From Groups to Groupoids: a Brief Survey

A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range of applications, so that even for purely group-theoretical work, it can be useful to take a path through the world of groupoids. A succinct definition is that a groupoid G is a small category in which every morphism is an isomorphism. Thus G has a set of morphisms, which we shall call just elements of G, a set Ob(G) of objects or vertices, together with functions s, t : G→ Ob(G), i : Ob(G) → G such that si = ti = 1. The functions s, t are sometimes called the source and target maps respectively. If a,b ∈ G and ta = sb, then a product or composite ab exists such that s(ab) = sa, t(ab) = tb. Further, this product is associative; the elements ix, x ∈ Ob(J), act as identities; and each element a has an inverse a with s(a) = ta, t(a) = sa,aa = isa,aaa = ita. An element a is often written as an arrow a : sa→ ta.