摘要
Topos概念的提出(即elementary topos的公理化)是在研究了如下几类范畴之后:Grothendieck topos,集合的范畴S,以及集合论的布尔值模型。 Grothendieck提出用拓扑空间X上的set valued sheaf的全体做成的范畴Sh(X)作为推广了的拓补空间X,用以研究空间X上的上同调.他把拓扑的概念推广到小范畴(smallcategory)C上,称为一个site(或称为Grothendieck拓扑,参看第2节)。这样就可以考虑给出的site(C,J)上的set valued sheaf的全体做成的范畴Sh(C,J),用它来研究site(C,J)
A topos is a special category, it satisfies several axioms which are written in the first order language. A topos is a universe of variable sets instead of a universe of constant sets. It takes various mathematical structrues as direct objects. A topos has both geometrical and logical characters. Some topos can be viewed as a generalized space(e. g. Sh( AT), all sheaves over a topological space X)and some topos can be viewed as a category of spaces(e.g. Johnstone topos) . Topos theory provides a base ground for studying continuum physics, synthetic differential geometry, algebraic geometry and categorical logic, etc.In this paper, the fundamental properties of topos, sheaf theory and Grothendieck topos, the internal structures of a topos, internal logic and its applications, classifying topos are briefly introduced.
出处
《数学进展》
CSCD
北大核心
1992年第1期1-24,共24页
Advances in Mathematics（China）
