simplicial category SimplicialCategory 2002-08-27 19:50:37 2004-04-16 11:14:33 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % making logically defined graphics %\usepackage{xypic} % my maths package \newcommand*{\Nset}{\mathbb{N}} \newcommand*{\Zset}{\mathbb{Z}} \newcommand*{\Qset}{\mathbb{Q}} \newcommand*{\Rset}{\mathbb{R}} \newcommand*{\Cset}{\mathbb{C}} \newcommand*{\Hset}{\mathbb{H}} \newcommand*{\Oset}{\mathbb{O}} \newcommand*{\Bset}{\mathbb{B}} \newcommand*{\Kset}{\mathbb{K}} \newcommand*{\Sset}{\mathbb{S}} \newcommand*{\Tset}{\mathbb{T}} \newcommand*{\GLgrp}{\mathrm{GL}} \newcommand*{\SLgrp}{\mathrm{SL}} \newcommand*{\Ogrp}{\mathrm{O}} \newcommand*{\SOgrp}{\mathrm{SO}} \newcommand*{\Ugrp}{\mathrm{U}} \newcommand*{\SUgrp}{\mathrm{SU}} \newcommand*{\e}{\mathop{\mathrm{e}}\nolimits} \newcommand*{\im}{\mathord{\mathrm{i}}} \newcommand*{\identity}{\mathord{\mathrm{1\!\!\!\:I}}} \newcommand*{\tr}{\mathop{\mathrm{tr}}} \newcommand*{\Tr}{\mathop{\mathrm{Tr}}} \renewcommand*{\d}{\mathrm{d}} \newcommand*{\deriv}[2]{\frac{\d #1}{\d #2}} \newcommand*{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand*{\fderiv}[2]{\frac{\delta #1}{\delta #2}} % my noncommutative geometry package \newcommand*{\algebra}[1][A]{\mathord{\mathcal{#1}}} \newcommand*{\hilbert}[1][H]{\mathord{\mathcal{#1}}} \newcommand*{\hilbmod}[1][E]{\mathord{\mathcal{#1}}} \newcommand*{\Matrix}[2]{\mathord{\mathrm{M}_{#1}(#2)}} \newcommand*{\dixmier}{\mathop{\mathrm{Tr}_\omega}} \newcommand*{\Res}{\mathop{\mathrm{Res}}} \newcommand*{\Wres}{\mathop{\mathrm{Wres}}} \newcommand*{\Aut}{\mathop{\mathrm{Aut}}\nolimits} \newcommand*{\Inn}{\mathop{\mathrm{Inn}}\nolimits} \newcommand*{\Out}{\mathop{\mathrm{Out}}\nolimits} \newcommand*{\Diff}{\mathop{\mathrm{Diff}}\nolimits} \newcommand*{\Ker}{\mathop{\mathrm{Ker}}\nolimits} \newcommand*{\Coker}{\mathop{\mathrm{Coker}}\nolimits} \newcommand*{\Img}{\mathop{\mathrm{Im}}\nolimits} \newcommand*{\End}{\mathop{\mathrm{End}}\nolimits} \newcommand*{\spin}{\mathop{\mathrm{spin}}\nolimits} \newcommand*{\Ind}{\mathop{\mathrm{Ind}}\nolimits} \newcommand*{\KK}{\mathit{KK}} \newcommand*{\HH}{\mathit{HH}} \newcommand*{\HC}{\mathit{HC}} \newcommand*{\ch}{\mathop{\mathrm{ch}}\nolimits} % my category theory package \newcommand*{\mathcat}[1]{\mathord{\mathbf{#1}}} \newcommand*{\id}{\mathrm{id}} \newcommand*{\op}{\mathrm{op}} \newcommand*{\boxprod}{\mathbin{\square}} % my environments \newtheoremstyle{inlinedefn}{}{0pt}{}{}{\bfseries}{.}{0.5em}{} \theoremstyle{inlinedefn} \newtheorem{definition}{Definition} \newtheoremstyle{break}{\baselineskip}{\baselineskip}{\itshape}{}{\bfseries}{}{\newline}{} \theoremstyle{break} \newtheorem{example}{Example} % misc commands \newcommand*{\defn}[1]{\textbf{#1}} The \textbf{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 \defn{face maps}, and the $\sigma^n_i$ morphisms are called \defn{degeneracy maps}. They satisfy the following relations, \begin{eqnarray} \delta^{n+1}_j\,\delta^n_i & = & \delta^{n+1}_i\,\delta^n_{j-1} \quad\mbox{for\ } i<j, \\ \sigma^{n-1}_j\,\sigma^n_i & = & \sigma^{n-1}_i\,\sigma^n_{j+1} \quad\mbox{for\ } i\leq j, \\ \sigma^n_j\,\delta^{n+1}_i & = & \left\{ \begin{array}{ll} \delta^n_i\,\sigma^{n-1}_{j-1} & \mbox{if\ } i<j, \\ \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. \end{eqnarray} 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 \begin{eqnarray} [m]+[n] & = & [m+n+1], \\ (f+g)(i) & = & \left\{ \begin{array}{ll} f(i) & \mbox{if\ } 0 \leq i \leq m, \\ g(i-m-1)+m'+1 & \mbox{if\ } m < i \leq (m+n+1), \end{array}\right. \end{eqnarray} 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 $a \leq b$.