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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, Théorie 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.
Reviewer: A.Kock (Aarhus)

18B25 Topoi
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18-02 Research exposition (monographs, survey articles) pertaining to category theory