Books & notes

Higher Topos Theory · Appendix A · Section A.1

Factorization systems

A.1. Factorization systems

We begin with the definition of orthogonality.

Definition A.1.

Let \(C\) be a category. A morphism \(l \colon A \to B\) is said to be left orthogonal to a morphism \(r \colon X \to Y\), written \(l \perp r\), if the following square is a pullback square:

Commutative diagram generated from the LaTeX source

Equivalently, for every commutative square

Commutative diagram generated from the LaTeX source

the anima of dashed fillers making both triangles commute is contractible. In this case, we also say that \(r\) is right orthogonal to \(l\).

Given a collection of morphisms \(S\), we denote by \(S^\perp\) the collection of all morphisms that are right orthogonal to every morphism in \(S\), and by \({}^\perp S\) the collection of those that are left orthogonal to every morphism in \(S\).

Definition A.2.

A factorization system on a category \(C\) consists of two classes of morphisms \(L\) and \(R\) of \(C\) satisfying the following two conditions:

  1. Existence of factorization: Every morphism \(f\colon A\to B\) in \(C\) admits a factorization

    \[A\xrightarrow{l}X\xrightarrow{r}B\]

    with \(l\in L\) and \(r\in R\).

  2. Orthogonality: Every morphism \(l \in L\) is left orthogonal to every morphism \(r \in R\).

The classes of maps in a factorization system satisfy various useful closure properties.

Proposition A.3.

Let \((L,R)\) be a factorization system on \(C\).

  1. We have \(L = {}^{\perp}R\) and \(R = L^{\perp}\).

  2. The classes \(L\) and \(R\) are closed under composition and contain all isomorphisms.

  3. The classes \(L\) and \(R\) are closed under retracts.

  4. The class \(R\) is closed under base change (pullbacks).

  5. The class \(L\) is closed under cobase change (pushouts).

  6. The class \(R\) has the left cancellation property: if \(g \circ f \in R\) and \(g \in R\), then \(f \in R\).

  7. The class \(L\) has the right cancellation property: if \(g \circ f \in L\) and \(f \in L\), then \(g \in L\).

  8. If \(C\) admits small limits, the class \(R\) is closed under small limits in \(\Ar(C)\).

  9. If \(C\) admits small colimits, the class \(L\) is closed under small colimits in \(\Ar(C)\).

Proof
The inclusions \(L\subseteq{}^{\perp}R\) and \(R\subseteq L^{\perp}\) are part of the definition. Conversely, suppose that \(f\in{}^{\perp}R\) and choose a factorization of \(f\) into \(l\in L\) followed by \(r\in R\). Orthogonality of \(f\) and \(r\) gives a section \(s\) of \(r\). Comparing the endomorphism \(sr\) of the middle object with its identity, using \(l\perp r\), shows that \(s\) is also a retraction. Thus \(r\) is an isomorphism and \(f\in L\). This proves \(L={}^\perp R\); the proof that \(R=L^\perp\) is dual.Orthogonality classes contain the isomorphisms and are closed under composition and retracts, which proves (2) and (3). A lifting problem against a base change of a morphism in \(R\) is equivalently a lifting problem against the original morphism; hence \(R\) is closed under base change. Dually, \(L\) is closed under cobase change. The cancellation properties follow by applying orthogonality successively to the two composable morphisms.Finally, if \((r_i)_i\) is a small diagram of morphisms in \(R\), then for every \(l\in L\) the anima of fillers from \(l\) to \(\lim_i r_i\) is the limit of the contractible animae of fillers from \(l\) to the \(r_i\). It is therefore contractible, so \(\lim_i r_i\in R\). The colimit statement for \(L\) is dual. Compare [Lurie 2009, Propositions 5.2.8.6 and 5.2.8.11].

Lemma A.4.

Let \((L,R)\) be a factorization system on \(C\). Any morphism \(f\colon X \to Y\) which is both in \(L\) and in \(R\) is an isomorphism.

Proof
Consider the commutative square:
Commutative diagram generated from the LaTeX source
Since \(f \in L\) as the left vertical map and \(f \in R\) as the right vertical map, orthogonality provides a unique filler \(g\). The two triangles say precisely that \(gf=\id_X\) and \(fg=\id_Y\), so \(g\) is an inverse to \(f\).

Lemma A.5.

Let \((L,R)\) be a factorization system on \(C\), and consider morphisms \(X \xrightarrow{f} Y \xrightarrow{g} Z\).

  1. Assume \(C\) admits pullbacks. If \(gf \in R\) and \(\Delta_g \in R\), then \(f \in R\).

  2. Assume \(C\) admits pushouts. If \(gf \in L\) and \(\nabla_f \in L\), then \(g \in L\).

Proof
We prove (1); the proof for (2) is dual. We may factor \(f\) as the composite
\[X \xrightarrow{(\id_X, f)} X \times_Z Y \xrightarrow{\pr_Y} Y.\]
The second map is a base change of \(gf\colon X \to Z\) along \(g\), hence lies in \(R\) because \(gf \in R\). The first map is a base change of the diagonal \(\Delta_g\colon Y \to Y \times_Z Y\) along the map \(f \times \id_Y \colon X \times_Z Y \to Y \times_Z Y\). Since \(\Delta_g \in R\), it follows that \((\id_X, f) \in R\). Since \(R\) is closed under composition, we conclude \(f \in R\).

Lemma A.6. ([Anel et al. 2022, Lemma 3.1.8])

Let \((L,R)\) be a factorization system on a category \(C\). Then for every object \(X \in C\) the slice \(C_{/X}\) admits a factorization system in which a morphism lies in the left, resp. right, class if and only if its underlying morphism in \(C\) lies in \(L\), resp. \(R\).

Proof
Given a morphism \(A\to B\) in \(C_{/X}\), factor its underlying morphism in \(C\) as \(A\xrightarrow{l}M\xrightarrow{r}B\). Giving \(M\) the structure map \(M\xrightarrow{r}B\to X\) makes this a factorization in \(C_{/X}\). Orthogonality in the slice follows from orthogonality in \(C\): the unique filler of the underlying square automatically commutes with the structure maps to \(X\).

Lemma A.7.

Let \((L,R)\) be a factorization system on \(C\).

  1. Let \(\Ar^R(C) \subseteq \Ar(C)\) denote the full subcategory spanned by the morphisms in \(R\). Then the inclusion \(\Ar^R(C) \hookrightarrow \Ar(C)\) admits a left adjoint. Given a morphism \(f\colon A \to B\) with factorization \(A \xrightarrow{l} C \xrightarrow{r} B\), this left adjoint sends \(f\) to \(r\).

  2. Similarly, the inclusion \(\Ar^L(C) \hookrightarrow \Ar(C)\) of the morphisms in \(L\) admits a right adjoint, sending \(f\) to \(l\).

  3. For every object \(X\in C\), let \(R[X]\subseteq C_{/X}\) be the full subcategory spanned by the morphisms in \(R\). Then \(R[X]\) is reflective in \(C_{/X}\), and the reflection sends a morphism \(f\colon A\to X\) to its right factor \(r\colon C\to X\).

Proof
By symmetry, it suffices to prove statement (1). The morphism
Commutative diagram generated from the LaTeX source
is the unit of the reflection. Indeed, for any \(r'\in\Ar^R(C)\), the map
\[\Hom_{\Ar(C)}(r, r') \to \Hom_{\Ar(C)}(f,r')\]
is an isomorphism as a consequence of the fact that \(l \perp r'\). This proves (1), and (2) follows dually. Applying the same argument in the slice factorization system of Lemma A.6 proves (3).

References

  1. Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.
  2. Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of ∞-topoi. I: Higher sheaves. Adv. Math., 400, 64. 2022.