What is a category?
A category, C, is a class of objects, Ob C, together with a class of morphisms, Mor C, between objects. The morphisms satisfy the following properties: 1) For every two objects X, Y in Ob C, there is a subset, Hom(X, Y), of Mor C. These subsets are all pairwise disjoint. When first referring to an element, f, of Hom(X, Y), we will usually adopt the usual function notation: f: X Y 2) For every X, Y, Z in Ob C, and morphisms f: X Y and g: Y Z, there is a morphism h: X Z called the composition of g with f. Again, we will adopt the usual functional notation: h = g ○ f. 3) Composition is associative, that is, for f: W X, g: X Y, and h: Y Z, h○(g○f) = (h○g)○f. 4) For every object, Y, there is a morphism 1 in Hom(Y, Y) such that for all f: X Y and g: Y Z, 1○f = f and g○1 = g. This morphism is called the identity. There are many familiar examples of categories. The category of all sets and relations is one such example. Another is the category of groups and homomorphisms, or the category or ve
