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Higher Topos Theory Section A.1: Factorization systems

Factorization induces adjoints on arrow categories

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).