Proving Epimorphism: Understanding Surjective Morphisms

  • Context: Graduate 
  • Thread starter Thread starter jhendren
  • Start date Start date
  • Tags Tags
    Proof
Click For Summary

Discussion Overview

The discussion centers around the concept of epimorphisms in category theory, specifically examining the relationship between epimorphisms and surjective morphisms (onto functions). Participants explore definitions and implications within different mathematical structures.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant asks how to prove that a morphism is an epimorphism if and only if it is surjective.
  • Another participant questions the definition of epimorphism if it is not equated with surjectivity.
  • A participant provides a definition of epimorphism from category theory, noting that it involves morphisms where equality holds under composition with other morphisms, but emphasizes that epimorphisms are not necessarily surjective in all categories.
  • There is a suggestion that the discussion must be focused on sets, supported by a link to a Wikipedia page discussing surjective functions as epimorphisms.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between epimorphisms and surjectivity, indicating that there is no consensus on whether epimorphisms are always surjective, as it may depend on the category in question.

Contextual Notes

The discussion highlights the dependence of definitions on the specific mathematical structures being considered, and the implications of category theory on the understanding of morphisms.

jhendren
Messages
34
Reaction score
0
How would one show a morphism is an epimorphism iff it is surjective (ONTO)?
 
Physics news on Phys.org
What is your definition of epimorphism if it's not that it is a surjective map?
 
Office_Shredder said:
What is your definition of epimorphism if it's not that it is a surjective map?

An epimorphism in category theory is a morphism ##f:X\rightarrow Y## such that if ##g,h:Y\rightarrow Z## are morphisms such that ##g\circ f = h\circ f##, then ##g=h##.

It is certainly not always true that an epimorphism is surjective. It depends on what category you work in. So, to the OP, you are talking about an epimorphism between which structures? Sets? Groups?
 

Similar threads

Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Negation of statement involving cardinalities
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad What is the Definition and Understanding of Surjective Functions?
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Show a functions inverse is injective iff f is surjective
  • · Replies 10 ·
Replies
10
Views
4K
MHB Definition of epimorphism and equivalence to 'surjectivity'
  • · Replies 7 ·
Replies
7
Views
3K
MHB Prove Injectivity & Surjectivity of Composite Application f
  • · Replies 2 ·
Replies
2
Views
2K
Given surjective ##T:V\to W##, find isomorphism ##T|_U## of U onto W.
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Equivalence classes of proper classes
  • · Replies 10 ·
Replies
10
Views
5K
Undergrad How to prove that a function is well-defined
  • · Replies 3 ·
Replies
3
Views
17K
Undergrad Determining if a function is surjective
  • · Replies 3 ·
Replies
3
Views
2K