DiscreteCategory 2006-09-16 11:30:59 2008-10-01 19:37:42 Definition trivial category empty category empty functor \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} \usepackage{pst-plot} \usepackage{psfrag} % define commands here A category $\mathcal{C}$ is said to be a \emph{discrete category} if the only morphisms in $\mathcal{C}$ are the identity morphisms associated with each of the objects in $\mathcal{C}$. For example, every set can be regarded as a discrete category. The objects are just the elements of the set. Furthermore, $\hom(a,a)$ is identified with $\lbrace a\rbrace$, and $\hom(a,b)=\varnothing$ if $a\ne b$. \textbf{Remarks}. \begin{itemize} \item A discrete category with one object is called a \emph{trivial category}. For every category $\mathcal{C}$, there is only one functor from $\mathcal{C}$ to a trivial category. Hence, any trivial category is a terminal object in \textbf{Cat}, the category of small categories. \item A discrete category with no objects is called the \emph{empty category}. For every category $\mathcal{C}$, there is only one functor from the empty category to $\mathcal{C}$. This functor is called the \emph{empty functor}, where both the object and morphism functions are the empty set $\varnothing$. Thus, the empty category is the initial object in \textbf{Cat}. \item Given any category $\mathcal{C}$, the smallest subcategory consisting of all objects in $\mathcal{C}$ is discrete, which is also the largest discrete subcategory in $\mathcal{C}$ (largest in the sense that it contains all objects of $\mathcal{C}$). For every object $X\in \mathcal{C}$, we can associate the trivial category $\mathcal{C}_X$ consisting of one object, $X$, and one morphism $1_X$. \end{itemize}