From contradiction to conciliation: a way to "dualize" sheaves
  Our aim is to give some insights about how to approach the formal description of situations where one has to conciliate several contradictory statements, rules, laws or ideas. We show that such a conciliation structure can be naturally defined on a topological space endowed with the set of its closed sets and that this specific structure is a kind of "dualization" of the sheaf concept where "global" and "local" levels are reversed. 
Dominique Lambert and Bertrand Hespel
math.LO