factorization system over a subcategory
k-ary factorization system, ternary factorization system
factorization system in a 2-category
factorization system in an (∞,1)-category
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 $C$ be a subcategory, and let $J$, $E$, and $M$ be wide subcategories of $C$ with $J\subseteq E$ and $J\subseteq M$. Given a morphism $f\colon x\to y$ in $C$, let $Fact^{E,M}_J(f)$ denote the non-full subcategory of the over-under-category (double comma category) $(x/C/y)$:
We say that $(E,M)$ is a factorization system over $J$ if $Fact^{E,M}_J(f)$ is connected (and thus, in particular, inhabited).
If $J$ consists of only the identities in $C$, then a factorization system over $J$ is a strict factorization system.
If $J$ is the core of $C$, then a factorization system over $J$ is an orthogonal factorization system
If $J$ is the canonical inclusion of (a skeleton of) $FinSet^{op}$ into a Lawvere theory $C$, then a factorization system over $J$ is a decomposition of $C$ into a distributive law of two other Lawvere theories.
Suppose given a category $J$. Then to give a category $C$ equipped with an identity-on-objects functor $J\to C$ and a factorization system over $J$ is the same as to give a distributive law between two monads on $J$ in the bicategory Prof. The two monads are the categories $E$ and $M$, and their composite is $C$.