Definition 17.1.4 (Factorization system). A factorization system on an \(\infty \)-category \(C\) consists of a pair of wide subcategories \((\Ll , \Rr )\) of \(C\) satisfying the following conditions:
- (1)
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Orthogonality: Every morphism \(l \in \Ll \) is left orthogonal to every morphism \(r \in \Rr \).
- (2)
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Factorization: Every morphism \(f\) in \(C\) admits a factorization \(f \simeq r \circ l\), where \(l \in \Ll \) and \(r \in \Rr \).
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