nLab stable factorization system

Definition

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.

Properties

In terms of indexed left adjoints

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 $f^* \colon C/y \to C/x$.

Stable reflective factorization systems

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.

Examples

• 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.

Related concepts

• reflective factorization system

• closure operator

References

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:

• J. Hughes and Bart Jacobs, Factorization systems and fibrations: Toward a fibred Birkhoff variety theorem, Electr. Notes in Theor. Comp. Sci., 69 (2002)

Discussion of the example (epi, mono) factorization system in toposes (for more see at regular category, here):

• Stephen Lack, Pawel Sobocínski, Lemma 18 in: Toposes are adhesive, in: Graph Transformations ICGT 2006, Lecture Notes in Computer Science 4178, Springer (2006) 184-198 [doi:10.1007/11841883_14, pdf]

Discussion of reflective stable factorization systems in the context of (∞,1)-category theory (and with an eye towards cohesive homotopy type theory):

• Mike Shulman, Axiomatic cohesion in HoTT (2011) (web)

Stable factorisation systems are called complete factorisation systems in the following, where they are related to comprehension schemes:

• Clemens Berger, Ralph M. Kaufmann, Comprehensive Factorization Systems, Tbilisi Math. J. 10 (2017), 255-277 (doi:10.1515/tmj-2017-0112)