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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 $ [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 $ \mathord{\mathbf{Cat}}$, which sends each object $ [n]$ to a pre-order $ \mathord{\mathbf{n+1}}$. The pre-order $ \mathord{\mathbf{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, monoid, unit, monoidal category, strict, category, initial object, empty set, bifunctor, terminal, factor, relations, generated by, maps, 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 3510 times total.

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

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