Lecture notes on topoi and quasitopoi.

*(English)*Zbl 0727.18001
Singapore etc.: World Scientific. xi, 290 p. $ 32.00; £21.00 (1991).

Quasi-toposes came into mathematics by examples like the category of quasi-topological spaces of E. Spanier [Duke Math. J. 30, 1-14 (1963; Zbl 0114.387)], the category of bornological sets [H. Hogbe- Nlend, ThĂ©orie des bornologies et applications (Lect. Notes Math. 213) (1971; Zbl 0225.46005)], or the category of subsequential spaces of P. T. Johnstone [Proc. Lond. Math. Soc., III. Ser. 38, 237-271 (1979; Zbl 0402.18006)], but also as categories of separated presheaves for an arbitrary (Grothendieck-) site.

Penon, the author, Dubuc, and others, realized in the mid 70s that these categories have many features in common with toposes, whence the name “quasi-topos” (Penon). The present book is the first coherent account of the theory of quasi-toposes, stressing the similarity with topos theory; in fact, by leaving ‘quasi’ aside, the book even provides a handy introduction to topos theory itself.

On the other hand, in so far as quasi-topos theory properly is concerned, the book has, to the reviewer’s opinion, an emphasis on a generality that leads into technicalities, on the expense of the applications in the significant examples. Less than half a page is devoted to the categories of subsequential spaces, and to that of bornological sets. Also, the structure theory for quasi-toposes, identifying them with categories of separated presheaves (due, for concrete quasi-toposes, to E. Dubuc [Lect. Notes Math. 753, 239-254 (1979; Zbl 0423.18006)] and to F. Borceux and M. C. Pedicchio [Category Theory Conference Como 1990, to appear] for the general case) is not to be found.

A final chapter claims to deal with fuzzy sets; but it is really about D. Higgs’ “sets with Heyting-algebra-valued equality”; this feature adds considerably to the value of the book, since this important notion seems not to be well documented in the literature.

Penon, the author, Dubuc, and others, realized in the mid 70s that these categories have many features in common with toposes, whence the name “quasi-topos” (Penon). The present book is the first coherent account of the theory of quasi-toposes, stressing the similarity with topos theory; in fact, by leaving ‘quasi’ aside, the book even provides a handy introduction to topos theory itself.

On the other hand, in so far as quasi-topos theory properly is concerned, the book has, to the reviewer’s opinion, an emphasis on a generality that leads into technicalities, on the expense of the applications in the significant examples. Less than half a page is devoted to the categories of subsequential spaces, and to that of bornological sets. Also, the structure theory for quasi-toposes, identifying them with categories of separated presheaves (due, for concrete quasi-toposes, to E. Dubuc [Lect. Notes Math. 753, 239-254 (1979; Zbl 0423.18006)] and to F. Borceux and M. C. Pedicchio [Category Theory Conference Como 1990, to appear] for the general case) is not to be found.

A final chapter claims to deal with fuzzy sets; but it is really about D. Higgs’ “sets with Heyting-algebra-valued equality”; this feature adds considerably to the value of the book, since this important notion seems not to be well documented in the literature.

Reviewer: A.Kock (Aarhus)