I am confused about the definition of pre-category.
One definition seems to say the (A,B)!=(C,D) ==> hom(A,B) intersected with hom(C,D)={} is required.
Another definition, in section 1.8 on page 10 says that if the uniqueness requirement (section 1.3 on page 8) is dropped, the we get a pre-category.
These requirements look the same to me, but one person is requiring this and the other is dropping it.
Can someone explain this?
Edit: The reason I am interested in something that is less than a category is because of this question. This thing is a category, but some things are not like the other examples given in the reference.
I have some code that fools around with some categories (those with a finite number of things and a finite number of morphisms). It would seem that the examples given fail because some composition pair is not defined or because associativity of composition fails.
In the first case, we can always add to the composition table. So i would like to use failing associativity as an criterion for a pre-categeory.
Is this a reasonable thing to do?
One of the things that interest me is to classify small graphs into things that are categories and things that are not. So maybe I should use some kind of directed graph as a base class in my code instead of pre-category?
