In the book Category Theory for Programmers one of the first question of the author is "Is Facebook a category, considering friendship as a morphism ?".
There isn't an official solutions, and most of the answer I saw on the internet say that it is not, considering A->B and B->C doesn't mean A->C in this context. I understand that as "The friend of my friend is not necessarily my friend".
Personally I think there is a misunderstanding in those answers. The composed morphism doesn't represent direct friendship.
Let say:
A the object "People on Facebook"
f : A -> A the morphism "Is friend with"
And the 3 elements of the object A (a, b, c)
If we have:
a -f> b and b-f>c
We can compose them
a -f.f> c
So c "is the friend of a friend of" a
It seems to me that this morphism respect composability and associativity. So this system can be considered as a category.
I don't have enought confidence to say which interpretation is the right one.
What do you think of this ? Am I making a mistake ?