- #1
Arian.D
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- 0
I know it may sound idiotic to ask questions about definition of something, but I'm going to do that now. I've seen the definition of categories in several different contexts, in all of them categories consisted of objects like groups, rings, R-modules (in particular, vector spaces), topological spaces and many other mathematical objects that I don't know and the arrows were group homomorphisms, ring homomorphisms, R-homomorphisms, linear transformations, continuous maps, etc that have the following properties:
1. The composition of arrows gives us a third.
2. There exists an identity arrow that if it's composed with another arrow it gives it back.
3. The composition of arrows is associative (which sounds a little bit obvious to me because composition of functions is associative and we think of arrows as functions acting on mathematical objects that preserve some structure. Right?).
Now my book adds an additional condition on categories. I need you to clarify things a bit please:
A category [itex]\mathbb{C}[/itex] is a class of mathematical objects that we usually denote them by A,B,C,D,... with following properties:
1) For any two objects A and B, a set [itex]Hom_{\mathbb{C}}(A,B)[/itex] is associated to them with the property that for any four objects A,B,C and D that [itex](A,B)\neq (C,D)[/itex], [itex]Hom_{\mathbb{C}}(A,B) \cap Hom_{\mathbb{C}}(C,D) = \emptyset[/itex]
Other conditions are understandable for me.
Why they have put this condition here? What does it want to tell me? Isn't the trivial homomorphism that sends A to B by f(A)=0 in the intersection?
I mean suppose that we have (A,B)≠(A,C) (which is true because one of the components of the ordered pairs is different). Then if we define f(A)=0, isn't this homomorphism in the intersection?
1. The composition of arrows gives us a third.
2. There exists an identity arrow that if it's composed with another arrow it gives it back.
3. The composition of arrows is associative (which sounds a little bit obvious to me because composition of functions is associative and we think of arrows as functions acting on mathematical objects that preserve some structure. Right?).
Now my book adds an additional condition on categories. I need you to clarify things a bit please:
A category [itex]\mathbb{C}[/itex] is a class of mathematical objects that we usually denote them by A,B,C,D,... with following properties:
1) For any two objects A and B, a set [itex]Hom_{\mathbb{C}}(A,B)[/itex] is associated to them with the property that for any four objects A,B,C and D that [itex](A,B)\neq (C,D)[/itex], [itex]Hom_{\mathbb{C}}(A,B) \cap Hom_{\mathbb{C}}(C,D) = \emptyset[/itex]
Other conditions are understandable for me.
Why they have put this condition here? What does it want to tell me? Isn't the trivial homomorphism that sends A to B by f(A)=0 in the intersection?
I mean suppose that we have (A,B)≠(A,C) (which is true because one of the components of the ordered pairs is different). Then if we define f(A)=0, isn't this homomorphism in the intersection?