Kerodon

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Construction 3.1.4.7 (The Homotopy Category of Kan Complexes). We define a category $\mathrm{h} \mathit{\operatorname{Kan}}$ as follows:

  • The objects of $\mathrm{h} \mathit{\operatorname{Kan}}$ are Kan complexes.

  • If $X_{}$ and $Y_{}$ are Kan complexes, then $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( X_{}, Y_{} ) = [ X_{}, Y_{} ] = \pi _0( \operatorname{Fun}( X_{}, Y_{} ) )$ is the set of homotopy classes of morphisms from $X_{}$ to $Y_{}$.

  • If $X_{}$, $Y_{}$, and $Z_{}$ are Kan complexes, then the composition law

    \[ \circ : \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( Y_{}, Z_{} ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( X_{}, Y_{} ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}}}( X_{}, Z_{} ) \]

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

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