Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Construction 2.4.6.1 (The Homotopy Category of a Simplicial Category). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We define an ordinary category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:

  • The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, we have

    \[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) = \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } ). \]
  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition map

    \[ \circ : \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y,Z) \times \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Z ) \]

    is given by the composition

    \begin{eqnarray*} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y,Z) \times \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) & = & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } ) \times \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet } ) \\ & \xleftarrow {\sim } & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } ) \\ & \rightarrow & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } ) \\ & = & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z). \end{eqnarray*}

We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the homotopy category of $\operatorname{\mathcal{C}}$.