simplicial objectSimplicialObject2002-08-27 20:00:402005-10-28 19:04:28Definitionsimplicial set\usepackage{amssymb}
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\newcommand*{\defn}[1]{\textbf{#1}}A \textbf{simplicial object} in a category $C$ is a contravariant functor from
the simplicial category $\Delta$ to $C$.
Such a functor $X$ is uniquely specified by the morphisms
$X(\delta^n_i)\colon X([n]) \to X([n-1])$ and $X(\sigma^n_i)\colon X([n]) \to X([n+1])$,
which satisfy
\begin{eqnarray}
X(\delta^{n-1}_i)\,X(\delta^n_j) & = & X(\delta^{n-1}_{j-1})\,X(\delta^n_i)
\quad\mbox{for\ } i<j, \\
X(\sigma^{n+1}_i)\,X(\sigma^n_j) & = & X(\sigma^{n+1}_{j+1})\,X(\sigma^n_i)
\quad\mbox{for\ } i\leq j, \\
X(\delta^{n+1}_i)\,X(\sigma^n_j) & = & \left\{
\begin{array}{ll}
X(\sigma^{n-1}_{j-1})\,X(\delta^n_i) & \mbox{if\ } i<j, \\
\id_n & \mbox{if\ } i=j \mbox{\ or\ } i=j+1, \\
X(\sigma^{n-1}_j)\,X(\delta^n_{i-1}) & \mbox{if\ } i>j+1.
\end{array}\right.
\end{eqnarray}
In particular, a \textbf{simplicial set} is a simplicial object in $\mathcat{Set}$.
Equivalently, one could say that a simplicial set is a presheaf on $\Delta$.
The object $X([n])$ of a simplicial set is a set of $n$-simplices,
and is called the $n$-skeleton.