PlanetMath: simplicial category

(more info)

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simplicial category

(Definition)

The simplicial category $\Delta$ is defined as the small category whose objects are the totally ordered finite sets

$\displaystyle [n] = \{0<1<2<\ldots<n\}, \quad n\geq0,$

(1)

and whose morphisms are monotonic non-decreasing (order-preserving) maps. It is generated by two families of morphisms:

$\displaystyle \delta^n_i$	$\displaystyle \colon$	$\displaystyle [n-1] \to [n]$ is the injection missing $\displaystyle i\in[n],$
$\displaystyle \sigma^n_i$	$\displaystyle \colon$	$\displaystyle [n+1] \to [n]$ is the surjection such that $\displaystyle \sigma^n_i(i)=\sigma^n_i(i+1)=i\in[n].$

The $\delta^n_i$ morphisms are called face maps, and the $\sigma^n_i$ morphisms are called degeneracy maps. They satisfy the following relations,

$\displaystyle \delta^{n+1}_j\,\delta^n_i$	$\displaystyle =$	$\displaystyle \delta^{n+1}_i\,\delta^n_{j-1}$ for $\displaystyle i<j,$	(2)
$\displaystyle \sigma^{n-1}_j\,\sigma^n_i$	$\displaystyle =$	$\displaystyle \sigma^{n-1}_i\,\sigma^n_{j+1}$ for $\displaystyle i\leq j,$	(3)
$\displaystyle \sigma^n_j\,\delta^{n+1}_i$	$\displaystyle =$	$\begin{displaymath}\left\{ \begin{array}{ll} \delta^n_i\,\sigma^{n-1}_{j-1} & \m... ...^n_{i-1}\,\sigma^{n-1}_j & \mbox{if\ }i>j+1. \end{array}\right.\end{displaymath}$	(4)

All morphisms $[n] \to [0]$ factor through $\sigma^0_0$ , so [0] is terminal.

There is a bifunctor $+\colon \Delta\times\Delta \to \Delta$ defined by

$\displaystyle [m]+[n]$	$\displaystyle =$	$\displaystyle [m+n+1],$	(5)
$\displaystyle (f+g)(i)$	$\displaystyle =$	$\begin{displaymath}\left\{ \begin{array}{ll} f(i) & \mbox{if\ } 0 \leq i \leq m,... ...-m-1)+m'+1 & \mbox{if\ } m < i \leq (m+n+1), \end{array}\right.\end{displaymath}$	(6)

where $f\colon [m] \to [m']$ and $g\colon [n] \to [n']$ . Sometimes, the simplicial category is defined to include the empty set $[-1] = \emptyset$ , which provides an initial object for the category. This makes $\Delta$ a strict monoidal category as $\emptyset$ is a unit for the bifunctor: $\emptyset+[n] = [n] = [n]+\emptyset$ and $\mathrm{id}_\emptyset+f = f = f+\mathrm{id}_\emptyset$ . Further, $\Delta$ is then the free monoidal category on a monoid object (the monoid object being [0], with product $\sigma^0_0\colon [0]+[0] \to [0]$ ).

There is a fully faithful functor from $\Delta$ to $\mathord{\mathbf{Top}}$ , which sends each object to an oriented -simplex. The face maps then embed an -simplex in an -simplex, and the degeneracy maps collapse an -simplex to an -simplex. The bifunctor forms a simplex from the disjoint union of two simplicies by joining their vertices together in a way compatible with their orientations.

There is also a fully faithful functor from $\Delta$ to $\mathord{\mathbf{Cat}}$ , which sends each object to a pre-order $\mathord{\mathbf{n+1}}$ . The pre-order $\mathord{\mathbf{n}}$ is the category consisting of partially-ordered objects, with one morphism $a \to b$ if and only if $a \leq b$ .