chain homotopy equivalence ChainHomotopyEquivalence 2004-11-24 14:20:36 2004-11-24 14:31:26 Definition chain homotopic equivalent % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{equivalent} Let $C$ and $D$ be two objects from the abelian category of chain complexes. A morphism (or chain map) $f\colon C\to D$ is said to be a \emph{chain homotopy equivalence} if there is a morphism $g\colon D\to C$ such that \begin{enumerate} \item there is a chain homotopy between $fg$ and $1\colon D\to D$; and \item there is a chain homotopy between $gf$ and $1\colon C\to C$. \end{enumerate} If a chain homotopy equivalence from a chain complex $C$ to $D$ exists, then $C$ is said to be \emph{chain homotopy equivalent} to $D$. Chain homotopy equivalence is an equivalence relation among chain complexes.