# simplicial category

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

$[n]=\{0<1<2<\ldots<n\},\quad n\geq 0,$  (1)

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

|                                                                            |                                                                            |                                                                            |
|----------------------------------------------------------------------------|----------------------------------------------------------------------------|----------------------------------------------------------------------------|
| $\displaystyle\delta^{n}_{i}$ | $\displaystyle\colon$ | $\displaystyle[n-1]\to[n]\quad\mbox{is the injection missing\ }i\in[n],$ |
| $\displaystyle\sigma^{n}_{i}$ | $\displaystyle\colon$ | $\displaystyle[n+1]\to[n]\quad\mbox{is the surjection such that\ }\sigma^{n}_{i% }(i)=\sigma^{n}_{i}(i+1)=i\in[n].$ |

The $\delta^{n}_{i}$ morphisms are called face maps, and the $\sigma^{n}_{i}$ morphisms are called degeneracy maps. They satisfy the following relations,

|                                                                             |                                                                             |                                                                             |     |
|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------|-----|
| $\displaystyle\delta^{{n+1}}_{j}\,\delta^{n}_{i}$ | $\displaystyle=$ | $\displaystyle\delta^{{n+1}}_{i}\,\delta^{n}_{{j-1}}\quad\mbox{for\ }i<j,$ | (2) |
| $\displaystyle\sigma^{{n-1}}_{j}\,\sigma^{n}_{i}$ | $\displaystyle=$ | $\displaystyle\sigma^{{n-1}}_{i}\,\sigma^{n}_{{j+1}}\quad\mbox{for\ }i\leq j,$ | (3) |
| $\displaystyle\sigma^{n}_{j}\,\delta^{{n+1}}_{i}$ | $\displaystyle=$ | $\displaystyle\left\{\begin{array}[]{ll}\delta^{n}_{i}\,\sigma^{{n-1}}_{{j-1}}&% \mbox{if\ }i<j,\\ \mathrm{id}_{n}&\mbox{if\ }i=j\mbox{\ or\ }i=j+1,\\ \delta^{n}_{{i-1}}\,\sigma^{{n-1}}_{j}&\mbox{if\ }i>j+1.\end{array}\right.$ | (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) |
| $\displaystyle(f+g)(i)$ | $\displaystyle=$ | $\displaystyle\left\{\begin{array}[]{ll}f(i)&\mbox{if\ }0\leq i\leq m,\\ g(i-m-1)+m^{{\prime}}+1&\mbox{if\ }m<i\leq(m+n+1),\end{array}\right.$ | (6) |

where $f\colon[m]\to[m^{{\prime}}]$ and $g\colon[n]\to[n^{{\prime}}]$. 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 $a\leq b$.

Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

18G30 Simplicial sets, simplicial objects (in a category)

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## Info

Owner: mhale
Added: 2002-08-27 - 23:50
Author(s): mhale

## Versions

(v8) by mhale 2013-03-22
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