nLab
orthogonal factorization system

Definition

An orthogonal factorization system can be defined as a weak factorization system in which solutions to lifting problems are unique. It can also be defined more directly as a pair $(E,M)$ of classes of maps in a category $C$ such that

• Every morphism in $C$ factors as an $E$-morphism followed by an $M$-morphism, and
• $E$ is precisely the class of morphisms that are left orthogonal to every morphism in $M$.
• $M$ is precisely the class of morphisms that are right orthogonal to every morphism in $E$.

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.

A prefactorization systems $(E,M)$ (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.

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

• (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 Catlab 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