A category is an abstract collection of objects constituents collection of objects, where if X is an object of C we write X \in C for a pair of objects X, Y \in C, a set of morphisms acting upon the objects which we call the homset additional information requirements there exists the identity morphism; that is, \forall X \in C, \exists I_{X}: X\to X morphisms are always composable: given f: X\to Y, and g: Y\to Z, exists gf: X \to Z the identity morphism can compose in either direction: given f: X \to Y, then f I_{x} = f = I_{y} f morphism composition is associative: (hg)f=h(gf)