Namespaces
Variants
Actions

Difference between revisions of "Epimorphism"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (TeX done)
 
Line 1: Line 1:
A concept reflecting the algebraic properties of surjective mappings of sets. A [[Morphism|morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358901.png" /> in a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358902.png" /> is called an epimorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358903.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358904.png" />. In other words, an epimorphism is a morphism that can be cancelled on the right.
+
A concept reflecting the algebraic properties of surjective mappings of sets. A [[Morphism|morphism]] $\pi : A \to B$ in a [[Category|category]] $\mathfrak{N}$ is called an epimorphism if $\alpha \, \pi = \beta \, \pi$ implies $\alpha = \beta$. In other words, an epimorphism is a morphism that can be cancelled on the right.
  
Every isomorphism is an epimorphism. The product of two epimorphisms is an epimorphism. Therefore, all epimorphisms of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358905.png" /> form a subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358906.png" /> (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358907.png" />).
+
Every isomorphism is an epimorphism. The product of two epimorphisms is an epimorphism. Therefore, all epimorphisms of a category $\mathfrak{N}$ form a subcategory of $\mathfrak{N}$ (denoted by $\operatorname{Epi} \mathfrak{N}$).
  
 
In the categories of sets, vector spaces, groups, and Abelian groups, the epimorphisms are precisely the surjective mappings, i.e. the linear mappings and the homomorphisms of one set, vector space or group onto another set, vector space or group. However, in the categories of topological spaces or associative rings there are non-surjective epimorphisms (that is, mappings that are not  "onto" ).
 
In the categories of sets, vector spaces, groups, and Abelian groups, the epimorphisms are precisely the surjective mappings, i.e. the linear mappings and the homomorphisms of one set, vector space or group onto another set, vector space or group. However, in the categories of topological spaces or associative rings there are non-surjective epimorphisms (that is, mappings that are not  "onto" ).
Line 10: Line 10:
  
 
====Comments====
 
====Comments====
In the article above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358908.png" /> are assumed to be morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e0358909.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e03589010.png" />. If composition of morphism is written from left to right, as is sometimes done, so that the composite of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e03589011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e03589012.png" /> is written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e03589013.png" />, then epimorphisms are of course morphisms that cancel on the left.
+
In the article above, $\alpha$, $\beta$ are assumed to be morphisms $B \to C$ for some $C$. If composition of morphism is written from left to right, as is sometimes done, so that the composite of $\pi : A \to B$ and $\alpha : B \to C$ is written $\pi \, \alpha$, then epimorphisms are of course morphisms that cancel on the left.
  
The inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035890/e03589014.png" /> in the category of rings is an example of an epimorphism that is not surjective.
+
The inclusion $\mathbf{Z} \to \mathbf{Q}$ in the category of rings is an example of an epimorphism that is not surjective.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Mitchell,  "Theory of categories" , Acad. Press  (1965)</TD></TR></table>
 +
 +
{{TEX|done}}

Latest revision as of 05:53, 12 January 2017

A concept reflecting the algebraic properties of surjective mappings of sets. A morphism $\pi : A \to B$ in a category $\mathfrak{N}$ is called an epimorphism if $\alpha \, \pi = \beta \, \pi$ implies $\alpha = \beta$. In other words, an epimorphism is a morphism that can be cancelled on the right.

Every isomorphism is an epimorphism. The product of two epimorphisms is an epimorphism. Therefore, all epimorphisms of a category $\mathfrak{N}$ form a subcategory of $\mathfrak{N}$ (denoted by $\operatorname{Epi} \mathfrak{N}$).

In the categories of sets, vector spaces, groups, and Abelian groups, the epimorphisms are precisely the surjective mappings, i.e. the linear mappings and the homomorphisms of one set, vector space or group onto another set, vector space or group. However, in the categories of topological spaces or associative rings there are non-surjective epimorphisms (that is, mappings that are not "onto" ).

The concept of an epimorphism is dual to that of a monomorphism.


Comments

In the article above, $\alpha$, $\beta$ are assumed to be morphisms $B \to C$ for some $C$. If composition of morphism is written from left to right, as is sometimes done, so that the composite of $\pi : A \to B$ and $\alpha : B \to C$ is written $\pi \, \alpha$, then epimorphisms are of course morphisms that cancel on the left.

The inclusion $\mathbf{Z} \to \mathbf{Q}$ in the category of rings is an example of an epimorphism that is not surjective.

References

[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965)
How to Cite This Entry:
Epimorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Epimorphism&oldid=40172
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article