Abstract
This paper introduces a topological approach to decoherence that can be seen as an extension of the consistent histories approach to quantum mechanics. The approach uses the formal tools of categorical algebraic topology and sheaf theory to capture the relationship between a global description of a quantum system in terms non-commutative algebras of quantum observables and a local description in terms of local commutative algebras associated with particular measurement contexts. We claim that the difference between the algebraic structure of quantum observables and the algebraic structure of quasi-classical observables at suitable coarse-grained scales is essentially ignored in the consistent histories approach to decoherence, and that the notorious problems of the approach can be overcome by an appropriate topological treatment of this difference. We describe decoherence as a process, in which a global quantum description is reduced to local quasi-classical descriptions according to specific topological compatibility conditions .