nLab orthogonal factorization system




Orthogonal factorization systems


Let CC be a category and let (E,M)(E,M) be two classes of morphisms in CC. We say that (E,M)(E,M) is an orthogonal factorization system if (E,M)(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)(E,M) is an orthogonal factorization system if every morphism ff in CC factors f=rf = r\circ \ell as a morphism E\ell \in E followed by a morphism rMr \in M; and the following equivalent conditions hold

  1. We have:

    a. EE is precisely the class of morphisms that are left orthogonal to every morphism in MM;

    b. MM is precisely the class of morphisms that are right orthogonal to every morphism in EE.

  2. We have:

    a. The factorization is unique up to unique isomorphism.

    b. EE and MM both contain all isomorphisms and are closed under composition.

  3. We have:

    a. EE and MM are replete subcategories of the arrow category C IC^I.

    b. Every morphism in EE is left orthogonal to every morphism in MM.

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 EE is an epimorphism and every morphism in MM is a monomorphism.

Prefactorization systems

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

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




The different characterization in def. are indeed all equivalent.



For the moment see (Joyal).



A weak factorization system (L,R)(L,R) is an orthogonal factorization system precisely if LRL \perp R.



For the moment see (Joyal).



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


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

f=Aid AAfB f = A \stackrel{id_A}{\to} A \stackrel{f}{\to} B


f=AfBid BB f = A \stackrel{f}{\to} B \stackrel{id_B}{\to} B

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

A id A A f f¯ f B id B B. \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 f¯\bar f is a left and right inverse of ff.

Closure properties

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

  • MM contains the isomorphisms and is closed under composition and pullback (insofar as pullbacks exist in CC).
  • If a composite fgf g is in MM, and ff is either in MM or a monomorphism, then gg is in MM.
  • MM is closed under all limits in the arrow category Arr(C)Arr(C).

If CC is a locally presentable category, then for any small set of maps AA, the prefactorization system ( (A ),A )({}^\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)(E,M) is any prefactorization system for which MM consists of monomorphisms and CC is complete and well-powered, then (E,M)(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


For (L,R)(L,R) an orthogonal factorization system. Let

Y f g X gf Z \array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X && \stackrel{g \circ f}{\to} && Z }

be two composable morphisms. Then

  • If ff and gfg \circ f are in LL, then so is gg.

  • If gg and gfg\circ f are in RR, then so is ff.


Consider the first case. The second is directly analogous.

Choose an (L,R)(L,R)-factorization of gg

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

With this we have lifting diagrams of the form

X gf Z f id Z Y r id Z I r ZX f Y I gf r 1 r Z id Z id Z Z \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 rr. Therefore rr is an isomorphism, hence is in LL, by prop. , hence so is the composite g=rg = r \circ \ell.

Characterization as Eilenberg-Moore algebras

Orthogonal factorization systems are equivalently described by the (appropriately defined) Eilenberg-Moore algebras with respect to the monad which belongs to the endofunctor 𝒦𝒦 2\mathcal{K} \mapsto \mathcal{K}^2 of (the 2-category) Cat (Korostenski-Tholen, Thrm B).


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


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.


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:

  • Saunders Mac Lane. Duality for groups. Bulletin of the American Mathematical Society 56.6 (1950): 485-516.

The definition of orthogonal factorisation system essentially appears under the name “factorization” in:

and under the name “factorization system” in:

See also:

Introductory texts:

A connection to double categories may be found in:

  • Miloslav Štěpán, Factorization systems and double categories, arXiv:2305.06714.

Last revised on July 4, 2024 at 08:30:23. See the history of this page for a list of all contributions to it.