Let be a category with a terminal object . If is an (orthogonal) factorization system on , then the full subcategory (consisting of those objects for which is in ) is reflective. The reflection of is obtained by the -factorization . (e.g. (Rosicky-Tholen 08, 2.10))
In fact, in this we do not need to be a factorization system; only a prefactorization system with the property that any morphism with terminal codomain admits an -factorization. For the nonce, let us call such a prefactorization system favorable.
Conversely, suppose that is a reflective subcategory, and define to be the class of morphisms inverted by the reflector , and define . Then is a favorable prefactorization system. In this way we obtain an adjunction
The unit of this adjunction is easily seen to be an isomorphism. That is, given a reflective subcategory , if we construct from it as above, then . 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 — 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 , we have a reflective prefactorization system , called the reflective interior of . Dualizing, it also has a coreflective closure.
The following is Theorem 2.3 in CHK.
Let be the reflective interior of . Then:
That (1) implies (2) is obvious, so we prove (1).
Since is, by definition, the class of maps inverted by the reflector into , it satisfies the 2-out-of-3 property. Since , it follows that and imply .
Conversely, if is in , then we have by naturality, where is the reflector into and its unit. But by construction of , and are in , and by assumption, is invertible; hence we can take .
Note that the left class in any orthogonal factorization system is automatically closed under composition, contains the isomorphisms, and satisfies the property that and together imply . Therefore, is reflective precisely when 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 is finitely complete and -complete for some factorization system , where consists of monomorphisms and contains the split monics. Then any reflective prefactorization system on 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 is finitely complete and that is a reflective prefactorization system on such that -morphisms are stable under pullback along -morphisms. Then is a factorization system.
Write for the corresponding reflection. Now given , let be the pullback of along :
By closure properties of prefactorization systems, any morphism in lies in , so in particular . Since is stable under pullback (being, again, the right class of a prefactorization system), we have .
But factors through , by the universal property of the pullback applied to the naturality square for at . Thus we have and it suffices to show . However, we also have , where by definition, and by assumption (being the pullback of along ). By the characterization theorem above, since is reflective this implies , as desired.
A reflection satisfying the condition of the preceeding theorem is called semi-left-exact. It is shown in Theorem 4.3 of CHK that this condition is equivalent to the reflector preserving pullbacks of -morphisms. (Saying that -morphisms are stable under all pullbacks is equivalent to saying that 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.)
Semi-left-exactness of a reflection of into is also equivalent to saying that for any , the right adjoint of the induced functor (which is given by pullback along ) is fully faithful. In this form it is equivalent to (a particular case of) the notion of admissible reflection in categorical Galois theory.
For any favorable prefactorization system , it is easy to show that is the localization of at . If is the reflective interior of , then since is the class of maps inverted by the reflector into , it is precisely the saturation of in the sense of localization (the class of maps inverted by the localization at ).
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.
On the other hand, many commonly encountered factorization systems are not reflective.
The basic theory is developed in
Discussion of “simple” reflective factorization systems and of simultaneously reflective and coreflective factorization systems is in