nLab reflective factorization system

Idea

A reflective factorization system is an orthogonal factorization system $(E,M)$ that is determined by the reflective subcategory $M/1$.

Definition

Let $C$ be a category with a terminal object $1$. If $(E,M)$ is an (orthogonal) factorization system on $C$, then the full subcategory $M/1 \subseteq C$ (consisting of those objects $X$ for which $X\to 1$ is in $M$) is reflective. The reflection of $Y\in C$ is obtained by the $(E,M)$-factorization $Y \xrightarrow{e} \ell Y \xrightarrow{m} 1$. (e.g. (Rosicky-Tholen 08, 2.10))

In fact, in this we do not need $(E,M)$ to be a factorization system; only a prefactorization system with the property that any morphism with terminal codomain admits an $(E,M)$-factorization. For the nonce, let us call such a prefactorization system favorable.

Conversely, suppose that $A\hookrightarrow C$ is a reflective subcategory, and define $E$ to be the class of morphisms inverted by the reflector $\ell\colon C\to A$, and define $M = E^\perp$. Then $(E,M)$ is a favorable prefactorization system. In this way we obtain an adjunction

$$\Phi : \text{reflective subcategories} \; \rightleftarrows \; \text{favorable prefactorization systems} : \Psi.$$

Here subcategories form a (possibly large) poset ordered by inclusion, and prefactorization systems form a poset ordered by inclusion of the right classes $M$.

The unit of this adjunction is easily seen to be an isomorphism. That is, given a reflective subcategory $A$, if we construct $(E,M)$ from it as above, then $A \simeq M/1$. Therefore, the adjunction allows us to identify reflective subcategories with certain favorable prefactorization systems.

The prefactorization systems arising in this way — equivalently, those for which $(E,M) = \Phi \Psi(E,M)$ — are called the reflective prefactorization systems. A reflective factorization system is a reflective prefactorization system which happens to be a factorization system.

More generally, for any favorable factorization system $(E,M)$, we have a reflective prefactorization system $\Phi \Psi(E,M)$, called the reflective interior of $(E,M)$. Dualizing, it also has a coreflective closure.

Properties

Characterization

The following is Theorem 2.3 in CHK.

Theorem

Let $(E',M')$ be the reflective interior of $(E,M)$. Then:

1. $f\in E'$ precisely when there exists a $g\in E$ such that $g f \in E$.
2. $(E,M)$ is reflective precisely when $g f\in E$ and $g\in E$ together imply $f\in E$.

Proof

That (1) implies (2) is obvious, so we prove (1).

Since $E'$ is, by definition, the class of maps inverted by the reflector into $M/1$, it satisfies the 2-out-of-3 property. Since $E\subseteq E'$, it follows that $f g\in E$ and $g\in E$ imply $f\in E'$.

Conversely, if $f\colon X\to Y$ is in $E'$, then we have $\eta_Y \circ f = \ell(f) \circ \eta_X$ by naturality, where $\ell$ is the reflector into $M/1$ and $\eta$ its unit. But by construction of $\ell$, $\eta_Y$ and $\eta_X$ are in $E$, and by assumption, $\ell(f)$ is invertible; hence we can take $g = \eta_Y$.

Note that the left class in any orthogonal factorization system is automatically closed under composition, contains the isomorphisms, and satisfies the property that $g f \in E$ and $f\in E$ together imply $g\in E$. Therefore, $(E,M)$ is reflective precisely when $E$ is a system of weak equivalences. See Relation to Localization, below.

Construction of factorizations

The following is a slightly generalized version of Corollary 3.4 from CHK.

Theorem

Suppose that $C$ is finitely complete and M-complete for some factorization system $(E,M)$, where $M$ consists of monomorphisms and contains the split monics. Then any reflective prefactorization system on $C$ is a factorization system.

Proof

This follows directly from this theorem applied to the reflection adjunction.

The following is a consequence of Theorems 4.1 and 4.3 from CHK.

Theorem

Suppose that $C$ is finitely complete and that $(E,M)$ is a reflective prefactorization system on $C$ such that $E$-morphisms are stable under pullback along $M$-morphisms. Then $(E,M)$ is a factorization system.

Proof

Write $\ell$ for the corresponding reflection. Now given $f\colon A\to B$, let $m$ be the pullback of $\ell(f)$ along $\eta_B\colon B \to \ell B$:

$$\array{ Y & \overset{g}{\to} & \ell A \\ ^m \downarrow & & \downarrow^{\ell(f)}\\ B & \underset{\eta_B}{\to} & \ell B}$$

By closure properties of prefactorization systems, any morphism in $M/1$ lies in $M$, so in particular $\ell(f)\in M$. Since $M$ is stable under pullback (being, again, the right class of a prefactorization system), we have $m\in M$.

But $f$ factors through $m$, by the universal property of the pullback applied to the naturality square for $\eta$ at $f$. Thus we have $f = m e$ and it suffices to show $e\in E$. However, we also have $g e = \eta_A$, where $\eta_A\in E$ by definition, and $g\in E$ by assumption (being the pullback of $\eta_B\in E$ along $\ell(f)\in M$). By the characterization theorem above, since $(E,M)$ is reflective this implies $e\in E$, as desired.

A reflection satisfying the condition of Theorem is called semi-left-exact (which see, for more equivalent characterizations). Note that saying that $E$-morphisms are stable under all pullbacks is equivalent to saying that $\ell$ preserves all pullbacks, hence all finite limits—i.e. it is left-exact. In this case the factorization system is called stable. Thus the terminology “semi-left-exact” for the weaker assumption.

However, semi-left-exactness is not necessary for the “one-step” construction in the proof of Theorem to work. The necessary and sufficient condition is that the reflection is simple; one characterization of this is that a morphism $f$ is in $M$ if and only if its naturality square for the reflector $\ell:C\to A$ is a pullback. For others, see CHK, Theorem 4.1.

Relation to localizations

For any favorable prefactorization system $(E,M)$, it is easy to show that $M/1$ is the localization of $C$ at $E$. If $(E',M')$ is the reflective interior of $(E,M)$, then since $E'$ is the class of maps inverted by the reflector into $M/1$, it is precisely the saturation of $E$ in the sense of localization (the class of maps inverted by the localization at $E$).

Reflective stable factorization systems

A reflective factorization system on a finitely complete category is a stable factorization system if and only if its corresponding reflector preserves finite limits. A stable reflective factorization system is sometimes called local.

Examples

Obviously, any reflective subcategory gives rise to a reflective factorization system. Here are a few examples.

• The category of complete metric spaces is reflective in the category of all metric spaces; the reflector is completion. In the corresponding factorization system, $E$ is the class of dense embeddings.

• Given a small site $S$, the sheaf topos $Sh(S)$ is a reflective subcategory of the presheaf topos $Psh(S)$. In the corresponding factorization system, $E$ is the class of local isomorphisms.

On the other hand, many commonly encountered factorization systems are not reflective.

• The factorization system $(Epi, Mono)$ on Set is not reflective. If $(E',M')$ is its reflective interior, then $E'$ is the class of morphisms $e\colon X\to Y$ such that if $Y$ is inhabited, so is $X$, while $M'$ is the class of morphisms $m\colon X\to Y$ such that if $X$ is inhabited, then $m$ is an isomorphism.

Related concepts

• semi-left-exact reflection

• stable factorization system

• closure operator

• reflective localization

References

The basic theory is developed in

• C. Cassidy, M. Hébert, Max Kelly, Reflective subcategories, localizations, and factorization systems, J. Austral. Math Soc. (Series A) 38 (1985) 287-329 [doi:10.1017/S1446788700023624]

• Aurelio Carboni, George Janelidze, Max Kelly, Robert Paré, On localization and stabilization for factorization systems, Appl. Categ. Structures 5 (1997) 1-58 [doi:10.1023/A:1008620404444]

Discussion of “simple” reflective factorization systems and of simultaneously reflective and coreflective factorization systems:

• Jiri Rosicky, Walter Tholen, Factorization, Fibration and Torsion, Journal of Homotopy and Related Structures, 2 2 (2007) 295-314 [arXiv:0801.0063, publisher]

Discussion from the perspective of modalities and modal homotopy type theory:

• Egbert Rijke, Mike Shulman, Bas Spitters, Section A.2 in: Modalities in homotopy type theory, Logical Methods in Computer Science, 16 1 (2020) [arXiv:1706.07526, doi:10.23638/LMCS-16(1:2)2020]