Kerodon

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

Remark 5.1.3.2. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S. & } \]

The following conditions are equivalent:

  • The morphism $f$ is a homotopy equivalence relative to $S$ (Definition 5.1.3.1).

  • For every object $B$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, postcomposition with $f$ induces a homotopy equivalence of simplicial sets $\operatorname{Fun}_{/S}(B,X) \rightarrow \operatorname{Fun}_{/S}(B,Y)$.

  • For every object $B$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, postcomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}_{/S}(B,X) ) \rightarrow \pi _0( \operatorname{Fun}_{/S}(B,Y) )$.

  • For every object $Z$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, precomposition with $f$ induces a homotopy equivalence of simplicial sets $\operatorname{Fun}_{/S}(Y,Z) \rightarrow \operatorname{Fun}_{/S}(X,Z)$.

  • For every object $Z$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, precomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}_{/S}(Y,Z) )\rightarrow \pi _0( \operatorname{Fun}_{/S}(X,Z) )$.