# Kerodon

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

Notation 8.1.1.6 (Projection Maps). Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is equipped with projection maps

$\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \quad \quad \lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}.$

Here $\lambda _{+}$ carries an $n$-simplex $\sigma$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to the $n$-simplex of $\operatorname{\mathcal{C}}$ given by the composition

$\Delta ^ n = \operatorname{N}_{\bullet }( [n] ) \hookrightarrow \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \xrightarrow {\sigma } \operatorname{\mathcal{C}},$

while $\lambda _{-}$ carries $\sigma$ to the $n$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}$ given by the composite map

$(\Delta ^ n)^{\operatorname{op}} = \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} ) \hookrightarrow \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \xrightarrow {\sigma } \operatorname{\mathcal{C}}.$

Concretely, $\lambda _{-}$ and $\lambda _{+}$ are given on vertices by the formulae $\lambda _{-}( f: X \rightarrow Y ) = X$ and $\lambda _{+}( f: X \rightarrow Y ) = Y$.