nLab
reflective factorization system

Reflective factorization systems

Idea

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

Definition

Let CC be a category with a terminal object 11. If (E,M)(E,M) is an (orthogonal) factorization system on CC, then the full subcategory M/1CM/1 \subseteq C (consisting of those objects XX for which X1X\to 1 is in MM) is reflective. The reflection of YCY\in C is obtained by the (E,M)(E,M)-factorization YeYm1Y \xrightarrow{e} \ell Y \xrightarrow{m} 1. (e.g. (Rosicky-Tholen 08, 2.10))

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

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

Φ:reflective subcategoriesfavorable prefactorization systems:Ψ. \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 MM.

The unit of this adjunction is easily seen to be an isomorphism. That is, given a reflective subcategory AA, if we construct (E,M)(E,M) from it as above, then AM/1A \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)=ΦΨ(E,M)(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)(E,M), we have a reflective prefactorization system ΦΨ(E,M) \Phi \Psi(E,M), called the reflective interior of (E,M)(E,M). Dualizing, it also has a coreflective closure.

Properties

Characterization

The following is Theorem 2.3 in CHK.

Theorem

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

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

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

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

Conversely, if f:XYf\colon X\to Y is in EE', then we have η Yf=(f)η X\eta_Y \circ f = \ell(f) \circ \eta_X by naturality, where \ell is the reflector into M/1M/1 and η\eta its unit. But by construction of \ell, η Y\eta_Y and η X\eta_X are in EE, and by assumption, (f)\ell(f) is invertible; hence we can take g=η Yg = \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 gfEg f \in E and fEf\in E together imply gEg\in E. Therefore, (E,M)(E,M) is reflective precisely when EE 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 CC is finitely complete and MM-complete for some factorization system (E,M)(E,M), where MM consists of monomorphisms and contains the split monics. Then any reflective prefactorization system on CC 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 CC is finitely complete and that (E,M)(E,M) is a reflective prefactorization system on CC such that EE-morphisms are stable under pullback along MM-morphisms. Then (E,M)(E,M) is a factorization system.

Proof

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

Y g A m (f) B η B 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/1M/1 lies in MM, so in particular (f)M\ell(f)\in M. Since MM is stable under pullback (being, again, the right class of a prefactorization system), we have mMm\in M.

But ff factors through mm, by the universal property of the pullback applied to the naturality square for η\eta at ff. Thus we have f=mef = m e and it suffices to show eEe\in E. However, we also have ge=η Ag e = \eta_A, where η AE\eta_A\in E by definition, and gEg\in E by assumption (being the pullback of η BE\eta_B\in E along (f)M\ell(f)\in M). By the characterization theorem above, since (E,M)(E,M) is reflective this implies eEe\in E, 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 \ell preserving pullbacks of MM-morphisms. (Saying that EE-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.)

Semi-left-exactness of a reflection \ell of CC into ACA\subseteq C is also equivalent to saying that for any xCx\in C, the right adjoint of the induced functor :C/xA/(x)\ell\colon C/x \to A/\ell(x) (which is given by pullback along η x\eta_x) is fully faithful. In this form it is equivalent to (a particular case of) the notion of admissible reflection in categorical Galois theory.

Relation to localizations

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

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, EE is the class of dense embeddings.

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

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

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

References

The basic theory is developed in

  • Cassidy and Hébert and Kelly, “Reflective subcategories, localizations, and factorization systems”. J. Austral. Math Soc. (Series A) 38 (1985), 287–329 (pdf)
  • Carboni and Janelidze and Kelly and Paré, “On localization and stabilization for factorization systems”, Appl. Categ. Structures 5 (1997), 1–58

Discussion of “simple” reflective factorization systems and of simultaneously reflective and coreflective factorization systems is in

Revised on December 2, 2014 12:57:32 by Tim Porter (2.31.52.186)