# Kerodon

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Construction 3.1.5.13 (The Homotopy $2$-Category of Kan Complexes). We define a strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ as follows:

• The objects of $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ are Kan complexes.

• If $X$ and $Y$ are Kan complexes, then $\underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{Kan}} }(X,Y)$ is the fundamental groupoid of the Kan complex $\operatorname{Fun}(X,Y)$.

• If $X$, $Y$, and $Z$ are Kan complexes, then the composition law on $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ is given by

\begin{eqnarray*} \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{Kan}}}( Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{Kan}}}( X,Y) & = & \pi _{\leq 1}( \operatorname{Fun}(Y,Z) ) \times \pi _{\leq 1}( \operatorname{Fun}(X,Y) ) \\ & \simeq & \pi _{\leq 1}( \operatorname{Fun}(Y,Z) \times \operatorname{Fun}(X,Y) ) \\ & \xrightarrow {\circ } & \pi _{\leq 1}( \operatorname{Fun}(X,Z) ) \\ & = & \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{Kan}}}( X, Z ). \end{eqnarray*}

We will refer to $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ as the homotopy $2$-category of Kan complexes.