# nLab orthogonal factorization system in an (infinity,1)-category

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## Definition

###### Definition

Let $C$ be an (∞,1)-category and $f:A\to B$ and $g:X\to Y$ two morphisms in $C$. Write ${C}_{A⫽Y}$ for the under-over-(∞,1)-category.

We say that $f$ is left orthogonal to $g$ and that $g$ is right orthogonal to $f$ and write

$f\perp g$f \perp g

if for every commuting diagram

$\begin{array}{ccc}A& \to & X\\ {}^{f}↓& {⇙}_{\simeq }& {↓}^{g}\\ B& \to & Y\end{array}$\array{ A &\to& X \\ {}^{\mathllap{f}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{g}} \\ B &\to& Y }

in $C$ we have that ${C}_{A⫽Y}\left(B,X\right)\simeq *$ is contractible.

Note that the notation ${C}_{A⫽Y}\left(B,X\right)$ subtly includes the given commuting diagram, since ${C}_{A⫽Y}$ is only defined relative to a particular given morphism $A\to Y$. Here we take that to be the common composite of the given commuting square, with $B$ and $X$ regarded as objects of ${C}_{A⫽Y}$ via the resulting commuting triangles.

###### Definition

Let $C$ be an (∞,1)-category. An orthogonal factorization system on $C$ is a pair $\left({S}_{L},{S}_{R}\right)$ of classes of morphisms in $C$ that satisfy the following axioms.

1. Both classes are stable under retracts.

2. Every morphism in ${S}_{L}$ is left orthogonal to every morphism in ${S}_{R}$;

3. For every morphism $h:X\to Z$ in $C$ there exists a commuting triangle

$\begin{array}{ccc}& & Y\\ & {}^{f}↗& & {↘}^{g}\\ X& & \stackrel{h}{\to }& & Z\end{array}$\array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z }

with $f\in {S}_{L}$ and $g\in {S}_{R}$.

## Properties

### Closure properties

###### Proposition

For $\left(L,R\right)$ a factorization system in an (∞,1)-category $𝒞$, the full sub-(∞,1)-category of the arrow category $\mathrm{Func}\left({\Delta }^{1},𝒞\right)$ on the morphisms in $R$ is closed under (∞,1)-limits of shapes that exist in $𝒞$. Similarly the full subcategory on $L$ is closed under (∞,1)-colimits that exist in $𝒞$.

This is (Lurie, prop. 5.2.6.8 (7), (8)).

### Reflection

###### Definition

Let $\left(L,R\right)$ be an orthogonal factorization system on an $\left(\infty ,1\right)$-category $𝒞$. Write ${𝒞}_{R}^{I}↪{𝒞}^{I}$ for the full sub-(∞,1)-category of the arrow category on the morphisms in $R$.

Then

1. this is a reflective sub-(∞,1)-category

${𝒞}_{R}^{I}\stackrel{\stackrel{}{←}}{↪}{𝒞}^{I}$\mathcal{C}^I_R \stackrel{\stackrel{}{\leftarrow}}{\hookrightarrow} \mathcal{C}^I
2. The adjunction units ${\eta }_{f}:f\to \overline{f}$ are of the form

$\begin{array}{ccc}X& \stackrel{\in L}{\to }& \overline{X}\\ {}^{f}↓& & {↓}^{\overline{f}\in R}\\ Y& \stackrel{\simeq }{\to }& \overline{Y}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X &\stackrel{\in L}{\to}& \bar X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{\bar f \in R} } \\ Y &\stackrel{\simeq}{\to}& \bar Y } \,.

(In words: the reflection into ${𝒞}_{R}^{I}$ is given by the factorization in $\left(L,R\right)$).

This is (Lurie, lemma 5.2.8.19).

## Examples

• In an (∞,1)-topos the classe of n-connected and that of n-truncated morphisms form an orthogonal factorization system, for all $\left(-2\right)\le n\le \infty$.

## References

Section 5.2.8 of

Revised on November 12, 2012 23:49:06 by Urs Schreiber (89.204.137.233)