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.
$(E,M)$ 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
We have:
a. $E$ is precisely the class of morphisms that are left orthogonal to every morphism in $M$;
b. $M$ is precisely the class of morphisms that are right orthogonal to every morphism in $E$.
We have:
a. The factorization is unique up to unique isomorphism.
b. $E$ and $M$ both contain all isomorphisms and are closed under composition.
We have:
a. $E$ and $M$ are replete subcategories of the arrow category $C^I$.
b. 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.
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 weak factorization system $(L,R)$ is an orthogonal factorization system precisely if $L \perp R$.
For $(L,R)$ an orthogonal factorization system in a category $C$, the intersection $L \cap R$ is precisely the class of isomorphisms in $C$.
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
and
are both $(L,R)$-factorization. Therefore there is a unique morphism $\tilde f$ in the commuting diagram
This says precisely that $\bar f$ is a left and right inverse of $f$.
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.
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.
For $(L,R)$ an orthogonal factorization system. Let
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 $f$.
Consider the first case. The second is directly analogous.
Choose an $(L,R)$-factorization of $g$
With this we have lifting diagrams of the form
exhibiting an inverse of $r$. Therefore $r$ is an isomorphism, hence is in $L$, by prop. , hence so is the composite $g = r \circ \ell$.
Orthogonal factorization systems are equivalently described by the (appropriately defined) Eilenberg-Moore algebras with respect to the monad which belongs to the endofunctor $\mathcal{K} \mapsto \mathcal{K}^2$ of (the 2-category) Cat (Korostenski-Tholen, Thrm B).
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 (This is called the comprehensive factorization system.)
(Street) in $\mathrm{Cat}$, $E$ = 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.
There is a categorified notion of a factorization system on a 2-category, in which lifts are only required to exist and be unique up to isomorphism. Some examples include:
Similarly, we can have a factorization system in an (∞,1)-category, and so on; see the links below for other generalizations.
orthogonal factorization system