Abstract
We consider pseudo-descent in the context of 2-fibrations. A 2-category of descent data is associated to a 3-truncated simplicial object in the base 2-category. A morphism q in the base induces (via comma-objects and pullbacks) an internal category whose truncated simplicial nerve induces in turn the 2-category of descent data for q. When the 2-fibration admits direct images, we provide the analogous of the Beck–Bénabou–Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad q *Σ q . We introduce a notion of strong 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations c od:F ib(R)→R admits direct images. In this context, we show that essentially-surjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem.
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Hermida, C. Descent on 2-Fibrations and Strongly 2-Regular 2-Categories. Applied Categorical Structures 12, 427–459 (2004). https://doi.org/10.1023/B:APCS.0000049311.17100.da
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DOI: https://doi.org/10.1023/B:APCS.0000049311.17100.da