Zilber, Jorge C. Local analytic rings. (English) Zbl 0705.32002 Cah. Topologie Géom. Différ. Catég. 31, No. 1, 21-52 (1990). The author develops some aspects of the theory of analytic rings in the sense of E. J. Dubuc and G. Taubin [Cah. Topologie Géom. Différ. 24, 225-265 (1983; Zbl 0575.32004)]. If C is the category of open subsets of \({\mathbb{C}}^ n\) (with holomorphic functions as morphisms) and E is a category, an analytic ring in E is a functor which preserves transversal pullbacks and terminal objects (or, equivalently, preserves independent equalizers, finite products and open inclusions). If A is an analytic ring in E, it is called local iff for each open covering \((U_{\alpha})_{\alpha \in I}\) of an open set \(U\subset {\mathbb{C}}^ n\), the family \((A(U_{\alpha})\to A(U))_{\alpha \in J}\) is a universal effective epimorphic family in E. This definition is equivalent to the one considered in the above quoted paper. The author explicitely constructs the classifying topos of the theory of local analytic rings. Further results, examples and open problems are presented. Reviewer: E.Pascu Cited in 2 ReviewsCited in 3 Documents MSC: 32B05 Analytic algebras and generalizations, preparation theorems 18B25 Topoi 18F05 Local categories and functors Keywords:local analytic ring; classifying topos Citations:Zbl 0575.32004 PDF BibTeX XML Cite \textit{J. C. Zilber}, Cah. Topologie Géom. Différ. Catégoriques 31, No. 1, 21--52 (1990; Zbl 0705.32002) Full Text: Numdam EuDML OpenURL References: [1] 1 Artin , M. , Grothendieck , A. & Verdier , J.L. , Théorie des topos et cohomologie étale des schèmes (S.G.A. 4) , Lecture Notes in Math. 269 , Springer ( 1972 ). MR 354653 [2] 2 Bunge , M. & Dubuc , E. , Archimedian local C\infty -rings and models of Synthetic Differential Geometry , Cahiers Top. et Géom. Diff. Cat. XXVII - 3 ( 1986 ). Numdam | Zbl 0614.18007 · Zbl 0614.18007 [3] 3 Dubuc , E. & Taubin , G. , Analytic rings , Cahiers Top. et Géom. Diff. XXIV - 3 ( 1983 ). Numdam | MR 728632 | Zbl 0575.32004 · Zbl 0575.32004 [4] 4 Guillemin , V. & Pollack , A. , Differential Topology , Prentice-Hall Inc. , New Jersey , 1974 . MR 348781 | Zbl 0361.57001 · Zbl 0361.57001 [5] 5 Mac Lane , S. , Categories for the working mathematician , Springer 1971 . MR 354798 | Zbl 0906.18001 · Zbl 0906.18001 [6] 6 Malgrange , B. , Analytic spaces. Monographie 17, l ’ Enseignement Math. , Genève 1968 . MR 237824 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.