Characterizing Van Kampen Squares via Descent Data

TL;DR

This paper investigates Van Kampen squares using descent data, providing algebraic characterizations of their properties and the conditions under which pushouts are Van Kampen squares.

Contribution

It offers a new algebraic characterization of Van Kampen squares and the conditions for pushouts to be Van Kampen, using descent data and equivalence relations.

Findings

A simple condition characterizes the reachable part of the functor in terms of liftings.

A necessary and sufficient algebraic condition for a pushout to be Van Kampen.

Enhanced understanding of exactness conditions in categories related to computer science.

Abstract

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.