Abstract
An elementary topological approach to Grothendieck's idea of descent is given. While being motivated by the idea of localization which is central in Sheaf Theory, we show how the theory of monads (=triples) provides a direct categorical approach to Descent Theory. Thanks to an important observation by Bénabou and Roubaud and by Beck, the monadic description covers descent also in the abstract context of a bifibred category satisfying the Beck-Chevalley condition. We present the fundamentals of fibrational descent theory without requiring any prior knowledge of fibred categories. The paper contains a number of new topological descent results as well as some new examples in the context of regular categories which demonstrate the subtlety of the descent problem in concrete situations.
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Dedicated to Nico Pumplün on the occasion of his sixtieth birthday
The second author acknowledges the hospitality of the Georgian Academy of Sciences and of the University of L'Aquila (Italy). The work presented in this paper was partially carried out during visits at these institutions. Both authors acknowledge partial financial support from the Natural Sciences and Engineering Council of Canada for a visit of the first author at the second author's institution.
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Janelidze, G., Tholen, W. Facets of descent, I. Appl Categor Struct 2, 245–281 (1994). https://doi.org/10.1007/BF00878100
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DOI: https://doi.org/10.1007/BF00878100