-full subcategory of the over-under-category (double comma category) (x/C/y)(x/C/y):

  • whose objects are pairs xzyx\to z \to y such that xzx\to z is in EE, zyz\to y is in MM, and the composite xyx\to y is ff;
  • whose morphisms from xzyx\to z \to y to xzyx\to z' \to y are morphisms zzz\to z' which are in JJ and make the two evident triangles commute.

We say that (E,M)(E,M) is a factorization system over JJ if Fact J E,M(f)Fact^{E,M}_J(f) is connected (and thus, in particular, inhabited).

Examples

Relation to distributive laws

Suppose given a category JJ. Then to give a category CC equipped with an identity-on-objects functor JCJ\to C and a factorization system over JJ is the same as to give a distributive law between two monads on JJ in the bicategory Prof. The two monads are the categories EE and MM, and their composite is CC.

References

Revised on January 27, 2012 19:13:38 by Urs Schreiber (131.174.40.111)