factorization system over a subcategory
k-ary factorization system, ternary factorization system
factorization system in a 2-category
factorization system in an (∞,1)-category
An orthogonal factorization system $(E,M)$ on a category $C$ with pullbacks is called stable if $E$ is stable under pullback.
For a general (orthogonal) factorization system $(E,M)$, the factorizations show that for all objects the full inclusion $M/x \to C/x$ (where $M/x$ consists of morphisms in $M$ with target $x$) has a left adjoint, hence is a reflective subcategory.
The factorization system is stable if and only if these left adjoints form an indexed functor — that is, they commute with the pullback functors $f^* \colon C/y \to C/x$.
A reflective factorization system on a finitely complete category is stable if and only if its corresponding reflector preserves finite limits (is a left exact functor).
The analogous statement also holds in (∞,1)-category theory, or rather at least in locally cartesian closed (∞,1)-categories. A discussion of this and formal proof in terms of homotopy type theory is in (Shulman).
A stable reflective factorization system is sometimes called local.
In an a topos, epimorphism are stable under pullback and hence the (epi, mono) factorization system in a topos is stable.
More generally, in an (∞,1)-topos for all $n \in \mathbb{N}$ the (n-epi, n-mono) factorization system (see there for more details) is a stable orthogonal factorization system in an (∞,1)-category.
The relation between stable factorization systems and the Beck-Chevalley condition of the associated fibrations is discussed in
The notion appears also for instance in
Max Kelly, A note on relations relative to a factorization system, Lecture Notes in Mathematics, 1991, Volume 1488 (1991)
Stefan Milius, Relations in categories, PhD thesis (pdf)
Discussion of epimorphisms in toposes is for instance in
Discussion of reflective stable factorization systems in the context of (∞,1)-category theory (and with an eye towards cohesive homotopy type theory) is in