Isomorphism2 2002-02-13 16:42:27 2005-11-30 10:20:39 Definition isomorphic automorphism % 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 A morphism $f: A \longrightarrow B$ in a category is an \emph{isomorphism} if there exists a morphism $f^{-1}: B \longrightarrow A$ which is its inverse. The objects $A$ and $B$ are \emph{isomorphic} if there is an isomorphism between them. A morphism which is both an isomorphism and an endomorphism is called an \emph{automorphism}. The set of automorphisms of an object $A$ is denoted $\operatorname{Aut}(A)$. Examples: \begin{itemize} \item In the category of sets and functions, a function $f: A \longrightarrow B$ is an isomorphism if and only if it is bijective. \item In the category of groups and group homomorphisms (or rings and ring homomorphisms), a homomorphism $\phi: G \longrightarrow H$ is an isomorphism if it has an inverse map $\phi^{-1}: H \longrightarrow G$ which is also a homomorphism. \item In the category of vector spaces and linear transformations, a linear transformation is an isomorphism if and only if it is an invertible linear transformation. \item In the category of topological spaces and continuous maps, a continuous map is an isomorphism if and only if it is a homeomorphism. \end{itemize}