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

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

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

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

Note that the notation $C_{A⫽Y}(B,X)$ 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 $(S_L,S_R)$ 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

   $$\begin{array}{ccc}& & Y\\ & {}^f\nearrow & & {\searrow }^g\\ X& & \stackrel{h}{\to }& & Z\end{array}$$\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 $(L,R)$ a factorization system in an (∞,1)-category $𝒞$, the full sub-(∞,1)-category of the arrow category $\mathrm{Func}({\Delta }^1,𝒞)$ 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 $(L,R)$ be an orthogonal factorization system on an $(\mathrm{\infty },1)$-category $𝒞$. Write ${𝒞}_R^I\hookrightarrow {𝒞}^I$ for the full sub-(∞,1)-category of the arrow category on the morphisms in $R$.

Then

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

   $${𝒞}_R^I\overleftarrow{\hookrightarrow }{𝒞}^I$$\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

   $$\begin{array}{ccc}X& \stackrel{\in L}{\to }& \overline{X}\\ {}^f\downarrow & & {\downarrow }^{\overline{f}\in R}\\ Y& \stackrel{\simeq }{\to }& \overline{Y}\end{array}.$$\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 $(L,R)$).

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 $(-2)\le n\le \mathrm{\infty }$.

  See (n-connected, n-truncated) factorization system.

Related concepts

• factorization system

  • weak factorization system

  • orthogonal factorization system

• factorization system in a 2-category

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

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

References

Section 5.2.8 of

• Jacob Lurie, Higher Topos Theory