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orthogonal factorization system

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Definition

Orthogonal factorization systems

Definition

Let CC be a category and let (E,M)(E,M) be two classes of morphisms in CC. We say that (E,M)(E,M) is an orthogonal factorization system if (E,M)(E,M) is a weak factorization system in which solutions to lifting problems are unique.

We spell out several equivalent explicit formulation of what this means.

Definition

(E,M)(E,M) is an orthogonal factorization system if every morphism ff in CC factors f=rf = r\circ \ell as a morphism E\ell \in E followed by a morphism rMr \in M; and the following equivalent conditions hold

  1. We have:

    1. EE is precisely the class of morphisms that are left orthogonal to every morphism in MM;

    2. MM is precisely the class of morphisms that are right orthogonal to every morphism in EE.

  2. We have:

    1. The factorization is unique up to unique isomorphism.

    2. EE and MM both contain all isomorphisms and are closed under composition.

  3. We have:

    1. EE and MM are replete subcategories of the arrow category C IC^I.

    2. Every morphism in EE is left orthogonal to every morphism in MM.

OFS’s are traditionally called just factorization systems. See the Catlab for more of the theory.

An orthogonal factorization system is called proper if every morphism in EE is an epimorphism and every morphism in MM is a monomorphism.

Prefactorization systems

For any class EE of morphisms in CC, we write E E^\perp for the class of all morphisms that are right orthogonal to every morphism in EE. Dually, given MM we write M{}^\perp M for the class of all morphisms that are left orthogonal to every morphism in MM. The second condition in the definition of an OFS then says that E= ME= {}^\perp M and M=E M= E^\perp.

In general, () (-)^\perp and (){}^\perp(-) form a Galois connection on the poset of classes of morphisms in CC. A pair (E,M)(E,M) such that E= ME= {}^\perp M and M=E M= E^\perp is sometimes called a prefactorization system. Note that by generalities about Galois connections, for any class AA of maps we have prefactorization systems ( (A ),A )({}^\perp(A^\perp),A^\perp) and ( A,( A) )({}^\perp A, ({}^\perp A)^\perp). We call these generated and cogenerated by AA, respectively.

Properties

General

Proposition

The different characterization in def. 2 are indeed all equivalent.

Proof

(…)

For the moment see (Joyal).

(…)

Proposition

A weak factorization system (L,R)(L,R) is an orthogonal factorization system precisely if LRL \perp R.

Proof

(…)

For the moment see (Joyal).

(…)

Proposition

For (L,R)(L,R) an orthogonal factorization system in a category CC, the intersection LRL \cap R is precisely the class of isomorphisms in CC.

Proof

If is clear that every isomorphism is in LRL \cap R. Conversely, let f:ABf : A \to B be a morphism in LRL \cap R. This implies that the two trivial factorizations

f=Aid AAfB f = A \stackrel{id_A}{\to} A \stackrel{f}{\to} B

and

f=AfBid BB f = A \stackrel{f}{\to} B \stackrel{id_B}{\to} B

are both (L,R)(L,R)-factorization. Therefore there is a unique morphism f˜\tilde f in the commuting diagram

A id A A f f¯ f B id B B. \array{ A &\stackrel{id_A}{\to}& A \\ \downarrow^{\mathrlap{f}} &\nearrow_{\bar f}& \downarrow^{\mathrlap{f}} \\ B &\stackrel{id_B}{\to}& B } \,.

This says precisely that f¯\bar f is a left and right inverse of ff.

Closure properties

A prefactorization system (E,M)(E,M) (and hence, also, a factorization system) satisfies the following closure properties. We state them for MM, but EE of course satisfies the dual property.

  • MM contains the isomorphisms and is closed under composition and pullback (insofar as pullbacks exist in CC).
  • If a composite fgf g is in MM, and ff is either in MM or a monomorphism, then gg is in MM.
  • MM is closed under all limits in the arrow category Arr(C)Arr(C).

If CC is a locally presentable category, then for any small set of maps AA, the prefactorization system ( (A ),A )({}^\perp(A^\perp),A^\perp) is actually a factorization system. The argument is by a transfinite construction similar to the small object argument.

On the other hand, if (E,M)(E,M) is any prefactorization system for which MM consists of monomorphisms and CC is complete and well-powered, then (E,M)(E,M) is actually a factorization system. (Of course, there is a dual statement as well.) In fact something slightly more general is true; see M-complete category for this and other related ways to construct factorization systems.

Cancellation properties

Proposition

For (L,R)(L,R) an orthogonal factorization system. Let

Y f g X gf Z \array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X && \stackrel{g \circ f}{\to} && Z }

be two composable morphisms. Then

  • If ff and gfg \circ f are in LL, then so is gg.

  • If gg and gfg\circ f are in RR, then so is ff.

Proof

Consider the first case. The second is directly analogous.

Choose an (L,R)(L,R)-factorization of gg

g:YIrZ. g : Y \stackrel{\ell}{\to} I \stackrel{r}{\to} Z \,.

With this we have lifting diagrams of the form

X gf Z f id Z Y r id Z I r ZX f Y I gf r 1 r Z id Z id Z Z \array{ X &\stackrel{g \circ f}{\to}& Z \\ \downarrow^{\mathrlap{f}} && \downarrow^{id_Z} \\ Y & \nearrow_r& \\ \downarrow^{\mathrlap{\ell}} && \downarrow^{id_Z} \\ I &\stackrel{r}{\to}& Z } \;\;\;\;\;\;\; \;\;\;\;\;\;\; \array{ X &\stackrel{f}{\to}& Y &\stackrel{\ell}{\to}& I \\ {}^{\mathllap{g \circ f}}\downarrow & & \nearrow_{r^{-1}}& & \downarrow^{\mathrlap{r}} \\ Z &\underset{id_Z}{\to}& &\underset{id_Z}{\to}& Z }

exhibiting an inverse of rr. Therefore rr is an isomorphism, hence is in LL, by prop. 3, hence so is the composite f=rf = r \circ \ell.

Examples

Several classical examples of OFS (E,M)(E,M):

References

Revised on November 28, 2012 15:17:04 by Urs Schreiber (82.169.65.155)