Contents

category theory

# Contents

## 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 $f = r \circ \ell$.

### 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, $E$ = bo functors, $M$ = fully faithful functors: the bo-ff factorization system

• (Street) in Cat, $E$ = 0-final functors, $M$ = discrete fibrations

• (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.

## 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:

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

• Mareli Korostenski, Walter Tholen, Factorization systems as Eilenberg-Moore algebras, (doi)