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
Factorisation systems appear to have been first studied by Mac Lane in the following paper under the term bicategory (not to be confused with bicategory), though this definition imposed extra conditions that are now not considered:
See also:
Last revised on December 14, 2022 at 18:29:57. See the history of this page for a list of all contributions to it.