##
**Algebraic theories. A categorical introduction to general algebra. With a foreword by F. W. Lawvere.**
*(English)*
Zbl 1209.18001

Cambridge Tracts in Mathematics 184. Cambridge: Cambridge University Press (ISBN 978-0-521-11922-1/hbk). xvii, 249 p. (2011).

This book is an updated account of categorical general (or universal) algebra. The subject, initiated by F. W. Lawvere in the 1960s, has witnessed much development since then. The last decade in particular has produced more refined duality and characterization results, due in no small part to the authors and to Lawvere himself, revealing the clearer and more complete picture skillfully presented here. Much effort in the book has been devoted to make explicit the connection with the “classical algebra” point of view, so that one who is more familiar with the latter can fully appreciate the power and elegance of the categorical framework. In the same spirit, the comparison between the presentation and results in this book and the ones in the language of monads is fully explained (in an appendix) with detailed proofs.

In Part I (Abstract algebraic categories), an algebraic theory \(\mathbf{T}\) is defined as a small category with finite products, and an algebraic category is one equivalent to the category of finite product preserving functors from some such \(\mathbf{T}\) to \(\mathbf{Set}\) with their natural transformations (the algebras and their homomorphisms). The necessary ingredients to formulate various characterizations of the latter, as well as a duality theorem between canonical (= idempotent closed) algebraic theories and algebraic categories are progressively defined, peppered with examples from classical algebra. Very useful parallels are regularly drawn with the duality between limit theories and locally presentable categories, the role of filtered colimits and finitely presentable objects being now played by the recently defined concepts of sifted colimits and perfectly presentable objects. Subtle differences are carefully described in a 2-categorical context. The final chapter is devoted to Birkhoff’s variety theorem, with an explanation for the necessity of the added closure under directed unions in this many-sorted context.

In Part II (Concrete algebraic categories), algebraic categories are considered together with their forgetful functor, to \(\mathbf{Set}\) in the one-sorted case, and then to \(\mathbf{Set^S}\) in the \(S\)-sorted one. Again dualities and characterizations are presented, together with precise relationship with the classical algebra point of view.

The special topics of Part III present a categorical generalization of the classical results on Morita equivalences, characterizations (and constructions) of algebraic categories as free exact completions of their regular projectives, and a characterization of finitary localizations of algebraic categories as the exact, locally finitely presentable categories.

In an interesting short Postcript, the authors “explain somewhat the position [their] book has in the literature on algebra and category theory”, and “mention some of the important topics that [they] decided not to deal with” in their book. Among the latter they give reference for extensions in the enriched context, as well as theories based on categories different from \(\mathbf{Set^S}\), in particular for recent work on “homotopy varieties” by the second author and others.

Basic definitions and facts on monads and abelian categories are recalled in appendices, before to explain the links with the concepts and results in the book.

Short useful historical comments are added at the end of each chapter. No exercises are included.

The book is very well written and made as self-contained as it is reasonable for the intended audience of graduate students and researchers.

In Part I (Abstract algebraic categories), an algebraic theory \(\mathbf{T}\) is defined as a small category with finite products, and an algebraic category is one equivalent to the category of finite product preserving functors from some such \(\mathbf{T}\) to \(\mathbf{Set}\) with their natural transformations (the algebras and their homomorphisms). The necessary ingredients to formulate various characterizations of the latter, as well as a duality theorem between canonical (= idempotent closed) algebraic theories and algebraic categories are progressively defined, peppered with examples from classical algebra. Very useful parallels are regularly drawn with the duality between limit theories and locally presentable categories, the role of filtered colimits and finitely presentable objects being now played by the recently defined concepts of sifted colimits and perfectly presentable objects. Subtle differences are carefully described in a 2-categorical context. The final chapter is devoted to Birkhoff’s variety theorem, with an explanation for the necessity of the added closure under directed unions in this many-sorted context.

In Part II (Concrete algebraic categories), algebraic categories are considered together with their forgetful functor, to \(\mathbf{Set}\) in the one-sorted case, and then to \(\mathbf{Set^S}\) in the \(S\)-sorted one. Again dualities and characterizations are presented, together with precise relationship with the classical algebra point of view.

The special topics of Part III present a categorical generalization of the classical results on Morita equivalences, characterizations (and constructions) of algebraic categories as free exact completions of their regular projectives, and a characterization of finitary localizations of algebraic categories as the exact, locally finitely presentable categories.

In an interesting short Postcript, the authors “explain somewhat the position [their] book has in the literature on algebra and category theory”, and “mention some of the important topics that [they] decided not to deal with” in their book. Among the latter they give reference for extensions in the enriched context, as well as theories based on categories different from \(\mathbf{Set^S}\), in particular for recent work on “homotopy varieties” by the second author and others.

Basic definitions and facts on monads and abelian categories are recalled in appendices, before to explain the links with the concepts and results in the book.

Short useful historical comments are added at the end of each chapter. No exercises are included.

The book is very well written and made as self-contained as it is reasonable for the intended audience of graduate students and researchers.

Reviewer: Michel Hébert (Cairo)

### MSC:

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

08-02 | Research exposition (monographs, survey articles) pertaining to general algebraic systems |

18C10 | Theories (e.g., algebraic theories), structure, and semantics |

08C05 | Categories of algebras |

18C05 | Equational categories |