factorization system over a subcategory in nLab

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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 category, and let JJ, EE, and MM be wide subcategories of CC with J⊆EJ\subseteq E and J⊆MJ\subseteq M. Given a morphism f:x→yf\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 x→z→yx\to z \to y such that x→zx\to z is in EE, z→yz\to y is in MM, and the composite x→yx\to y is ff;
• whose morphisms from x→z→yx\to z \to y to x→z′→yx\to z' \to y are morphisms z→z′z\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

• If JJ consists of only the identities in CC, then a factorization system over JJ is a strict factorization system.

• If JJ is the core of CC, then a factorization system over JJ is an orthogonal factorization system

• If JJ is the canonical inclusion of (a skeleton of) FinSet opFinSet^{op} into a Lawvere theory CC, then a factorization system over JJ is a decomposition of CC into a distributive law of two other Lawvere theories.

Relation to distributive laws

Suppose given a category JJ. Then to give a category CC equipped with an identity-on-objects functor J→CJ\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

• Eugenia Cheng, Distributive laws for Lawvere theories, arXiv:1112.3076