Kerodon

Remark 3.2.2.2. Let $B$ be a simplicial set and let $A$ be a simplicial subset. Then the simplicial set $B/A$ can be described more informally as follows: it is obtained from $B$ by collapsing the simplicial subset $A \subseteq B$ to a single vertex $q$. Beware that this informal description is a bit misleading when $A = \emptyset $: in this case, the natural map $B \rightarrow B/A$ is not surjective (instead, $B/A$ can be described as the coproduct $B_{+} = B \coprod \{ q\} $, obtained from $B$ by adding a new base point).

Remark 3.2.2.9. Let $n$ be a nonnegative integer. By virtue of Corollary 3.1.7.2, there exists an anodyne morphism $f: \Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow Q$, where $Q$ is a Kan complex. Let $q \in Q$ denote the image of the base point of $\Delta ^ n / \operatorname{\partial \Delta }^ n$. If $(X,x)$ is a pointed Kan complex, then precomposition with $f$ induces a trivial Kan fibration $\operatorname{Fun}( Q, X ) \rightarrow \operatorname{Fun}( \Delta ^ n / \operatorname{\partial \Delta }^ n, X)$ (Theorem 3.1.3.5), hence also a trivial Kan fibration

Passing to connected components, we see that $f$ induces a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }}( Q, X ) \simeq \pi _{n}(X,x)$. In other words, the functor $(X,x) \mapsto \pi _{n}(X,x)$ is corepresentable (in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$) by the pointed Kan complex $(Q,q)$ (which can be regarded as a combinatorial incarnation of the $n$-sphere).

Remark 3.2.2.15. Let $(X,x)$ be a pointed Kan complex. For $n \geq 2$, the homotopy group $\pi _{n}(X,x)$ is abelian. We will generally emphasize this by using additive notation for the group structure on $\pi _{n}(X,x)$: that is, we denote the group law by

With this convention, we can restate property $(b)$ of Theorem 3.2.2.10 as follows:

$(b)$

Let $f: \operatorname{\partial \Delta }^{n+1} \rightarrow X$ be a morphism of simplicial sets, corresponding to a tuple $(\sigma _0, \sigma _1, \ldots , \sigma _{n+1})$ of $n$-simplices of $X$. Then $f$ extends to an $(n+1)$-simplex of $X$ if and only if the sum $\sum _{i = 0}^{n+1} (-1)^{i} [ \sigma _ i ]$ vanishes in $\pi _{n}(X,x)$.

Remark 3.2.2.16 (Functoriality). Let $f: X \rightarrow Y$ be a morphism of Kan complexes, let $x$ be a vertex of $X$, and set $y = f(x)$. For each $n \geq 1$, the morphism $f$ induces a homomorphism $\pi _{n}(f): \pi _{n}(X,x) \rightarrow \pi _{n}(Y,y)$, characterized by the formula $\pi _{n}(f)( [\sigma ] ) = [ f(\sigma ) ]$ for each $n$-simplex $\sigma $ of $X$ for which $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$. We can therefore regard the construction $(X,x) \mapsto \pi _{n}(X,x)$ as a functor from the category of pointed Kan complexes to the category of groups. Moreover, this functor preserves filtered colimits.

Remark 3.2.2.17 (Homotopy Invariance). In the situation of Remark 3.2.2.16, suppose that $f: X \rightarrow Y$ is a homotopy equivalence. It follows from Proposition 3.2.1.13 that the homotopy class $[f]$ determines an isomorphism from $(X,x)$ to $(Y,y)$ in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$. In particular, the induced map $\pi _{n}(X,x) \rightarrow \pi _{n}(Y,y)$ is an isomorphism of groups for all $n > 0$ (and a bijection of sets for $n = 0$).

Remark 3.2.2.20. Let $X$ be a Kan complex and let $x$ be a vertex of $X$. Then $x$ can also be regarded as a vertex of the opposite simplicial set $X^{\operatorname{op}}$, which is also a Kan complex. For $n \geq 1$, we have an evident bijection $\varphi : \pi _{n}(X,x) \simeq \pi _{n}(X^{\operatorname{op}}, x)$. If $n \geq 2$, then this bijection is an isomorphism of abelian groups. Beware that, in the case $n=1$, it is generally not an isomorphism of groups: instead, it is an anti-isomorphism (that is, it satisfies the identity $\varphi ( \xi \xi ' ) = \varphi (\xi ') \varphi (\xi )$ for $\xi , \xi ' \in \pi _{1}(X,x)$; see Warning 3.2.2.14 above).

Remark 3.2.2.21. Let $(X,x)$ be a pointed Kan complex and let $n$ be a positive integer. Suppose that $\sigma , \sigma ': \Delta ^{n} \rightarrow X$ are $n$-simplices of $X$ for which $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ and $\sigma '|_{ \operatorname{\partial \Delta }^{n} }$ are equal to the constant map $\operatorname{\partial \Delta }^{n} \rightarrow \{ x\} \subseteq X$. It follows from Theorem 3.2.2.10 that the equality $[ \sigma ] = [\sigma ']$ holds (in the homotopy group $\pi _{n}(X,x)$) if and only if there exists an $(n+1)$-simplex $\tau $ of $X$ such that $d^{n+1}_0(\tau ) = \sigma $, $d^{n+1}_1(\tau ) = \sigma '$, and $d^{n+1}_{i}(\tau )$ is the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X$ for $2 \leq i \leq n+1$.