Definiton of a category question

Gold Member

Main Question or Discussion Point

"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

HallsofIvy
Homework Helper
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.

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.