PreservationAndReflection 2007-06-04 19:40:15 2007-06-07 15:43:25 Definition preserve reflect \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} \usepackage{psfrag} % define commands here \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}} In mathematics, the word ``preserve'' usually means the ``preservation of properties''. Loosely speaking, whenever a mathematical construct $A$ has some property $P$, after $A$ is somehow ``transformed'' into $A'$, the transformed object $A'$ also has property $P$. The constructs usually refer to sets and the transformations typically are functions or something similar. Here is a simple example, let $f$ be a function from a set $A$ to $B$. Let $A$ be a finite set. Let $P$ be the property of a set being finite. Then $f$ preserves $P$, since $f(A)$ is finite. Note that we are not saying that $B$ is finite. We are merely saying that the portion of $B$ that is the \emph{image} of $A$ (the transformed portion) is finite. Here is another example. The property of being connected in a topological space is preserved under a continuous function. Here, the constructs are topological spaces, and the transformation is a continuous function. In other words, if $f:X\to Y$ is a continuous function from $X$ to $Y$. If $X$ is connected, so is $f(X)\subseteq Y$. Many more examples can be found in abstract algebra. Group homomorphisms, for example, preserve commutativity, as well as the property of being finitely generated. The word ``reflect'' is the dual notion of ``preserve''. It means that if the transformed object has property $P$, then the original object also has property $P$. This usage is rarely found outside of category theory, and is almost exclusively reserved for functors. For example, a faithful functor reflects isomorphism: if $F$ is a faithful functor from $\mathcal{C}$ to $\mathcal{D}$, and the object $F(A)$ is isomorphic to the object $F(B)$ in $\mathcal{D}$, then $A$ is isomorphic to $B$ in $\mathcal{C}$.