Kerodon

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

Notation 1.3.2.1. Let $\operatorname{Lin}$ denote the category whose objects are finite linearly ordered sets and whose morphisms are nondecreasing functions. Let $I$ be an object of $\operatorname{Lin}$, regarded as a set with a linear ordering $\leq _{I}$. We let $I^{\operatorname{op}}$ denote the same set with the opposite ordering, so that

\[ ( i \leq _{I^{\operatorname{op}} } j ) \Leftrightarrow (j \leq _{I} i ). \]

The construction $I \mapsto I^{\operatorname{op}}$ determines an equivalence from the category $\operatorname{Lin}$ to itself.

Recall that the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.1.2 is the full subcategory of $\operatorname{Lin}$ spanned by objects of the form $[n] = \{ 0 < 1 < \cdots < n \} $, and is equivalent to the full subcategory of $\operatorname{Lin}$ spanned by those linearly ordered sets which are finite and nonempty (Remark 1.1.1.3). There is a unique functor $\mathrm{Op}: \operatorname{{\bf \Delta }}\rightarrow \operatorname{{\bf \Delta }}$ for which the diagram

\[ \xymatrix { \operatorname{{\bf \Delta }}\ar [r] \ar [d]^{\mathrm{Op} } & \operatorname{Lin}\ar [d]^{I \mapsto I^{\operatorname{op}} } \\ \operatorname{{\bf \Delta }}\ar [r] & \operatorname{Lin}} \]

commutes up to isomorphism, where the horizontal maps are given by the inclusion. The functor $\mathrm{Op}$ can be described more concretely as follows:

  • For each object $[n] \in \operatorname{{\bf \Delta }}$, we have $\mathrm{Op}([n]) = [n]$ (note that the construction $i \mapsto n-i$ determines an isomorphism of $[n]$ with the opposite linear ordering $[n]^{\operatorname{op}}$).

  • For each morphism $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, the morphism $\mathrm{Op}( \alpha ): [m] \rightarrow [n]$ is given by the formula $\mathrm{Op}( \alpha )(i) = n - \alpha (m-i)$.