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Categories in which cocones satisfy certain exactness conditions w.r.t.pullbacks are subject to current research activities in theoretical computerscience. Usually, exactness is expressed in terms of properties of the pullbackfunctor associated with the cocone. Even in the case of non-exactness,researchers in model semantics and rewriting theory inquire an elementarycharacterization of the image of this functor. In this paper we willinvestigate this question in the special case where the cocone is a cospan,i.e. part of a Van Kampen square. The use of Descent Data as the dominantcategorical tool yields two main results: A simple condition whichcharacterizes the reachable part of the above mentioned functor in terms ofliftings of involved equivalence relations and (as a consequence) a necessaryand sufficient condition for a pushout to be a Van Kampen square formulated ina purely algebraic manner.

Characterizing Van Kampen Squares via Descent Data