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preservation and reflection


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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 $\mathcal{C}$.