Proposition 4.6.9.12 (01PT): Associativity

Proposition 4.6.9.12 (Associativity). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $W$, $X$, $Y$, and $Z$. Then the diagram

4.70

\begin{equation} \begin{gathered}\label{equation:homotopy-composition-diagram} \xymatrix@R =50pt@C=30pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \ar [r]^-{\circ } \ar [d]^{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \ar [d]^{\circ } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y) \ar [r]^-{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z) } \end{gathered} \end{equation}

commutes (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).

Proof. By virtue of Corollary 4.6.9.5, (4.70) is isomorphic to the diagram of restriction maps

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X,Y,Z) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y,Z) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z), } \]

which commutes in the category of simplicial sets (and therefore also in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). $\square$