PartiallyOrderedCategory 2008-08-09 15:14:06 2009-08-14 23:02:45 Definition linearly ordered category directed category \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}} Let $\mathcal{C}$ be a category in which for every pair of objects $A,B$, the collection $\hom(A,B)$ is a set. $\mathcal{C}$ is called a \emph{partially ordered category} if $\hom(A,B)\cup \hom(B,A)$ has at most one element. The reason why it is called \emph{partially ordered} is because we can put a partial order on the objects of $\mathcal{C}$, as follows: $$A\le B \qquad \mbox{iff} \qquad \hom(A,B)\ne \varnothing.$$ It is easily verified that $\le$ is a partial order on $\operatorname{Ob}(\mathcal{C})$: clearly, $\hom(A,A)$ is a singleton as it contains the identity morphism, so that $\le$ is reflexive; if $\hom(A,B)\ne \varnothing$ and $A\ne B$, then $\hom(B,A)=\varnothing$ by definition, and so $\le$ is antisymmetric; finally, if neither $\hom(A,B)$ nor $\hom(B,C)$ are empty, then $\hom(A,C)$ can not be empty, as it contains the composition of the elements in $\hom(A,B)$ and $\hom(B,C)$, and thus $\le$ is transitive. It is easy to see that any poset can be realized as categroy, and in fact, a partially ordered category. Given a partially ordered category $\mathcal{C}$, the partial order defined on the objects of $\mathcal{C}$ may be extended to a partial order on the morphisms of $\mathcal{C}$. Formally, let $f,g$ be morphisms in $\mathcal{C}$. Define $f\le g$ iff $\operatorname{dom}(f)\le \operatorname{dom}(g)$ and $\operatorname{codom}(f)\le \operatorname{codom}(g)$. This binary relation can be immediately verified as a partial order (on the \emph{class} of all morphisms), and an extension of the partial order on objects of $\mathcal{C}$, since every object is identified the corresponding identity morphism. Furthermore, the partial order respects composition, if $f\le g$, then \begin{itemize} \item $f\circ h \le g\circ h$, provided that $\operatorname{dom}(f)=\operatorname{dom}(g)=\operatorname{codom}(h)$, \item $h\circ f \le h\circ g$, provided that $\operatorname{codom}(f)=\operatorname{codom}(g)=\operatorname{dom}(h)$. \end{itemize} Combining the two above, we see that $f_1\le g_1$ and $f_2\le g_2$ imply $f_1\circ f_2 \le g_1\le g_2$, provided that $f_1\circ f_2$ and $g_1\circ g_2$ are both defined. \textbf{Remarks}. From a partially ordered category, one may also define \begin{itemize} \item a \emph{linearly ordered category}, which is partially category in which $\hom(A,B)\cup \hom(B,A)$ has at least one element for every pair $A,B$ of objects \item a \emph{directed category}, which is a partially ordered category in which for every pair $A,B$ of objects, there is an object $C$ such that $\hom(A,C)$ and $\hom(B,C)$ are both non-empty. \item The definition given above of a partially ordered category can be generalized. Instead of having a partial order on the objects of the category, the partial order is now on the morphisms. Let $\mathcal{C}$ be a category such that a partial order $\le$ is given on the class of morphisms of $\mathcal{C}$. This partial order induces a partial order on the objects: $A\le B$ iff $1_A\le 1_B$. Call $\mathcal{C}$ a \emph{partially ordered category} if \begin{enumerate} \item if $f\le g$, then $\operatorname{dom}(f)\le \operatorname{dom}(g)$ and $\operatorname{codom}(f)\le \operatorname{codom}(g)$, \item if $f_1\le g_1$ and $f_2\le g_2$ such that $f_1\circ f_2$ and $g_1\circ g_2$ are defined, then $f_1\circ f_2 \le g_1\circ g_2$, \item for any $A$ and $f$ such that $A\le \operatorname{dom}(f)$, there exists a unique $g$ such that $g\le f$ with $\operatorname{dom}(g)=A$, \item for any $A$ and $f$ such that $A\le \operatorname{codom}(f)$, there exists a unique $g$ such that $g\le f$ with $\operatorname{codom}(g)=A$. \end{enumerate} It's not hard to see that the above definition does generalize the one given at the beginning of the entry. For example, to see 3., let $f:B\to C$. So $A\le B$, which means there is a unique $h:A\to B$. Then $g= f\circ h:A\to C$ is the desired morphism: $g$ is unique, $\operatorname{dom}(g)=A$, and $g\le f$. \end{itemize} \begin{thebibliography}{9} \bibitem{BK} A. J. Berrick, M. E. Keating, {\em Categories and Modules, with K-theory in View}, Cambridge (2000). \bibitem{CH} C. Hollins, {\em \PMlinkexternal{Extending The Ehresmann-Schein-Nambooripad Theorem}{http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.4702v3.pdf}}, (2009) \end{thebibliography}