  Categories in which cocones satisfy certain exactness conditions w.r.t.
pullbacks are subject to current research activities in theoretical computer
science. Usually, exactness is expressed in terms of properties of the pullback
functor associated with the cocone. Even in the case of non-exactness,
researchers in model semantics and rewriting theory inquire an elementary
characterization of the image of this functor. In this paper we will
investigate 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 dominant
categorical tool yields two main results: A simple condition which
characterizes the reachable part of the above mentioned functor in terms of
liftings of involved equivalence relations and (as a consequence) a necessary
and sufficient condition for a pushout to be a Van Kampen square formulated in
a purely algebraic manner.
