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.