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



Factorization systems

(,1)(\infty,1)-Category theory




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

We say that ff is left orthogonal to gg and that gg is right orthogonal to ff and write

fg f \perp g

if for every commuting diagram

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

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

Note that the notation C AY(B,X)C_{A\sslash Y}(B,X) subtly includes the given commuting diagram, since C AYC_{A\sslash Y} is only defined relative to a particular given morphism AYA\to Y. Here we take that to be the common composite of the given commuting square, with BB and XX regarded as objects of C AYC_{A\sslash Y} via the resulting commuting triangles.


Let CC be an (∞,1)-category. An orthogonal factorization system on CC is a pair (S L,S R)(S_L, S_R) of classes of morphisms in CC that satisfy the following axioms.

  1. Both classes are stable under retracts.

  2. Every morphism in S LS_L is left orthogonal to every morphism in S RS_R;

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

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

    with fS Lf \in S_L and gS Rg \in S_R.


Closure properties


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

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

In fact:


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

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

Moreover, the adjunction units η f:ff¯\eta_f \colon f \to \bar f are of the form

X L X¯ f f¯R Y Y¯. \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 (𝒞 Δ[1]) R\big(\mathcal{C}^{\Delta[1]}\big)_R is given by the factorization in (L,R)(L,R).)

(Lurie 2009, Lemma



Section 5.2.8 of

Formalization in homotopy type theory is discussed in

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

Last revised on November 2, 2021 at 14:55:10. See the history of this page for a list of all contributions to it.