JOURNAL OF ALGEBRA 107, 316375 (1987) Families Parametrized by Coalgebras LUZIUS GRUNENFELDER AND ROBERT PARI? Dalhousie University, Halifax, Nova Scoria B3H 4H8, Canada Communicated by Saunders MacLane Received November 11, 1984 I. INTRODUCTION Since Hopf [H] first introduced the coalgebra structure on the homology ring of a grouplike manifold, making it into what is now called a Hopf algebra, coalgebras have been appearing more and more frequently in many branches of mathematics, in particular in algebraic topology, homological algebra and in algebraic geometry. This prompted fundamen- tal work on the structure of coalgebras themselves, a good part of which is outlined in the influential article of Milnor and Moore [M-M], in Sweedler’s book [S] and in articles mentioned later in this text. At first glance, coalgebras are strange objects. Although they are defined in algebraic terms, our algebraic intuition breaks down when trying to understand them. They really are more geometric than algebraic. This is somewhat explained by the partial duality that exists between algebras and coalgebras (with scalars from a field). The dual of a coalgebra is an algebra and although the dualizing functor is not an equivalence of categories, it does have an adjoint on the right ( )” [S, p. 1091. These two functors restrict to a contravariant equivalence between finite dimensional coalgebras and finite dimensional algebras. The category of cocommutative coalgebras is similar in many respects to the opposite of the category of commutative k-algebras, and this is the same as the category of afftne schemes over k. So, it is not surprising that cocommutative coalgebras are geometric entities. In fact, they correspond to formal schemes [C, T]. Due to this partial duality between coalgebras and algebras, many definitions in the theory of coalgebras were suggested by the corresponding concepts for algebras (such as the cotensor 0 <. of [M-M]), and the statements of many theorems are inspired by the corresponding results for algebras. But in many respects, the category of cocommutative coalgebras is much better than the category of algebras. It is Cartesian closed, i.e., there is a coalgehra of morphisms from one coalgebra to another, with the appropriate universal property (see [ML, p. 953 for the definition of “car- tesian closed”). This makes Coalg into a monoidal category [K, p. 213 and 0021-8693/87 $3.00 Copyright c) 1987 by Academc Press, Inc. All rights of reproduction in any form reserved 316