Definition 12.7.
An object \(X\) of a model category \(\mathcal{C}\) is cofibrant if the unique map \(\varnothing \to X\) is a cofibration. It is fibrant if the unique map \(X \to *\) is a fibration.
Section 12.3 Cofibrant and Fibrant Objects
Applying the functorial factorization \((\alpha, \beta)\) to the map \(\varnothing \to X\) produces
\begin{equation*}
\varnothing \;\xrightarrow{\;\alpha\;}\; QX \;\xrightarrow{\;\beta\;}\; X,
\end{equation*}
where \(QX\) is cofibrant and \(\beta \colon QX \xrightarrow{\;\simeq\;} X\) is a (trivial) fibration, hence a weak equivalence. We call \(QX\) a cofibrant replacement of \(X\text{.}\) Dually, \((\gamma, \delta)\) applied to \(X \to *\) produces a fibrant replacement \(X \xrightarrow{\;\simeq\;} RX\text{.}\)
Example 12.8. Projective model structure on chain complexes.
Let \(R\) be a ring and let \(\mathrm{Ch}_{\geq 0}(R)\) denote the category of non-negatively graded chain complexes of \(R\)-modules. The projective model structure is given by:
-
weak equivalences are quasi-isomorphisms;
-
fibrations are maps \(f \colon C_\bullet \to D_\bullet\) that are surjective in every positive degree;
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cofibrations are maps \(g \colon C_\bullet \to D_\bullet\) that are degreewise injective with projective cokernel.
A complex is cofibrant if and only if each of its modules is projective. Cofibrant replacement is precisely projective resolution. Every object is fibrant.
