# Definiton of a category question

• MathematicalPhysicist
In summary, for each object A in a category, there is an arrow id_A whose source and target are both A. This does not mean that id_A is a point, but rather a specific arrow from A to itself. In general, arrows cannot be identified with objects in a category. The terms "objects" and "arrows" are also preferred over "points" and "relations" in category theory. A helpful way to visualize this is to think of objects as boxes and morphisms as arrows, similar to a flow chart.

#### MathematicalPhysicist

Gold Member
"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

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.

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.