Fundamentals and Rudiments of Homological Algebra

A category C consists of a class of C - Objects and a class of C - Morphisms, denoted respectfully by Ob(C) and Mor(C), for each ordered pair (A, B) of C - Objects which satisfies the axioms of Composition of Morphisms, Products of Morphisms, an Identity of Morphisms, as well as the Domain and the Co-domain (Range). In any case, each object A in the given category C is uniquely determined by the Identity Morphism. On the other hand, the Identity is uniquely determined by A. And so, by replacing as C, the Identity Morphisms, a category can then be defined alternatively and entirely in terms of Morphisms.