Kerodon

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

Remark 3.1.5.8. Let $\{ f_ i: X_ i \rightarrow Y_ i \} _{i \in I}$ be a collection of homotopy equivalences of simplicial sets indexed by a set $I$, and let $f: \prod _{i \in I} X_ i \rightarrow \prod _{i \in I} Y_ i$ be their product. Then:

  • If $I$ is finite, then $f$ is a homotopy equivalence. This follows from Remark 3.1.5.5 and Corollary 1.1.6.26.

  • If each of the simplicial sets $X_{i}$ and $Y_{i}$ is a Kan complex, then $f$ is a homotopy equivalence. This follows from Remark 3.1.5.5 and Corollary 1.1.9.11.

  • The morphism $f$ need not be a homotopy equivalence in general (see Warning 1.1.6.27).