Kerodon

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

Remark 1.4.2.3. Let $S_{\bullet }$ be a simplicial set. Then the opposite simplicial set $S_{\bullet }^{\operatorname{op}}$ can be described more concretely as follows:

  • For each $n \geq 0$, we have $S_{n}^{\operatorname{op}} = S_{n}$.

  • The face and degeneracy operators of $S_{\bullet }^{\operatorname{op}}$ are given by

    \[ (d^{n}_ i: S^{\operatorname{op}}_ n \rightarrow S^{\operatorname{op}}_{n-1}) = (d^{n}_{n-i}: S_ n \rightarrow S_{n-1}) \]
    \[ (s^{n}_ i: S^{\operatorname{op}}_ n \rightarrow S^{\operatorname{op}}_{n+1}) = (s^{n}_{n-i}: S_ n \rightarrow S_{n+1}). \]