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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 \begin{equation} [n] = \{0<1<2<\ldots<n\}, \quad n\geq0, \end{equation}and whose morphisms are monotonic non-decreasing (order-preserving) maps. It is generated by two families of morphisms: \begin{eqnarray*} \delta^n_i & \colon & [n-1] \to [n] \quad\mbox{is the injection missing\ } i\in[n], \\ \sigma^n_i & \colon & [n+1] \to [n] \quad\mbox{is the surjection such that\ } \sigma^n_i(i)=\sigma^n_i(i+1)=i\in[n]. \end{eqnarray*}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 $\displaystyle i<j,$ (1)
$\displaystyle \sigma^{n-1}_j\,\sigma^n_i$ $\displaystyle =$ $\displaystyle \sigma^{n-1}_i\,\sigma^n_{j+1}$   for $\displaystyle i\leq j,$ (2)
$\displaystyle \sigma^n_j\,\delta^{n+1}_i$ $\displaystyle =$ \begin{displaymath}\left\{ \begin{array}{ll} \delta^n_i\,\sigma^{n-1}_{j-1} & \m... ...a^n_{i-1}\,\sigma^{n-1}_j & \mbox{if\ }i>j+1. \end{array}right.\end{displaymath} (3)

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

$\displaystyle [m]+[n]$ $\displaystyle =$ $\displaystyle [m+n+1],$ (4)
$\displaystyle (f+g)(i)$ $\displaystyle =$ \begin{displaymath}\left\{ \begin{array}{ll} f(i) & \mbox{if\ } 0 \leq i \leq m,... ...i-m-1)+m'+1 & \mbox{if\ } m < i \leq (m+n+1), \end{array}right.\end{displaymath} (5)

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 $\id_\emptyset+f = f = f+\id_\emptyset$ . 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 $\mathcat{Top}$ , which sends each object $[n]$ to an oriented $n$ -simplex. The face maps then embed an $(n-1)$ -simplex in an $n$ -simplex, and the degeneracy maps collapse an $(n+1)$ -simplex to an $n$ -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 $\mathcat{Cat}$ , which sends each object $[n]$ to a pre-order $\mathcat{n+1}$ . The pre-order $\mathcat{n}$ is the category consisting of $n$ partially-ordered objects, with one morphism $a \to b$ if and only if $a \leq b$ .




"simplicial category" is owned by mhale.
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See Also: simplicial object, nerve

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Cross-references: pre-order, orientations, compatible, vertices, disjoint union, face, oriented, faithful functor, product, unit, monoidal category, strict, category, initial object, empty set, bifunctor, terminal, factor, relations, generated by, maps, order-preserving, monotonic, morphisms, finite sets, totally ordered, objects, small category
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This is version 5 of simplicial category, born on 2002-08-27, modified 2004-04-16.
Object id is 3367, canonical name is SimplicialCategory.
Accessed 4189 times total.

Classification:
AMS MSC18G30 (Category theory; homological algebra :: Homological algebra :: Simplicial sets, simplicial objects )

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