Definition of Category • Data General S (Set) A set of Objects of the Category In S, the Objects are Abstract Sets e.g. A,B,C,X_i, etc A set of Arrows between Objects In S, the arrows are functions The source is called the Domain (also called morphisms or maps) The target is called the Codomain ∀a∈A, ∃!b∈B s.t. f(a)=b e.g. f:A→B A special arrow called The Identity Arrow In S, this is the identity map: defined on each object in the Category ∀a∈A,1_A (a)=a 1_A:A→A • Rules ○ Compositions § Given two arbitrary arrows A →┴f B_1, and B_2 →┴g C § We can form the composite A→┴(g∘f) C (called g following f) iff B_1=B_2 § e.g. in the case where we have A→┴f B→┴g C ○ Associative § Compositions of arbitrary arrows A →┴f B, B →┴g C, and C→┴hD is associative § i.e. the following relations holds: h∘(g∘f)=(h∘g)∘f ○ Identity Laws § For arbitrary objects A and B § The arrows A→┴1_A A, B→┴1_B B, and A→┴f B must obey the following laws § f∘1_A=1_B∘f=f Objects • Objects in S ○ The Objects in S are Abstract Sets ○ We will represent them, for example, as: A,B,C,X_i,Y_j^′ • Elements in Abstract Sets ○ If we have a set A of size 2, and a set B of size 3, for example ○ We will use the following notation to refer to the unique
