Kerodon

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

Proposition 3.2.4.2. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes and let $n \geq 0$ be a nonnegative integer. Then there exists a unique function $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ with the following property:

$(\ast )$

Let $\sigma : \Delta ^{n} \rightarrow X_{s}$ and $\tau : \Delta ^{n+1} \rightarrow S$ be simplices having the property that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ and $\tau |_{ \operatorname{\partial \Delta }^{n+1} }$ are the constant maps taking the values $x$ and $s$, respectively. Then $\sigma $ is incident to $\tau $ (in the sense of Definition 3.2.4.1) if and only if $\partial ( [ \tau ] ) = [\sigma ]$.

Proof of Proposition 3.2.4.2. Fix a map $\tau : \Delta ^{n+1} \rightarrow S$ be an $(n+1)$-simplex for which $\tau |_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $s$. To prove Proposition 3.2.4.2, it will suffice to prove the following:

$(1)$

There exists an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X_ s$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ is the constant map taking the value $x$ and $\sigma $ is incident to $\tau $.

$(2)$

Let $\sigma ': \Delta ^{n} \rightarrow X_{s}$ and $\tau ': \Delta ^{n+1} \rightarrow S$ have the property that $\sigma '|_{ \operatorname{\partial \Delta }^{n} }$ and $\tau '|_{ \operatorname{\partial \Delta }^{n+1} }$ are the constant maps taking the values $x$ and $s$, respectively, and suppose that $[\tau ] = [\tau ']$ in $\pi _{n+1}(S,s)$. Then $\sigma '$ is incident to $\tau '$ if and only if $[ \sigma ] = [ \sigma ' ]$ in $\pi _{n}(X_ s, x)$.

Assertion $(1)$ follows from the solvability of the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{0} \ar [r] \ar [d] & X \ar [d]^{f} \\ \Delta ^{n+1} \ar [r]^-{\tau } \ar@ {-->}[ur]^{ \widetilde{\tau } } & S, } \]

where the upper horizontal map is constant taking the value $x$. Let $\sigma '$ and $\tau '$ be as in $(2)$, and let $\widetilde{\tau }'_{0}: \operatorname{\partial \Delta }^{n+1} \rightarrow X_{s}$ be the map given by the tuple of $n$-simplices $(\sigma ', e, \ldots , e)$ (see Exercise 1.1.2.8) where $e: \Delta ^{n} \rightarrow X_{s}$ denotes the constant map taking the value $x$. If $[ \sigma ] = [ \sigma ' ]$ in $\pi _{n}(X_ s, x)$, then we can choose a homotopy from $\sigma $ to $\sigma '$ (in the Kan complex $X_ s$) which is constant along the boundary $\operatorname{\partial \Delta }^{n}$, and therefore a homotopy $\widetilde{h}_0$ from $\widetilde{\tau }|_{ \operatorname{\partial \Delta }^{n+1} }$ to $\widetilde{\tau }'_{0}$ (also in the Kan complex $X_ s$) which is constant along the simplicial subset $\Lambda ^{n+1}_{0} \subset \operatorname{\partial \Delta }^{n+1}$. Let $h: \Delta ^{1} \times \Delta ^{n+1} \rightarrow S$ be a homotopy from $\tau $ to $\tau '$ which is constant on $\operatorname{\partial \Delta }^{n+1}$. Since $f$ is a Kan fibration, the homotopy extension lifting problem

\[ \xymatrix@R =50pt@C=50pt{ (\Delta ^1 \times \operatorname{\partial \Delta }^{n+1}) \coprod _{ \{ 0\} \times \operatorname{\partial \Delta }^{n+1} } (\{ 0\} \times \Delta ^{n+1} ) \ar [r]^-{ ( \widetilde{h}_0, \widetilde{\tau } ) } \ar [d] & X \ar [d]^{f} \\ \Delta ^{1} \times \Delta ^{n+1} \ar [r]^-{h} \ar@ {-->}[ur]^{ \widetilde{h} } & S } \]

admits a solution $\widetilde{h}: \Delta ^{1} \times \Delta ^{n+1} \rightarrow X$ (Remark 3.1.4.3), which we can regard as a homotopy from $\widetilde{\tau }$ to another $(n+1)$-simplex $\widetilde{\tau }': \Delta ^{n+1} \rightarrow X$. By construction, this $(n+1)$-simplex witnesses that $\sigma '$ is incident to $\tau '$.

For the converse, suppose that $\sigma '$ is incident to $\tau '$, so that there exists an $(n+1)$-simplex $\widetilde{\tau }': \Delta ^{n+1} \rightarrow X$ satisfying $d_0( \widetilde{\tau }') = \sigma '$, $f( \widetilde{\tau }' ) = \tau '$, and $\widetilde{\tau }'|_{ \Lambda ^{n+1}_{0} }$ is the constant map taking the value $x$. Since $f$ is a Kan fibration, the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ (\Delta ^1 \times \Lambda ^{n+1}_0) \coprod _{ \operatorname{\partial \Delta }^{1} \times \Lambda ^{n+1}_{0} } (\operatorname{\partial \Delta }^{1} \times \Delta ^{n+1} ) \ar [r]^-{ ( \overline{e}, ( \widetilde{\tau }, \widetilde{\tau }' )) } \ar [d] & X \ar [d]^{f} \\ \Delta ^{1} \times \Delta ^{n+1} \ar [r]^-{h} \ar@ {-->}[ur]^{ \widetilde{h} } & S } \]

admits a solution, where $\overline{e}: \Delta ^1 \times \Lambda ^{n+1}_0 \rightarrow X$ is the constant map taking the value $x$. Then $\widetilde{h}$ is a homotopy from $\widetilde{\tau }$ to $\widetilde{\tau }'$ (in the Kan complex $X$) which is constant along the horn $\Lambda ^{n+1}_{0} \subseteq \Delta ^{n+1}$, and it restricts to a homotopy from $\sigma = d_0( \widetilde{\tau } )$ to $\sigma ' = d_0( \widetilde{\tau }' )$ (in the Kan complex $X_ s$) which is constant along the boundary $\operatorname{\partial \Delta }^{n}$. It follows that $[\sigma ] = [\sigma ']$ in $\pi _{n}(X_ s,x)$. $\square$