nLab 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 $C_{A\sslash Y}$ 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

$$f \perp g$$

if for every commuting diagram

$$\array{ A &\to& X \\ {}^{\mathllap{f}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{g}} \\ B &\to& Y }$$

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

Note that the notation $C_{A\sslash Y}(B,X)$ subtly includes the given commuting diagram, since $C_{A\sslash Y}$ 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 $C_{A\sslash Y}$ 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

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

   with $f \in S_L$ and $g \in S_R$.

Properties

Closure properties

Proposition

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

(Lurie 2009, prop. 5.2.8.6 (7), (8))

In fact:

Proposition

The full sub-$\infty$-category of the arrow category on the right class is a reflective sub-$\infty$-category

$$\big(\mathcal{C}^{\Delta[1]}\big)_R \underoverset {\underset{}{\hookrightarrow}} {\overset{}{\longleftarrow}} {\;\; \bot \;\;} \mathcal{C}^{\Delta[1]}$$

Moreover, the adjunction units $\eta_f \colon f \to \bar f$ are of the form

$$\array{ X &\stackrel{\in L}{\longrightarrow}& \bar X \\ {}^{\mathllap{f}} \big\downarrow && \big\downarrow{}^{\mathrlap{\bar f \in R} } \\ Y &\stackrel{\simeq}{\longrightarrow}& \bar Y } \,.$$

(In words: the reflection into $\big(\mathcal{C}^{\Delta[1]}\big)_R$ is given by the factorization in $(L,R)$.)

(Lurie 2009, 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 $(-2) \leq n \leq \infty$.

  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

  • orthogonal factorization system in a derivator.

  • homotopy factorization system in a model category

References

Section 5.2.8 of

• Jacob Lurie, Higher Topos Theory

Formalization in homotopy type theory is discussed in

• Egbert Rijke, Orthogonal factorization in HoTT, talk at IAS, January 24, 2013 (video)