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Objets compacts dans les topos

Part of: Special categories

Published online by Cambridge University Press:  09 April 2009

Eduardo J. Dubuc  and
Jacques Penon
Eduardo J. Dubuc
Affiliation:
Departamento de Matematicas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Jacques Penon
Affiliation:
U.E.R. de Mathmatiques, Universit de Paris VII, Tours 4555, 2 Place Jussieu, 75005 Paris, France
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Abstract

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It is well known that compact topological spaces are those space K for which given any point x0 in any topological space X, and a neighborhood H of the fibre -1 {x0} KXX, then there exists a neighborhood U of x0 such that -1UH. If now is an object in an arbitrary topos, in the internal logic of the topos this property means that, for any A in and B in K, we have (-1AB)=AB. We introduce this formula as a definition of compactness for objects in an arbitrary topos. Then we prove that in the gross topoi of algebraic, analytic, and differential geometry, this property characterizes exactly the complete varieties, the compact (analytic) spaces, and the compact manifolds, respectively.

MSC classification

Secondary: 18B25: Topoi
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 2 , April 1986 , pp. 203 - 217
Copyright
Copyright Australian Mathematical Society 1986

References

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