CategoryIsomorphism 2004-05-11 13:15:39 2007-06-16 14:45:01 Definition % 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 Let $\mathcal{C}$ and $\mathcal{D}$ be categories. An \emph{isomorphism} $T:\mathcal{C}\to\mathcal{D}$ is a (covariant) functor which has a \emph{two-sided inverse}. In other words, there is a (covariant) functor $S:\mathcal{D}\to\mathcal{C}$ such that $T\circ S=I_{\mathcal{D}}$ and $S\circ T=I_{\mathcal{C}}$, where $I_{\mathcal{D}}$ and $I_{\mathcal{C}}$ are the identity functors of $\mathcal{D}$ and $\mathcal{C}$ respectively. Two categories $\mathcal{C}$ and $\mathcal{D}$ are \emph{isomorphic} if there exists a functor $T:\mathcal{C}\to\mathcal{D}$ that is an isomorphism. \textbf{Remarks} \begin{enumerate} \item An isomorphism (functor) from $\mathcal{C}$ to $\mathcal{D}$ is just an \PMlinkname{isomorphism}{Isomorphism2} (in the sense of morphism) in the functor category $\mathcal{D}^{\mathcal{C}}$. \item Two isomorphic categories are \PMlinkname{equivalent}{EquivalenceOfCategories}. The converse is not true. For example, the category of all finite sets is \PMlinkescapetext{equivalent} to its subcategory of all finite ordinals. But clearly these two categories are not isomorphic. Isomorphism has a ``size'' restriction, whereas natural equivalence does not. \end{enumerate} \PMlinkescapeword{isomorphism} \PMlinkescapeword{isomorphic} \PMlinkescapeword{restriction}