##
**Toposes, triples and theories.**
*(English)*
Zbl 0567.18001

Grundlehren der Mathematischen Wissenschaften, 278. New York etc.: Springer-Verlag. XIII, 345 p. DM 138.00 (1985).

This book gives a fairly good introduction to the three themes in the title with the specific purpose of exploiting some of their interconnections, while also exploring in detail some selected aspects of each of the three. The style is direct, the exercises plentiful and never trivial, and the notation standard enough to permit further reading. The book, however – unlike its counterpart by P. T. Johnstone [Topos theory. London etc.: Academic Press (1977; Zbl 0368.18001)] – is not intended as a reference work for topos theory, not only because several aspects of the latter are absent, but also because it is easier to work throughout the book in a systematic manner than it is to try to find something specific in it. In fact, the book can be of great value to someone trying to learn the subject of toposes and acquire some category theory useful in other contexts too, in the process. Now let us deal with its contents in some detail.

The book begins with a chapter on category theory. However, this does not render it self-contained from this point of view; as the authors themselves point out, the book might be ”tough going” for someone without previous exposure to categorical reasoning. The next three chapters are introductions, respectively, to toposes, triples, and theories.

Toposes are defined as left exact categories with power objects. Although this definition paves the way to proving that certain categories are toposes, it also obliterates the role of the internal logic in carrying out set-like constructions in universes of variable - rather than constant - sets; surely one of the main aspects of a theory which provided the bridge between logic and geometry. Logic is also absent from the treatment the authors make of theories; the latter are presented in the light of a variation of Ehresmann’s sketches [A. Bastiani, C. Ehresmann, Cah. Topologie Géom. Différ. 13, 105–214 (1972; Zbl 0263.18009)] although, unlike the latter, the sketches in this book are not even categories. As a justification for what seems like a step backwards in Lawvere’s program, one of the authors [M. Barr, ”Models of sketches”, Cah. Topol. Géom. Différ. Catégoriques 27, No. 2, 93–107 (1986; Zbl 0594.18001)] points out its virtues in that, not only does it allow an approach which – in its naiveté – is closer to mathematical practice, but also in that the almost ever present finiteness aspect renders such theories of potential importance in theoretical computer science. Although these claims have yet to be supported, it is indeed a fact that the appearance of the book being reviewed has provided an incentive for the study of various aspects of sketches – including its logical aspects – and that such research is undergoing and active.

The chapter on triples is fairly standard and – although much of it is available in print in the same form as here – its inclusion makes the book rather self contained beyond basic category theory. It must be noted that Prop. 3.6.1 in it contains an error in the statement (and proof); as noted by F. Gago-Couso, it is necessary to add to condition (4) the identity \(\mu '\cdot \sigma T'=\sigma \cdot T\mu\), for the statement to be correct. (The correction will be included in a subsequent edition of this book.) Also standard are the applications of triples in topos theory, e.g., Paré’s proof of Mikkelsen’s theorem on the existence of finite colimits; the proof on the existence of a left (respectively, right) adjoint to a logical functor of toposes possessing a right (respectively, left) adjoint – also due to Mikkelsen; and the characterization of sheaf categories over a base topos as categories of coalgebras for a left exact cotriple. One important omission here is any reference to the work of E. Dubuc [Lect. Notes Math. 61, 69–91 (1968; Zbl 0172.021)], while, instead, almost direct consequences of his early theorems are amply quoted (”Butler’s theorems”).

The most positive aspect of this book, in the reviewer’s opinion, is that - although bypassing a direct appeal to logic as such - it provides a far more accessible source to the beautiful and elegant ideas of P. Freyd [Bull. Aust. Math. Soc. 7, 1–76, 467–480 (1972; Zbl 0252.18001 and Zbl 0252.18002] than its original source ever was, and that, in addition, it exploits such ideas in order to give simple proofs to some of the important theorems in topos theory – to wit: Deligne’s theorem and its generalization by Makkai and Reyes, and Barr’s theorem. The embedding theorems are also used to prove Diaconescu’s result that (internal) choice implies Booleaness. Regardless of the fact that most of the constructivist point of view is surely lost by an appeal to non- elementary theorems as the embedding theorems - as is lost, too, if the metatheorem for coherent logic is employed - there is no doubt that these methods have brought about, in recent years, an enormous progress in the subject of topos theory and its applications in mathematics and that, if anything, that alone surely justifies its uses.

As mentioned earlier, this book lacks several of the standard topics in topos theory, among these the development and use of the internal language of a topos. Also missing - partly since the use of the metatheorems to prove, e.g., Deligne’s theorem, makes it unnecessary there – is a proof of Diaconescu’s theorem about presheaf toposes classifying flat functors, as well as other relative results of category theory over an arbitrary base topos. Equally conspicuous for their absence are applications of topos theory in mathematics, except briefly in some of the exercises. These, however, are not serious defects: the book certainly achieves well what it sets out to do and, as such, it can be warmly recommended. As compared to Johnstone’s book (op. cit.) for example, it has to its advantage a greater accessibility than the latter, if not its erudition and completeness. The book being reviewed does, indeed, fill a wide gap between a mostly trivial and full of historical (and otherwise) errors as is the book by R. Goldblatt [Topoi. Amsterdam etc.: North-Holland (1979; Zbl 0434.03050)], and Johnstone’s authoritative, yet difficult monograph.

The book begins with a chapter on category theory. However, this does not render it self-contained from this point of view; as the authors themselves point out, the book might be ”tough going” for someone without previous exposure to categorical reasoning. The next three chapters are introductions, respectively, to toposes, triples, and theories.

Toposes are defined as left exact categories with power objects. Although this definition paves the way to proving that certain categories are toposes, it also obliterates the role of the internal logic in carrying out set-like constructions in universes of variable - rather than constant - sets; surely one of the main aspects of a theory which provided the bridge between logic and geometry. Logic is also absent from the treatment the authors make of theories; the latter are presented in the light of a variation of Ehresmann’s sketches [A. Bastiani, C. Ehresmann, Cah. Topologie Géom. Différ. 13, 105–214 (1972; Zbl 0263.18009)] although, unlike the latter, the sketches in this book are not even categories. As a justification for what seems like a step backwards in Lawvere’s program, one of the authors [M. Barr, ”Models of sketches”, Cah. Topol. Géom. Différ. Catégoriques 27, No. 2, 93–107 (1986; Zbl 0594.18001)] points out its virtues in that, not only does it allow an approach which – in its naiveté – is closer to mathematical practice, but also in that the almost ever present finiteness aspect renders such theories of potential importance in theoretical computer science. Although these claims have yet to be supported, it is indeed a fact that the appearance of the book being reviewed has provided an incentive for the study of various aspects of sketches – including its logical aspects – and that such research is undergoing and active.

The chapter on triples is fairly standard and – although much of it is available in print in the same form as here – its inclusion makes the book rather self contained beyond basic category theory. It must be noted that Prop. 3.6.1 in it contains an error in the statement (and proof); as noted by F. Gago-Couso, it is necessary to add to condition (4) the identity \(\mu '\cdot \sigma T'=\sigma \cdot T\mu\), for the statement to be correct. (The correction will be included in a subsequent edition of this book.) Also standard are the applications of triples in topos theory, e.g., Paré’s proof of Mikkelsen’s theorem on the existence of finite colimits; the proof on the existence of a left (respectively, right) adjoint to a logical functor of toposes possessing a right (respectively, left) adjoint – also due to Mikkelsen; and the characterization of sheaf categories over a base topos as categories of coalgebras for a left exact cotriple. One important omission here is any reference to the work of E. Dubuc [Lect. Notes Math. 61, 69–91 (1968; Zbl 0172.021)], while, instead, almost direct consequences of his early theorems are amply quoted (”Butler’s theorems”).

The most positive aspect of this book, in the reviewer’s opinion, is that - although bypassing a direct appeal to logic as such - it provides a far more accessible source to the beautiful and elegant ideas of P. Freyd [Bull. Aust. Math. Soc. 7, 1–76, 467–480 (1972; Zbl 0252.18001 and Zbl 0252.18002] than its original source ever was, and that, in addition, it exploits such ideas in order to give simple proofs to some of the important theorems in topos theory – to wit: Deligne’s theorem and its generalization by Makkai and Reyes, and Barr’s theorem. The embedding theorems are also used to prove Diaconescu’s result that (internal) choice implies Booleaness. Regardless of the fact that most of the constructivist point of view is surely lost by an appeal to non- elementary theorems as the embedding theorems - as is lost, too, if the metatheorem for coherent logic is employed - there is no doubt that these methods have brought about, in recent years, an enormous progress in the subject of topos theory and its applications in mathematics and that, if anything, that alone surely justifies its uses.

As mentioned earlier, this book lacks several of the standard topics in topos theory, among these the development and use of the internal language of a topos. Also missing - partly since the use of the metatheorems to prove, e.g., Deligne’s theorem, makes it unnecessary there – is a proof of Diaconescu’s theorem about presheaf toposes classifying flat functors, as well as other relative results of category theory over an arbitrary base topos. Equally conspicuous for their absence are applications of topos theory in mathematics, except briefly in some of the exercises. These, however, are not serious defects: the book certainly achieves well what it sets out to do and, as such, it can be warmly recommended. As compared to Johnstone’s book (op. cit.) for example, it has to its advantage a greater accessibility than the latter, if not its erudition and completeness. The book being reviewed does, indeed, fill a wide gap between a mostly trivial and full of historical (and otherwise) errors as is the book by R. Goldblatt [Topoi. Amsterdam etc.: North-Holland (1979; Zbl 0434.03050)], and Johnstone’s authoritative, yet difficult monograph.

Reviewer: Marta Bunge (Montreal /Quebec)

### MSC:

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

18B25 | Topoi |

03G30 | Categorical logic, topoi |

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

18C15 | Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |