Let be a category and let be two classes of morphisms in . We say that is an orthogonal factorization system if is a weak factorization system in which solutions to lifting problems are unique.
We spell out several equivalent explicit formulation of what this means.
is an orthogonal factorization system if every morphism in factors as a morphism followed by a morphism ; and the following equivalent conditions hold
We have:
a. is precisely the class of morphisms that are left orthogonal to every morphism in ;
b. is precisely the class of morphisms that are right orthogonal to every morphism in .
We have:
a. The factorization is unique up to unique isomorphism.
b. and both contain all isomorphisms and are closed under composition.
We have:
a. and are replete subcategories of the arrow category .
b. Every morphism in is left orthogonal to every morphism in .
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 is an epimorphism and every morphism in is a monomorphism.
For any class of morphisms in , we write for the class of all morphisms that are right orthogonal to every morphism in . Dually, given we write for the class of all morphisms that are left orthogonal to every morphism in . The second condition in the definition of an OFS then says that and .
In general, and form a Galois connection on the poset of classes of morphisms in . A pair such that and is sometimes called a prefactorization system. Note that by generalities about Galois connections, for any class of maps we have prefactorization systems and . We call these generated and cogenerated by , respectively.
A weak factorization system is an orthogonal factorization system precisely if .
For an orthogonal factorization system in a category , the intersection is precisely the class of isomorphisms in .
If is clear that every isomorphism is in . Conversely, let be a morphism in . This implies that the two trivial factorizations
and
are both -factorization. Therefore there is a unique morphism in the commuting diagram
This says precisely that is a left and right inverse of .
A prefactorization system (and hence, also, a factorization system) satisfies the following closure properties. We state them for , but of course satisfies the dual property.
If is a locally presentable category, then for any small set of maps , the prefactorization system is actually a factorization system. The argument is by a transfinite construction similar to the small object argument.
On the other hand, if is any prefactorization system for which consists of monomorphisms and is complete and well-powered, then 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 an orthogonal factorization system. Let
be two composable morphisms. Then
If and are in , then so is .
If and are in , then so is .
Consider the first case. The second is directly analogous.
Choose an -factorization of
With this we have lifting diagrams of the form
exhibiting an inverse of . Therefore is an isomorphism, hence is in , by prop. , hence so is the composite .
Orthogonal factorization systems are equivalently described by the (appropriately defined) Eilenberg-Moore algebras with respect to the monad which belongs to the endofunctor of (the 2-category) Cat (Korostenski-Tholen, Thrm B).
Several classical examples of OFS :
in any topos or pretopos, = class of all epis, = class of all monos: the (epi, mono) factorization system;
more generally, in any regular category, = class of all regular epimorphisms, = class of all monos
in any quasitopos, = all epimorphisms, = all strong monomorphisms
In Cat, = identity-on-objects functors, = fully faithful functors (this is furthermore a strict factorisation system)
In Cat, = bo functors, = fully faithful functors: the bo-ff factorization system
in Cat, = final functors, = discrete fibrations (This is called the comprehensive factorization system.)
in Cat, = initial functors, = discrete opfibrations
in Cat, = ULF functors (see there for a reference)
in Cat, = conservative functors, = left orthogonal of (“iterated strict localizations” after A. Joyal)
in the category of small categories where morphisms are functors which are left exact and have right adjoints, = class of all such functors which are also localizations, = class of all such functors which are also conservative
if is a fibered category in the sense of Grothendieck, then admits a factorization system where = arrows whose projection to is invertible, = cartesian arrows in
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:
The definition of orthogonal factorisation system essentially appears under the name “factorization” in:
and under the name “factorization system” in:
See also:
Mareli Korostenski, Walter Tholen, Factorization systems as Eilenberg-Moore algebras, (doi)
Marco Grandis, On the monad of proper factorisation systems in categories, 2001, (doi, arxiv)
Introductory texts:
A connection to double categories may be found in:
Last revised on July 29, 2024 at 18:52:23. See the history of this page for a list of all contributions to it.