DiagonalFunctor 2007-01-24 23:58:51 2007-10-24 19:37:42 Definition 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 Let $\mathcal{C}$ be a category. A \emph{diagonal functor} on $\mathcal{C}$ is a functor $\delta:\mathcal{C}\to \mathcal{C}^I$ for some set $I$ given by $$\delta(A)=(A)_{i\in I}\quad\mbox{ and }\quad \delta(\alpha)=(\alpha)_{i\in I}.$$ Here, $\mathcal{C}^I$ denotes the \PMlinkname{$I$-fold direct product}{ProductCategory} of the category $\mathcal{C}$. For any given $I$, $\delta$ is unique. $\delta$ is \PMlinkname{faithful}{FaithfulFunctor}. Its image, $\delta(\mathcal{C})$, is the subcategory of $\mathcal{C}^I$ whose objects are $(A)_{i\in I}$ and morphisms are $(\alpha)_{i\in I}$. $\delta(\mathcal{C})$ is \PMlinkname{isomorphic}{CategoryIsomorphism} to $\mathcal{C}$, and may be pictured as the great diagonal of an $I$-dimensional ``cube''. More generally, when $I$ is a category, then the diagonal functor is just a functor $\delta$ that sends each object $A\in \mathcal{C}$ to the constant functor $\delta(A):I\to \mathcal{C}$ with fixed value $A$, and every morphism $\alpha:A\to B$ to the natural transformation $\delta(\alpha):\delta(A)\dot{\to} \delta(B)$, which sends every object $i\in I$ to $\alpha$. A routine verification shows that $\delta(\alpha)$ is indeed a natural transformation.