ZeroObject 2002-04-22 18:28:16 2002-07-11 10:22:55 Definition initial object terminal object % 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 An {\em initial object} in a category $\mathcal{C}$ is an object $A$ in $\mathcal{C}$ such that, for every object $X$ in $\mathcal{C}$, there is exactly one morphism $A \longrightarrow X$. A {\em terminal object} in a category $\mathcal{C}$ is an object $B$ in $\mathcal{C}$ such that, for every object $X$ in $\mathcal{C}$, there is exactly one morphism $X \longrightarrow B$. A {\em zero object} in a category $\mathcal{C}$ is an object $0$ that is both an initial object and a terminal object. All initial objects (respectively, terminal objects, and zero objects), if they exist, are isomorphic in $\mathcal{C}$.