CategoryOfSmallCategories 2008-10-01 16:41:26 2009-01-17 11:13:38 Definition Cat \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % 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} % define commands here \newcommand*{\abs}[1]{\left\lvert #1\right\rvert} \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}} The category \textbf{Cat} of small categories consists of all small categories as objects, and, functors between small categories as morphisms. The composition of morphisms in \textbf{Cat} is the functor composition, and, associated with each small category, the identity functor acts as the identity morphism. Now, \textbf{Cat} is indeed a category, since $\hom(\mathcal{C},\mathcal{D})$, the class of all functors from $\mathcal{C}$ to $\mathcal{D}$ is a set. The proof of this fact can be found \PMlinkname{here}{FunctorCategory}. Here are some of the basic properties of \textbf{Cat}: \begin{enumerate} \item It has \PMlinkname{arbitrary products}{ProductOfCategories} \item It has \PMlinkname{arbitrary coproducts}{DisjointUnionOfCategories} \item Initial object exists: the initial object is the empty category and the associated empty functor. \item Terminal object exists: the terminal object is any trivial category and the associated constant functor into the trival category. \item It has pullbacks. See \PMlinkname{this entry}{ExamplesOfPullbacks}. So it has equalizers, and therefore, it is complete. \item however, it does have coequalizers. This, together with 2 above, shows that it is cocomplete. \end{enumerate} \textbf{Remarks}. \begin{itemize} \item If we replace functors in $\hom(\mathcal{C},\mathcal{D})$ by natural transformations between pairs of functors from $\mathcal{C}$ to $\mathcal{D}$, and composition of morphisms the horizontal composition $\circ$ of natural transformations, then we again end up with a category (provided that both $\mathcal{C}$ and $\mathcal{D}$ are small). Indeed, every natural transformation $\eta$ between two functors from $\mathcal{C}$ to $\mathcal{D}$ is a set function from the \emph{set} of objects of $\mathcal{C}$ to the \emph{set} of morphisms of $\mathcal{D}$. As a result, $\hom(\mathcal{C},\mathcal{D})$ is a subcollection of the \emph{set} of \emph{all} functions from $\operatorname{Ob}(\mathcal{C})$ to $\operatorname{Mor}(\mathcal{D})$, and hence a set. For more detail, please see \PMlinkname{this entry}{CompositionsOfNaturalTransformations}. \item In fact, \textbf{Cat} has the structure of a 2-category, where the small categories are the $0$-cells, the functors between them are the $1$-cells, and the natural transformations between parallel functors are the $2$-cells. \item If we remove the requirement that each object in \textbf{Cat} be small, then $\hom(\mathcal{C},\mathcal{D})$ may no longer be a set, and we end up with a large category. \end{itemize}