forgetful functor ForgetfulFunctor 2002-05-17 14:50:53 2002-05-17 14:50:53 Definition forgetful % 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 $\mathcal{C}$ and $\mathcal{D}$ be categories such that each object $c$ of $\mathcal{C}$ can be regarded an object of $\mathcal{D}$ by suitably ignoring structures $c$ may have as a $\mathcal{C}$-object but not a $\mathcal{D}$-object. A functor $U:\mathcal{C} \to \mathcal{D}$ which operates on objects of $\mathcal{C}$ by ``forgetting'' any imposed mathematical structure is called a \emph{forgetful functor}. The following are examples of forgetful functors: \begin{enumerate} \item $U:\mathbf{Grp} \to \mathbf{Set}$ takes groups into their underlying sets and group homomorphisms to set maps. \item $U:\mathbf{Top} \to \mathbf{Set}$ takes topological spaces into their underlying sets and continuous maps to set maps. \item $U:\mathbf{Ab} \to \mathbf{Grp}$ takes abelian groups to groups and acts as identity on arrows. \end{enumerate} Forgetful functors are often instrumental in studying adjoint functors.