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Read aloud Readability settings Prev Up Next \(\newcommand{\N}{\mathbb N}
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Section 12.4 The Homotopy Category
Definition 12.9 .
Let \(\mathcal{M}\) be a model category. The homotopy category \(\mathrm{Ho}(\mathcal{M})\) has the same objects as \(\mathcal{M}\text{,}\) with morphism sets
\begin{equation*}
\mathrm{Ho}(\mathcal{M})(X, Y) \;=\; \mathcal{M}(QRX, QRY)\,/\,{\simeq},
\end{equation*}
where \(QR\) denotes a cofibrant–fibrant replacement and \(\simeq\) is the appropriate notion of homotopy.
