Kerodon

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

Construction 4.6.7.3. Fix an integer $n \geq 0$, let $[n]$ denote the linearly ordered set $\{ 0 < 1 < \cdots < n \} $, and let $\{ x\} \star [n]$ denote the linearly ordered set obtained from $[n]$ by adjoining a new least element $x$. Let $\operatorname{Path}[ \{ x\} \star [n] ]_{\bullet }$ denote the simplicial path category of Notation 2.4.3.1). We define a simplicial functor $\pi : \operatorname{Path}[ \{ x\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{E}}[\Delta ^ n]$ as follows:

  • On objects, the functor $\pi $ is given by the formula

    \[ \pi (i) = \begin{cases} x & \textnormal{ if $i = x$} \\ y & \textnormal{ if $0 \leq i \leq n$.} \end{cases} \]
  • For $0 \leq m \leq n$, the morphism of simplicial sets

    \[ \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star [n] ] }(x, m)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}[\Delta ^ n]}( x,y )_{\bullet } = \Delta ^ n \]

    is given by the map of partially ordered sets

    \[ \{ \textnormal{Subsets $S = \{ x < i_0 < \cdots < i_ k = m \} \subseteq \{ x\} \star [n]$} \} ^{\operatorname{op}} \rightarrow [n] \quad \quad S \mapsto i_0. \]

Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X$ and $Y$. Then every $n$-simplex $\sigma \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$ determines a simplicial functor $F_{\sigma }: \operatorname{\mathcal{E}}[\Delta ^ n] \rightarrow \operatorname{\mathcal{C}}$, given on objects by $F_{\sigma }(x) = X$ and $F_{\sigma }(y) = Y$. The composition $F_{\sigma } \circ \pi $ is a simplicial functor from $\operatorname{Path}[ \{ x\} \star [n]]_{\bullet }$ to $\operatorname{\mathcal{C}}$, which (by Proposition 2.4.4.15) we can view as a map of simplicial sets $f_{\sigma }: \{ x\} \star \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$. By construction, $f_{\sigma }$ carries $x$ to $X$, and the restriction $f_{\sigma }|_{ \operatorname{N}_{\bullet }( \{ 0 < 2 < \dots < n \} ) }$ is the constant map taking the value $Y$. We can therefore identify $f_{\sigma }$ with an $n$-simplex $\theta (\sigma )$ of the left-pinched morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y)$ introduced in Construction 4.6.6.1 (see Remark 4.6.6.2). The construction $\sigma \mapsto \theta (\sigma )$ depends functorially on the object $[n] \in \operatorname{{\bf \Delta }}$, and therefore determines a map of simplicial sets

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y), \]

which we will refer to as the comparison map.