ENRICHED CATEGORIES AND COHOMOLOGY To Margery, who typed the original preprint in Milan

From the outset, the theories of ordinary categories and of additive categories were developed in parallel. Indeed additive category theory was dominant in the early days. By additivity for a category I mean that each set of morphisms between two objects (each “hom”) is equipped with the structure of abelian group and composition on either side, with any morphism, distributes over addition: that is to say, the category is enriched in the monoidal category of abelian groups. “Enrichment” in this context is happening to the homs of the category. This enrichment in abelian groups is rather atypical since, for a category with finite products or finite coproducts, it is a property of the category rather than a structure. Linton, in [14], began developing the theory of categories enriched in monoidal categories of sets with structure. Independently of each other, Eilenberg and Kelly recognized the need for studying categories enriched in the monoidal category of chain complexes of abelian groups: differential graded categories (or DG-categories). This led to the collaboration [6] which began the theory of categories enriched in a general monoidal category, called the base. Soon after, in [1], Bénabou defined bicategories and morphisms between them. He observed that a bicategory with one object is the same as a monoidal category. He noted that a morphism of bicategories from the category 1 to Cat is what had been called (after [7]) a category together with a triple thereon; Bénabou called this a monad. For any set X he defined the term polyad in a bicategory W to mean a morphism from the chaotic category on X to W . This is important here since such a polyad is precisely a category A enriched in the bicategory W where the set of objects of A is X. Categories enriched in V are closely related to categories on which a monoidal category V acts (lately called “actegories” [15]) and the latter subject was pursued by Bénabou (in lectures I attended at Tulane University in 1969-70).