|
|
|
Revision difference : preservation and reflection |
| Version current |
Version 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. |
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 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 $f(X)\subseteq Y$. |
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. |
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 $\mathcal{C}$.
|
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 most commonly found in category theory, especially referring to 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 $\mathcal{C}$.
|
|
|
|
|
