Characterizing Van Kampen Squares via Descent Data

Harald König
Uwe Wolter
Michael Löwe

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.

In Ulrike Golas and Thomas Soboll: Proceedings Seventh ACCAT Workshop on Applied and Computational Category Theory (ACCAT 2012), Tallinn, Estonia, 01/04/2012, Electronic Proceedings in Theoretical Computer Science 93, pp. 61–81.
Published: 21st August 2012.

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