4 I. INTRODUCTION

that the functor GLn is representable. Thus it has a left adjoint; this is the functor

taking every group G to the ring RQ with a universal nXn matrix

representation of G (straightforward to construct by generators and relations).

Thus, the representable functors are doubly important, as a class of fundamental

constructions, and also as the context in which another fundamental type of

construction becomes possible. Hence for two given varieties of algebras C and

D, it is of interest to study, and where possible, describe completely the class of all

representable functors from C to D; or what is equivalent, and often more

convenient, the class of all co-D objects of C.

In algebraic geometry, cogroups in the category of commutative ^-algebras are

studied under the name affine group schemes. (We shall discuss these in

Chapter VIII; they include the "classical groups".) In algebraic topology, cogroups

in the category of topological spaces, with homotopy-classes of maps for morphisms

are considered [178, §1.6]. For example, the fact that the /^-sphere Sn has such a

structure for n0 is the "reason" the sets nn{X), for X a topological space,

may be made into groups in a functorial way.

The main new results of the present work, sketched in the next section, concern

coalgebras which represent functors from categories of associative rings to

categories whose objects have abelian group structures - abelian groups themselves,

rings, etc.. Incidentally, the concept called a coalgebra here should not be confused

with the concept given that name by Hopf algebraists; the two are related, but

neither is a case of the other. (We shall discuss the latter objects briefly in the last

paragraph of §9.1 below, and more extensively in §43.)

§2. Overview of results.

Let k be an associative ring with 1, and let ft-Ring denote the category

whose objects are associative rings S with 1 given with unital homomorphisms

k — 5, and whose morphisms are the obvious commuting triangles. (Thus, when

k is commutative, these include the k-algebras.) For any k-bimodule M, let

kM denote the tensor /c-ring

kM = k@ M 0 (M®kM) 0 ... .

The construction of kM is the left adjoint of the forgetful functor

/:-Ring — /c-Bimod. That is, for every k-ring S, if we loosely denote the

underlying /c-bimodule of S by the same symbol S (writing now as ring-theorists

rather than category-theorists), we have

(2.1) k-Ring\kM, S) = k-Bimod(M, S).

Fixing M, and regarding this as a functor in 5, we see from the right-hand-side

of (2.1) that the values of this functor have natural structures of abelian group

(since bimodule homomorphisms can be added and subtracted), and from the left-

hand side that this functor is representable. The fundamental result of this work,

proved in §§10-13, is that every representable functor from /c-Ring to abelian

groups has the form (2.1) for some M. This yields a contravariant equivalence

between the category of such representable functors and the category &-Bimod.

(The reader might stop to verify at this point that the "underlying additive group"