In this postscript, we intend to explain somewhat the position our book has in the literature on algebra and category theory, and we want to mention some of the important topics that we decided not to deal with in our book.

One-sorted algebraic theories provide a very convenient formalization, based on the concept of finite product, of the classical concept of “the collection of all algebraic operations” present in a given kind of algebras, for example, in groups or boolean algebras. These theories lead to concrete categories A of algebras, that is, to categories equipped with a faithful functor U: A → Set. They can also be used to find an algebraic information present in a given concrete category A: we can form the algebraic theory whose n-ary operations are precisely the natural transformations Un → U. In the case of groups (and in any one-sorted algebraic category), these “implicit” operations are explicit; that is, they correspond to operations of the theory of groups. But on finite algebras (e.g., finite semigroups), there exist implicit operations that are not explicit, and they are important in the theory of automata (see Almeida, 1994). The passages from one-sorted algebraic theories to one-sorted algebraic categories and back form a duality that is a biequivalence in general. And, as we will see in Appendix C, this passage is an equivalence if we restrict one-sorted algebraic categories to uniquely transportable ones.

F. W. Lawvere's introduction of the concept of an algebraic theory in 1963 proved to be a fundamental step toward developing a categorical view of general algebra in which varieties of algebras are formalized without details of equational presentations. An algebraic theory as originally introduced is, roughly speaking, a category whose objects are all finite powers of a given object. An algebra is then a set-valued functor preserving finite products, and a homomorphism between algebras is a natural transformation. In the almost half a century that has followed Lawvere's introduction, this idea has gone through a number of generalizations, ramifications, and applications in areas such as algebraic geometry, topology, and computer science. The generalization from one-sorted algebras to many-sorted algebras (of particular interest in computer science) leads to a simplification: an algebraic theory is now simply a small category with finite products.

Abstract algebraic categories

In Part I of this book, consisting of Chapters 1–10, we develop the approach in which algebraic theories are studied without reference to sorting. Consequently, algebraic categories are investigated as abstract categories. We study limits and colimits of algebras, paying special attention to the sifted colimits because they play a central role in the development of algebraic categories. For example, algebraic categories are characterized precisely as the free completions under sifted colimits of small categories with finite coproducts, and algebraic functors are precisely the functors preserving limits and sifted colimits.

The aim of this chapter is to fix some notation and recall well-known facts concerning basic concepts of category theory used throughout the book. The reader may well skip it and return to it when needed. Only the most usual definitions and results of the theory of categories are mentioned here; more about them can be found in any of the books mentioned at the end of this chapter.

Foundations In category theory, one needs to distinguish between small collections (sets) and large ones (classes). An arbitrary set theory making such a distinction possible is sufficient for our book. The category of (small) sets and functions is denoted by

Set.

All categories with which we work have small hom-sets. It follows that every object has only a set of retracts (see 0.16) up to isomorphism.

Properties of functors A functor F: A → B is

1. faithful if for every parallel pair of morphisms f, g: A ⇉ A′ in A, one has f = g whenever Ff = Fg

2. full if for every morphism b: FA → FA′ in B, there exists a morphism a: A → A′ in A such that Fa = b

3. essentially surjective if for every object B in B, there exists an object A in A with B isomorphic to FA

4. […]