CHAPTER I.

Introduction.

§0. General prerequisites.

In this work we shall assume familiarity with the elementary concepts of

noncommutative ring theory, including that of a tensor product of bimodules, and

with the basic terminology of category theory, including ''natural transformations"

(morphisms of functors). General references for ring theory are [115], [48], [49],

for category theory, [122].

More specialized results and concepts which are important in this work, in

particular, those of algebra and coalgebra objects in a category, and of representable

algebra-valued functors, are developed in Chapter II. Below, in §1, we give a brief

motivated sketch of some of this background material, followed in §2 by a summary

of the main results of Chapters III, V, and VI, and some indication of the subject

matter of the remaining chapters.

§1. Introductory sketch - what are coalgebras, and why?

Clearly, the following is a very basic type of construction in algebra: Given two

categories of algebraic objects C and D (e.g., commutative rings, and groups),

we construct from each object S of C an object V(S) of D by taking for the

elements of V(S) all X-tuples of elements of S (X a fixed set) that satisfy a fixed

system Y of equations in their coordinates, and defining the operations of V(S)

by formulas (in precise language, "terms") in these coordinates under the

operations of S. Examples include "forgetful functors", where X = l , Y= 0

and the operations of V(S) are a subset (possibly empty) of the operations of 5,

as well as less trivial constructions, such as the formal-power-series-ring functor

from rings to rings, and the functor SLn from commutative rings to groups. Note

that to establish that one has such a construction, one must verify that the operations

so defined will always take a family of Z-tuples satisfying the equations Y to an

X-tuple again satisfying these equations, and that the resulting objects V(S) will

satisfy the identities or other conditions defining membership in D.

A key tool in studying such functors is the following observation. Let

categories C and D, and a construction V based, as above, on families X and

Y be given. Then if C is a reasonable category of algebraic objects (e.g., a

variety of algebras, definition recalled in §6 below), we can form an object of C

presented by X and Y as generators and relations:

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