Kerodon

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

Remark 3.1.5.5. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. The following conditions are equivalent:

  • The morphism $f$ is a homotopy equivalence.

  • For every simplicial set $Z_{}$, composition with $f$ induces a bijection $\pi _0( \operatorname{Fun}(Y_{}, Z_{})) \rightarrow \pi _0( \operatorname{Fun}( X_{}, Z_{}) ).$.

  • For every simplicial set $W_{}$, composition with $f$ induces a bijection $\pi _0( \operatorname{Fun}(W_{}, X_{} ) ) \rightarrow \pi _0( \operatorname{Fun}( W_{}, Y_{} ))$.

In particular (taking $W_{} = \Delta ^{0}$), if $f$ is a homotopy equivalence, then the induced map $\pi _0(f): \pi _0( X_{} ) \rightarrow \pi _0( Y_{} )$ is a bijection.