Definition 12.4.
A model structure on a category \(\mathcal{C}\) consists of three subcategories — cofibrations \(\mathrm{Cof}\text{,}\) fibrations \(\mathrm{Fib}\text{,}\) and weak equivalences \(\mathcal{W}\) — together with two functorial factorizations \((\alpha, \beta)\) and \((\gamma, \delta)\text{,}\) satisfying:
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(Two-out-of-three.) If two of \(\{f, g, g \circ f\}\) are weak equivalences, then so is the third.
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(Retracts.) The classes \(\mathrm{Cof}\text{,}\) \(\mathrm{Fib}\text{,}\) and \(\mathcal{W}\) are each closed under retracts in \(\mathrm{Ar}(\mathcal{C})\text{.}\)
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(Lifting.) A trivial cofibration is a map that is both a cofibration and a weak equivalence; similarly for trivial fibrations. Trivial cofibrations have the LLP with respect to fibrations, and trivial fibrations have the RLP with respect to cofibrations.
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(Factorizations.) For every morphism \(f\text{,}\)\begin{equation*} \alpha(f) \in \mathrm{Cof}, \quad \beta(f) \in \mathrm{Fib} \cap \mathcal{W}, \quad \gamma(f) \in \mathrm{Cof} \cap \mathcal{W}, \quad \delta(f) \in \mathrm{Fib}. \end{equation*}
