DualityPrinciple 2002-02-25 10:22:11 2006-07-07 11:28:21 Definition self-dual statement % 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} \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 Let $\Sigma$ be any statement of the elementary theory of an abstract category. We form the dual of $\Sigma$ as follows: \begin{enumerate} \item Replace each occurrence of ``domain'' in $\Sigma$ with ``codomain'' and conversely. \item Replace each occurrence of $g \circ f =h$ with $f \circ g = h$ \end{enumerate} Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions. For example, consider the following statements about a category $\mathcal{C}$: \begin{itemize} \item $f:A \to B$ \item $f$ is monic, i.e. for all morphisms $g,h$ for which composition makes sense, $f \circ g = f \circ h$ implies $g=h$. \end{itemize} The respective dual statements are \begin{itemize} \item $f:B \to A$ \item $f$ is epi, i.e. for all morphisms $g,h$ for which composition makes sense, $g \circ f = h \circ f$ implies $g=h$. \end{itemize} The \emph{duality principle} asserts that if a statement is a theorem, then the dual statment is also a theorem. We take "theorem" here to mean provable from the axioms of the elementary theory of an abstract category. In practice, for a valid statement about a particular category $\mathcal{C}$, the dual statement is valid in the dual category $\mathcal{C}^{*}$ ($\mathcal{C}^{op}$). If the property $\Sigma$ is the same as its dual, then it is called \emph{self-dual}.