PowerOfAnObject 2008-10-31 02:58:04 2008-10-31 12:18:36 Definition categorical direct product power copower \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[all]{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 and $A$ an object in $\mathcal{C}$. Suppose $n$ is a non-negative integer. The \emph{$n$-th power} of $A$ is defined as the direct product of $A$ with itself $n$ times. In other words, the \emph{$n$-th power} of $A$ is an object $P$ in $\mathcal{C}$, together with $n$ parallel morphisms $\pi_1, \ldots, \pi_n \in \hom(P,A)$, such that if there are $n$ parallel morphisms $p_1, \ldots, p_n \in \hom(B,A)$, then there is a unique morphism $h:B\to P$ such that $\pi_i\circ h=p_i$, where $i=1,\ldots, n$. The commutative diagram below illustrates the situation: $$\xymatrix@C=0.5cm{& B \ar@{.>}[d]^{h} \ar@/_4ex/[dddl]_{p_1} \ar@/^4ex/[dddr]^{p_n} & \\ & P \ar[ddl]_{\pi_1}="1" \ar[ddr]^{\pi_n}="2" & \\ & & \\ A & \cdots & A \ar@{}"1";"2"|-{\cdots}}$$ The $n$-th power of $A$ is denoted by $A^n$. Below are some properties of the power of an object in a category: \begin{itemize} \item Each of the projection morphisms $\pi_i$ is a split epimorphism. \item $A^1 \cong A$. \item $A^0$ is a terminal object in $\mathcal{C}$. \item $A^{m+n}\cong A^n \times A^m$, if the product exists. \end{itemize} For example, in the category of sets, the $n$-th power of a set $A$ is the set of $n$-tuples where each entry is an element of $A$. \textbf{Remark}. The \emph{copower} of an object is defined dually. All of the properties above can be dualized. For example, the $0$-th copower of an object is an initial object. The $n$-th copower of an object $A$ in \textbf{Set} is the disjoint union of $n$-copies of $A$.