In this chapter we define groupoids and their morphisms and give the basic algebraic definitions and constructions of subgroupoids, quotient groupoids, kernels of morphisms, products of groupoids and other standard concepts. We do not address the algebraic theory of groupoids for its own sake, and we do not prove any of the deeper results from the algebraic theory.

An interesting algebraic theory of groupoids exists, and was begun by Brandt and by Baer in the 1920's, well before Ehresmann made the concept of groupoid central to his vision of differential geometry. However the algebraic theory is primarily concerned with problems which are largely trivial for categories of transitive groupoids and there is therefore no reason for us to treat it here. See Higgins (1971) for a full account and further references, and Brown (1968) for an account which is more accessible to the non-algebraist, though less comprehensive than Higgins. Much material on the algebraic theory of groupoids, from a different point of view to that of the work cited above, can be extracted from Ehresmann (1965). See also Clifford and Preston (1961, §3.3).

The examples given in this chapter are examples of topological or differentiable groupoids, presented without their topological, or smooth, structures. We have managed to avoid giving examples which can arise only in the purely algebraic setting.

The development of the algebraic theory of groupoids has been succinctly chronicled by Higgins (1971, pp. 171–172).