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"
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