A reflective factorization system is an orthogonal factorization system $(E,M)$ that is determined by the reflective subcategory $M/1$.
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
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.
The following is Theorem 2.3 in CHK.
Let $(E',M')$ be the reflective interior of $(E,M)$. Then:
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.
The following is a slightly generalized version of Corollary 3.4 from CHK.
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.
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.
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.
Write $\ell$ for the corresponding reflection. Now given $f\colon A\to B$, let $m$ be the pullback of $\ell(f)$ along $\eta_B\colon B \to \ell B$:
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.
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 $E$).
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.
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 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:
Discussion from the perspective of modalities and modal homotopy type theory:
Last revised on January 4, 2024 at 20:25:38. See the history of this page for a list of all contributions to it.