Kerodon

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

Warning 3.2.4.6. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes. Then $x$ and $s$ can also be regarded as vertices of the opposite simplicial sets $X_ s^{\operatorname{op}}$ and $S^{\operatorname{op}}$, respectively, and we have canonical bijections $\pi _{n+1}(S,s) \simeq \pi _{n+1}(S^{\operatorname{op}}, s)$ and $\pi _{n}(X_ s, x) \simeq \pi _{n}(X_ s^{\operatorname{op}}, x)$, respectively. However, these bijections are not necessarily compatible with the connecting homomorphisms Construction 3.2.4.3. The diagram

\[ \xymatrix@R =50pt@C=50pt{ \pi _{n+1}(S,s) \ar [r]^-{\sim } \ar [d]^{\partial } & \pi _{n+1}(S^{\operatorname{op}}, s) \ar [d]^{ \partial } \\ \pi _{n}(X_ s, x) \ar [r]^-{\sim } & \pi _{n}(X_ s^{\operatorname{op}}, x) } \]

commutes when $n$ is odd, but anticommutes if $n \geq 2$ is even. This phenomenon is also visible in the case $n = 0$: in this case, the connecting maps $\partial : \pi _{1}(S^{\operatorname{op}},s) \rightarrow \pi _0( X^{\operatorname{op}}_ s, x)$ determine a left action of the fundamental group $\pi _{1}(S^{\operatorname{op}},s)$ on $\pi _0(X^{\operatorname{op}}_ s, x) \simeq \pi _0( X_ s, x)$, which can be interpreted as a right action of the group $\pi _{1}(S, s)$ on $\pi _0( X_ s,x)$ (see Remark 3.2.2.16). To recover the left action of Variant 3.2.4.5, we must compose with the anti-homomorphism $\pi _{1}(S,s) \rightarrow \pi _{1}(S,s)$ given by $\eta \mapsto \eta ^{-1}$.