##
**Topos theory.**
*(English)*
Zbl 0858.18001

Hazewinkel, M. (ed.), Handbook of algebra. Volume 1. Amsterdam: North-Holland. 501-528 (1996).

This excellent expository article (actually a chapter in the Handbook of algebra, Vol. 1), does an outstanding job of discussing the main features of topos theory in under 25 pages. This is not surprising as the authors were natural candidates for such an article in light of their recent book on topos theory, “Sheaves in geometry and logic” [New York, etc.: Springer (1992; Zbl 0822.18001)].

Toposes are both of geometrical and of logical nature, arising from Grothendieck’s work in algebraic geometry as a notion of “generalized space” on which one can define cohomology groups, as well as from Lawvere’s work on providing a foundation for mathematics based on function composition rather than set membership. Of course, as Lawvere and Tierney showed, the geometrical and logical aspects of toposes are closely related.

The article begins with a brief presentation of the axioms for elementary topoi followed by discussion of sites, sheaves and Grothendieck topoi. Geometric morphisms are then defined with a brief glimpse of classifying topoi. The examples of \(G\)-sets and sheaves are used in each section to highlight and clarify the definitions. Locales are then introduced leading to a discussion of various representation theorems for topoi, namely those of Freyd, Tierney-Joyal and Barr. The following sections of the article discuss cohomology and the fundamental group of a topos and the final section outlines the connections between topos theory and intuitionistic logic.

This article is recommended for anyone desirous of obtaining a concise introduction to and overview of topos theory and even those familiar with the subject matter will find the presentation illuminating.

For the entire collection see [Zbl 0859.00011].

Toposes are both of geometrical and of logical nature, arising from Grothendieck’s work in algebraic geometry as a notion of “generalized space” on which one can define cohomology groups, as well as from Lawvere’s work on providing a foundation for mathematics based on function composition rather than set membership. Of course, as Lawvere and Tierney showed, the geometrical and logical aspects of toposes are closely related.

The article begins with a brief presentation of the axioms for elementary topoi followed by discussion of sites, sheaves and Grothendieck topoi. Geometric morphisms are then defined with a brief glimpse of classifying topoi. The examples of \(G\)-sets and sheaves are used in each section to highlight and clarify the definitions. Locales are then introduced leading to a discussion of various representation theorems for topoi, namely those of Freyd, Tierney-Joyal and Barr. The following sections of the article discuss cohomology and the fundamental group of a topos and the final section outlines the connections between topos theory and intuitionistic logic.

This article is recommended for anyone desirous of obtaining a concise introduction to and overview of topos theory and even those familiar with the subject matter will find the presentation illuminating.

For the entire collection see [Zbl 0859.00011].

Reviewer: K.I.Rosenthal (Schenectady)

### MSC:

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

03G30 | Categorical logic, topoi |

18B25 | Topoi |