Corollary 3.4.0.10 (04HS)

Corollary 3.4.0.10. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, where $Y$ is a Kan complex, and let $n$ be a positive integer. Then $f$ is $n$-connective if and only if the map $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is a surjection and the composite map

\[ \delta _{X/Y}^{\mathrm{h}}: X \rightarrow X \times _{Y} X \hookrightarrow X \times _ Y^{\mathrm{h}} X \]

is $(n-1)$-connective.

Proof. Without loss of generality, we may assume that $\pi _0(f)$ is a surjection. Using Proposition 3.1.7.1, we can factor $f$ as a composition $X \xrightarrow {u} X' \xrightarrow {f'} Y$, where $u$ is a weak homotopy equivalence and $f'$ is inner anodyne. Then $f$ is $n$-connective if and only if $f'$ is $n$-connective. Moreover, we have a commutative diagram

\[ \xymatrix { X \ar [r]^{ \delta _{X/Y}^{\mathrm{h}}} \ar [d]^{u} & X \times ^{\mathrm{h}}_{Y} X \ar [d] \\ X' \ar [r]^{\delta _{X/Y}^{\mathrm{h}}} & X' \times ^{ \mathrm{h}}_{ Y } X' } \]

where the horizontal maps are weak homotopy equivalences (Proposition 3.4.0.9), so that $\delta _{X/Y}^{\mathrm{h}}$ is $(n-1)$-connective if and only if $\delta _{f'}^{\mathrm{h}}$ is $(n-1)$-connective. We may therefore replace $f$ by $f'$ and thereby reduce to the case where $f$ is a Kan fibration. In this case, Proposition 3.4.0.7 guarantees that the inclusion map $X \times _{Y} X \hookrightarrow X \times _ Y^{\mathrm{h}} X$ is a homotopy equivalence. We are therefore reduced to showing that $f$ is $n$-connective if and only if the relative diagonal $X \hookrightarrow X \times _{Y} X$ is $(n-1)$-connective, which follows from Corollary 3.2.7.11. $\square$