Remark 3.1.6.20 (00X7): Coproducts of Weak Homotopy Equivalences

Remark 3.1.6.20 (Coproducts of Weak Homotopy Equivalences). Let $\{ f(i): X(i) \rightarrow Y(i) \} _{i \in I}$ be a collection of weak homotopy equivalences of simplicial sets indexed by a set $I$. For every Kan complex $Z$, we have a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \coprod _{i \in I} Y(i), Z) \ar [r] \ar [d]^{\sim } & \operatorname{Fun}( \coprod _{i \in I} X(i), Z) \ar [d]^{\sim } \\ \prod _{i \in I} \operatorname{Fun}( Y(i), Z) \ar [r] & \prod _{i \in I} \operatorname{Fun}(X(i), Z), } \]

where the vertical maps are isomorphisms. Passing to the connected components (and using the fact that the functor $Q \mapsto \pi _0(Q)$ preserves products when restricted to Kan complexes; see Corollary 1.2.5.11), we deduce that the map $\pi _0( \operatorname{Fun}( \coprod _{i \in I} Y(i), Z) ) \rightarrow \pi _0( \operatorname{Fun}(\coprod _{i \in I} X(i), Z) )$ is bijective. Allowing $Z$ to vary, we conclude that the induced map $\coprod _{i \in I} X(i) \rightarrow \coprod _{i \in I} Y(i)$ is also a weak homotopy equivalence.