Comparing the definitions The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then and become partial operations on G, and will in fact be defined everywhere. We define ∗ to be and −1 to be , which gives a groupoid in the algebraic sense. Explicit reference to G0 (and hence to ) can be dropped.
Comparing the definitions The algebraic and category-theoretic definitions are equivalent as we now show. Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then and become partial operations on G, and will in fact be defined everywhere. We define ∗ to be and −1 to be , which gives a groupoid in the algebraic sense. Explicit reference to G0 (and hence to ) can be dropped.