Coalgebra 2002-10-18 13:48:31 2005-02-02 10:10:16 Definition comultiplication counit coassociative cocommutative \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[all]{xy} % 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}} A \textbf{coalgebra} is a vector space $A$ over a field $\Kset$ with a $\Kset$-linear map $\Delta\colon A \to A\otimes A$, called the \textbf{comultiplication}, and a (non-zero) $\Kset$-linear map $\varepsilon\colon A \to \Kset$, called the \textbf{counit}, such that \begin{eqnarray*} (\Delta\otimes\id)\circ\Delta & = & (\id\otimes\Delta)\circ\Delta \quad\mbox{(coassociativity)}, \\ (\varepsilon\otimes\id)\circ\Delta & = \id = & (\id\otimes\varepsilon)\circ\Delta. \end{eqnarray*} In \PMlinkescapetext{terms} of commutative diagrams: \[\begin{xy} \xymatrix@R=20pt@C=20pt{ & *+<10pt>\txt{$A$} \ar_{\Delta}[dl] \ar^{\Delta}[dr] & \\ *+<10pt>\txt{$A\otimes A$} \ar_{\Delta\otimes\id}[dr] & & *+<10pt>\txt{$A\otimes A$} \ar^{\id\otimes\Delta}[dl] \\ & *+<10pt>\txt{$A\otimes A\otimes A$} & }\end{xy}\] \[\begin{xy} \xymatrix@R=20pt@C=20pt{ & *+<10pt>\txt{$A$} \ar_{\Delta}[dl] \ar^{\id}[dd] \ar^{\Delta}[dr] & \\ *+<10pt>\txt{$A\otimes A$} \ar_{\varepsilon\otimes\id}[dr] & & *+<10pt>\txt{$A\otimes A$} \ar^{\id\otimes\varepsilon}[dl] \\ & *+<10pt>\txt{$A$} & }\end{xy}\] Let $\sigma\colon A\otimes A \to A\otimes A$ be the flip map $\sigma(a\otimes b) = b\otimes a$. A coalgebra is said to be \textbf{cocommutative} if $\sigma\circ\Delta = \Delta$. Let $A$ and $B$ be two coalgebras over a field $\Kset$. A coalgebra homomorphism is a $\Kset$-linear map $f\colon A \to B$ such that $\Delta_B\circ f = (f\otimes f)\circ\Delta_A$ and $\varepsilon_B\circ f = \varepsilon_A$. \begin{example}[Coalgebra of a set] Let $S$ be a set. The free vector space $\Kset S$, with basis given by the elements of $S$, is a coalgebra with comultiplication $\Delta(s) = s \otimes s$ and counit $\varepsilon(s) = 1$. \end{example}