Kerodon

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

Construction 3.2.1.10 (The Homotopy Category of Pointed Kan Complexes). We define a category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ as follows:

  • The objects of $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ are pointed Kan complexes $(X,x)$.

  • If $(X_{},x)$ and $(Y_{},y)$ are Kan complexes, then $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (X,x), (Y,y) ) = [ X_{}, Y_{} ]_{\ast }$ is the set of pointed homotopy classes of morphisms from $(X,x)$ to $(Y,y)$.

  • If $(X_{},x)$, $(Y_{},y)$, and $(Z_{},z)$ are Kan complexes, then the composition law

    \[ \circ : \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (Y,y) , (Z,z) ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (X,x), (Y,y) ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( (X,x), (Z,z)) \]

    is characterized by the formula $[g] \circ [f] = [g \circ f]$.

We will refer to $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$ as the homotopy category of pointed Kan complexes.