| Summary: | This thesis investigates descent for the 2-fibration of cocomplete categories over toposes and geometric morphisms. Change of base within this 2-fibration is given by the left adjoint to the restriction functor. Pitts' pullback theorem (Pi) is important to descent in this context, and a new and more natural proof of it is obtained. As in (Pi), the proof herein depends on Pare's (P2) results on generated topologies. The present context is 2-categorical, and an abstract 2-descent theorem is obtained. Its first use is to show that a geometric morphism which is of effective descent for cocomplete categories remains so for toposes. === Studying toposes as cocomplete categories is analogous to studying locales as sup-lattices. Pure geometric morphisms are introduced in terms of the cocontinuous dual of a cocomplete category. They are shown to be of effective descent for cocomplete categories. Hence, a new proof of Moerdijk's (M5) version of a classification theorem for toposes originally due to Bunge (B4) is obtained.
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