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. ${\displaystyle \sim }$ ${\displaystyle x\sim y}$ ${\displaystyle x-y\in U}$ ${\displaystyle x\sim y}$ ${\displaystyle xy^{-1}\in N}$

• 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.

See also

• Factor ring
• Homomorphism theorem
• Quotient norm