# preservation and reflection

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 ${A}^{\prime}$, the transformed object ${A}^{\prime}$ 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 *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 $\mathcal{C}$.

Title | preservation and reflection |
---|---|

Canonical name | PreservationAndReflection |

Date of creation | 2013-03-22 17:12:18 |

Last modified on | 2013-03-22 17:12:18 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 5 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 00A35 |

Defines | preserve |

Defines | reflect |