  Things changed in the following ten years, when categories started to be used in algebraic topology and homological algebra. Mac Lane, Buchsbaum, Grothendieck and Heller were considering categories in which the collections of morphisms between two fixed objects have an additional structure. More specifically, given any two objects \(X\) and \(Y\) of a category \(\mathbf{C}\), the set \(\mathbf{Hom}(X, Y)\) of morphisms from \(X\) to \(Y\) form an abelian group. Furthermore, for reasons related to the ways homology and cohomology theories are linked, the definition of a category had to satisfy an additional formal property (which we will leave aside for the moment): it had to be self-dual. These requirements lead to the following definition.