Kerodon

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Notation 3.2.3.1. Let $\Sigma $ denote the collection of all $n$-simplices $\sigma : \Delta ^ n \rightarrow X$ having the property that the restriction $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$. We let $e \in \Sigma $ denote the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X$. Note that an $(n+2)$-tuple $\vec{\sigma } = ( \sigma _0, \sigma _1, \ldots , \sigma _{n+1} )$ of elements of $\Sigma $ can be identified with a map of simplicial sets $f: \operatorname{\partial \Delta }^{n+1} \rightarrow X$, having the property that the restriction of $f$ to the $(n-1)$-skeleton of $\operatorname{\partial \Delta }^{n+1}$ is equal to the constant map $\operatorname{sk}_{n-1}( \operatorname{\partial \Delta }^{n+1} ) \rightarrow \{ x\} \subseteq X$ (see Exercise 1.1.2.8). We will say that a tuple $\vec{\sigma }$ bounds if $f$ can be extended to an $(n+1)$-simplex of $X$: that is, if there exists an $(n+1)$-simplex $\tau $ of $X$ satisfying $\sigma _{i} = d_ i(\tau )$ for $0 \leq i \leq n+1$.

The construction $\sigma \mapsto [ \sigma ]$ determines a surjective map $\Sigma \twoheadrightarrow \pi _{n}(X,x)$. We will say that a pair of elements $\sigma , \sigma ' \in \Sigma $ are homotopic if $[\sigma ] = [ \sigma ' ]$ (that is, if there is a homotopy from $\sigma $ to $\sigma '$ which is constant along the boundary $\operatorname{\partial \Delta }^ n$).