An orthogonal factorization system $(E,M)$ on a category $C$ with pullbacks is called stable if also the left class $E$ 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 notion appears 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)
On the relation between stable factorization systems and the Beck-Chevalley condition of the associated fibrations:
Discussion of the example (epi, mono) factorization system in toposes (for more see at regular category, here):
Discussion of reflective stable factorization systems in the context of (∞,1)-category theory (and with an eye towards cohesive homotopy type theory):
Last revised on July 14, 2022 at 13:15:05. See the history of this page for a list of all contributions to it.