# Quotient mapping

Quotient mapping , canonical surjection or canonical projection is a mathematical term that occurs in many mathematical sub- areas. It is a mapping that assigns its equivalence class to each element of a set on which there is an equivalence relation . In category theory , the term for quotient objects is generalized.

## Examples

• Is a vector space and a subspace , it may be the quotient vector space form, which from all cosets with there. The mapping that maps the vector to is called the quotient mapping.${\ displaystyle V}$ ${\ displaystyle U \ subset V}$ ${\ displaystyle V / U}$ ${\ displaystyle x + U}$ ${\ displaystyle x \ in V}$ ${\ displaystyle V \ rightarrow V / U}$ ${\ displaystyle x \ in V}$ ${\ displaystyle x + U}$ • If a group with a normal divisor is more general , then the quotient group of the secondary classes can be formed, where . Again, the canonical mapping is called the quotient mapping .${\ displaystyle G}$ ${\ displaystyle N \ subset G}$ ${\ displaystyle G / N}$ ${\ displaystyle xN}$ ${\ displaystyle x \ in G}$ ${\ displaystyle G \ rightarrow G / N, x \ mapsto xN}$ Both examples are based on an equivalence relation . In the vector space example one has exactly if , and analogously in the group example exactly if . Therefore, the following construction generalizes the above examples. ${\ displaystyle \ sim}$ ${\ displaystyle x \ sim y}$ ${\ displaystyle xy \ in U}$ ${\ displaystyle x \ sim y}$ ${\ displaystyle xy ^ {- 1} \ in N}$ • Let there be a set and an equivalence relation . Then be the set of equivalence classes . The mapping is called the quotient mapping .${\ displaystyle X}$ ${\ displaystyle \ sim}$ ${\ displaystyle X}$ ${\ displaystyle X / \! \! \ sim}$ ${\ displaystyle [x]}$ ${\ displaystyle X \ rightarrow X / \! \! \ sim, \, x \ mapsto [x]}$ • If there is a surjective mapping, then is given by an equivalence relation. In this case the mapping is bijective. One then also calls a quotient mapping.${\ displaystyle f \ colon X \ rightarrow Y}$ ${\ displaystyle x \ sim y \;: \ Leftrightarrow \; f (x) = f (y)}$ ${\ displaystyle X / \! \! \ sim \, \ rightarrow Y, \, [x] \ mapsto f (x)}$ ${\ displaystyle f}$ • Is a surjective map on a topological space , so there is a finest topology on , respect. Of is continuous, called the quotient topology . This is why the mapping is called a quotient mapping in this case too.${\ displaystyle f \ colon X \ rightarrow Y}$ ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle f}$ These examples are generalized to so-called quotient objects in category theory . In fact, such quotient objects are certain epimorphisms , so that these are essentially the quotient maps presented here, but morphisms in category theory do not have to be mappings.

## Individual evidence

1. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , chap. 0, §1
2. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture , Bibliographisches Institut Mannheim (1978), ISBN 3-411-00121-6 , chapter 2.6.