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orthogonal factorization system

Definition

Orthogonal factorization systems

Definition

Let $C$ be a category and let $(E,M)$ be two classes of morphisms in $C$. We say that $(E,M)$ is an orthogonal factorization system if $(E,M)$ 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

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

1. We have:

   1. $E$ is precisely the class of morphisms that are left orthogonal to every morphism in $M$;

   2. $M$ is precisely the class of morphisms that are right orthogonal to every morphism in $E$.

2. We have:

   1. The factorization is unique up to unique isomorphism.

   2. $E$ and $M$ both contain all isomorphisms and are closed under composition.

3. We have:

   1. $E$ and $M$ are replete subcategories of the arrow category $C^I$.

   2. Every morphism in $E$ is left orthogonal to every morphism in $M$.

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 $E$ is an epimorphism and every morphism in $M$ is a monomorphism.

Prefactorization systems

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

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

Proof

(…)

For the moment see (Joyal).

(…)

Proposition

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

Proof

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

$$f=A\stackrel{{\mathrm{id}}_A}{\to }A\stackrel{f}{\to }B$$f = A \stackrel{id_A}{\to} A \stackrel{f}{\to} B

and

$$f=A\stackrel{f}{\to }B\stackrel{{\mathrm{id}}_B}{\to }B$$f = A \stackrel{f}{\to} B \stackrel{id_B}{\to} B

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

$$\begin{array}{ccc}A& \stackrel{{\mathrm{id}}_A}{\to }& A\\ {\downarrow }^f& {\nearrow }_{\overline{f}}& {\downarrow }^f\\ B& \stackrel{{\mathrm{id}}_B}{\to }& B\end{array}.$$\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 $\overline{f}$ is a left and right inverse of $f$.

Closure properties

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

• $M$ contains the isomorphisms and is closed under composition and pullback (insofar as pullbacks exist in $C$).
• If a composite $fg$ is in $M$, and $f$ is either in $M$ or a monomorphism, then $g$ is in $M$.
• $M$ is closed under all limits in the arrow category $\mathrm{Arr}(C)$.

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

On the other hand, if $(E,M)$ is any prefactorization system for which $M$ consists of monomorphisms and $C$ is complete and well-powered, then $(E,M)$ 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 $(L,R)$ an orthogonal factorization system. Let

$$\begin{array}{ccc}& & Y\\ & {}^f\nearrow & & {\searrow }^g\\ X& & \stackrel{g\circ f}{\to }& & Z\end{array}$$\array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X && \stackrel{g \circ f}{\to} && Z }

be two composable morphisms. Then

• If $f$ and $g\circ f$ are in $L$, then so is $g$.

• If $g$ and $g\circ f$ are in $R$, then so is $f$.

Proof

Consider the first case. The second is directly analogous.

Choose an $(L,R)$-factorization of $g$

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

With this we have lifting diagrams of the form

$$\begin{array}{ccc}X& \stackrel{g\circ f}{\to }& Z\\ {\downarrow }^f& & {\downarrow }^{{\mathrm{id}}_Z}\\ Y& {\nearrow }_r& \\ {\downarrow }^{\ell }& & {\downarrow }^{{\mathrm{id}}_Z}\\ I& \stackrel{r}{\to }& Z\end{array}\begin{array}{ccccc}X& \stackrel{f}{\to }& Y& \stackrel{\ell }{\to }& I\\ {}^{g\circ f}\downarrow & & {\nearrow }_{r^{-1}}& & {\downarrow }^r\\ Z& \underset{{\mathrm{id}}_Z}{\to }& & \underset{{\mathrm{id}}_Z}{\to }& Z\end{array}$$\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 $r$. Therefore $r$ is an isomorphism, hence is in $L$, by prop. 3, hence so is the composite $f=r\circ \ell$.

Examples

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

• in any topos or pretopos, $E$ = class of all epis, $M$ = class of all monos: the (epi, mono) factorization system;

• more generally, in any regular category, $E$ = class of all regular epimorphisms, $M$ = class of all monos

• in any quasitopos, $E$ = all epimorphisms, $M$ = all strong monomorphisms

• In Cat, $E$ = bo functors, $M$ = fully faithful functors: the bo-ff factorization system

• In Cat, (eso, fully faithful) factorization system

• In Cat, (eso+full, faithful) factorization system

• (Street) in Cat, $E$ = 0-final functors, $M$ = discrete fibrations

• (Street) in $\mathrm{Cat}$, $M$ = 0-initial functors, $M$ = discrete opfibrations

• in $\mathrm{Cat}$, $M$ = conservative functors, $E$ = left orthogonal of $M$ (“iterated strict localizations” after A. Joyal)

• in the category of small categories where morphisms are functors which are left exact and have right adjoints, $E$ = class of all such functors which are also localizations, $M$ = class of all such functors which are also conservative

• if $F\to C$ is a fibered category in the sense of Grothendieck, then $F$ admits a factorization system $(E,M)$ where $E$ = arrows whose projection to $C$ is invertible, $M$ = cartesian arrows in $F$

• See the (Joyal) for more examples.

Related concepts

• factorization system

  • weak factorization system

  • orthogonal factorization system

  • reflective factorization system

  • stable factorization system

• factorization system in a 2-category

• factorization system in an (∞,1)-category

  • orthogonal factorization system in an (∞,1)-category

• factorization structure for sinks

References

• André Joyal Factorization systems