nLab orthogonal factorization system

Definition

Orthogonal factorization systems

Definition

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.

Definition

$(E,M)$ 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

1. 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 $E$.

2. We have:

   a. The factorization is unique up to unique isomorphism.

   b. $E$ and $M$ both contain all isomorphisms and are closed under composition.

3. 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.

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.

Properties

General

Proposition

The different characterization in def. are indeed all equivalent.

Proof

(…)

For the moment see (Joyal).

(…)

Proposition

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

Proof

(…)

For the moment see (Joyal).

(…)

Proposition

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

Proof

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

$$f = A \stackrel{id_A}{\to} A \stackrel{f}{\to} B$$

and

$$f = A \stackrel{f}{\to} B \stackrel{id_B}{\to} B$$

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

$$\array{ A &\stackrel{id_A}{\to}& A \\ \downarrow^{\mathrlap{f}} &\nearrow_{\bar f}& \downarrow^{\mathrlap{f}} \\ B &\stackrel{id_B}{\to}& B } \,.$$

This says precisely that $\bar f$ is a left and right inverse of $f$.

Closure properties

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.

• $M$ contains the isomorphisms and is closed under composition and pullback (insofar as pullbacks exist in $C$).
• If a composite $f g$ 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 $Arr(C)$.

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.

Cancellation properties

Proposition

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

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

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 $f$.

Proof

Consider the first case. The second is directly analogous.

Choose an $(L,R)$-factorization of $g$

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

With this we have lifting diagrams of the form

$$\array{ X &\stackrel{g \circ f}{\to}& Z \\ \downarrow^{\mathrlap{f}} && \downarrow^{id_Z} \\ Y & \nearrow_r& \\ \downarrow^{\mathrlap{\ell}} && \downarrow^{id_Z} \\ I &\stackrel{r}{\to}& Z } \;\;\;\;\;\;\; \;\;\;\;\;\;\; \array{ X &\stackrel{f}{\to}& Y &\stackrel{\ell}{\to}& I \\ {}^{\mathllap{g \circ f}}\downarrow & & \nearrow_{r^{-1}}& & \downarrow^{\mathrlap{r}} \\ Z &\underset{id_Z}{\to}& &\underset{id_Z}{\to}& Z }$$

exhibiting an inverse of $r$. Therefore $r$ is an isomorphism, hence is in $L$, by prop. , hence so is the composite $g = r \circ \ell$.

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 $\mathcal{K} \mapsto \mathcal{K}^2$ of (the 2-category) Cat (Korostenski-Tholen, Thrm B).

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, $E$ = identity-on-objects functors, $M$ = fully faithful functors (this is furthermore a strict factorisation system)

• In Cat, $E$ = bo functors, $M$ = fully faithful functors: the bo-ff factorization system

• in Cat, $E$ = final functors, $M$ = discrete fibrations (This is called the comprehensive factorization system.)

• in Cat, $E$ = initial functors, $M$ = discrete opfibrations

• in Cat, $M$ = ULF functors (see there for a reference)

• in 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.

Generalizations

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:

• In Cat, (eso, fully faithful) factorization system

• In Cat, (eso+full, faithful) factorization system

Similarly, we can have a factorization system in an (∞,1)-category, and so on; see the links below for other generalizations.

Related concepts

• factorization system

  • weak factorization system

  • orthogonal factorization system

  • strict factorization system

  • reflective factorization system

  • stable factorization system

• factorization system in a 2-category

  • enhanced factorization system

• factorization system in an (∞,1)-category

  • orthogonal factorization system in an (∞,1)-category

  • orthogonal factorization system in a derivator

• factorization structure for sinks

References

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:

• Peter Freyd, Max Kelly, Categories of continuous functors, J. Pure. Appl. Algebra 2 (1972) 169-191 [doi:10.1016/0022-4049(72)90001-1]

and under the name “factorization system” in:

• Aldridge Bousfield, Constructions of factorization systems in categories, Journal of Pure and Applied Algebra 9 2-3 (1977) 207-220 [doi:10.1016/0022-4049(77)90067-6]

See also:

• André Joyal, Factorization systems

• 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:

• Introduction to Homotopy Theory
• Emily Riehl, Factorization Systems, 2008
• The following dissertation section is entirely written after learning from Riehl’s note above, but has complementary examples and may dive deeper into some proofs:
  • Andreas Nuyts, Contributions to Multimode and Presheaf Type Theory, section 2.4: Factorization Systems, PhD thesis, KU Leuven, Belgium, 2020

A connection to double categories may be found in:

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