Definiton of a category question

  • #1
MathematicalPhysicist
Gold Member
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"for each object A there is an arrow id_A called the identity of A whose source and target are both A."

wouldnt this definition imply that id_A is a point instead of an arrow?
if not help me visualise this.

btw this should be the notation id_A:A->A
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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Categories contain objects and "arrows" (I would call them "relations") but I've never heard of a "point" being defined for a general category.

You may be thinking of the object as a single "point" and then identifying the arrow (from the object to itself) with that "point".

Remember that in general, given two objects in a category, there may be many arrows from one to another so you cannot identify arrows with objects. In particular, there may be many arrows from a given category to itself. The arrow "id_A" is a specific one of those so you certainly cannot identify "id_A" with the object.
 
  • #3
phoenixthoth
1,605
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i wouldn't call them relations because a relation is generally used in reference to being some subset of the cartesian product of sets. arrows are not sets in general.
 
  • #4
selfAdjoint
Staff Emeritus
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Objects and morphism is the usual terminology. If it helps, you can think of morphisms as arrows and objects as boxes, as in a flow chart.
 

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