Contents

category theory

# Contents

## Definition

An orthogonal factorization system $(E,M)$ on a category $C$ with pullbacks is called stable if $E$ 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

The relation between stable factorization systems and the Beck-Chevalley condition of the associated fibrations is discussed in

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

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