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'duality principle'
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| Title of object: |
duality principle |
| Canonical Name: |
DualityPrinciple |
| Type: |
Definition |
| Created on: |
2002-02-25 10:22:11 |
| Modified on: |
2005-04-14 19:31:10 |
| Classification: |
msc:18A05 |
Revision comment (for changes between this and next version):
| Changes for correction #8616 ('self-dual'). |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
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 $\mathcal{C}^{*}$ ($\mathcal{C}^{op}$). |
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