The spectrum of a noncommutative C*-algebra in covariant quantum logic
In algebraic quantum theory a system is described by a C*-algebra (the selfadjoint part of this algebra represents the observable quantities). Inspired by Bohr's view on quantum mechanics we study this C*-algebra by looking at its commutative C*-subalgebras. More precise, these subalgebras make up a poset (partially ordered by inclusion), and we look at the topos of functors from this poset to the category of sets. The original non-commutative C*-algebra (in the topos Sets) defines in a simple way such a functor. This functor is a commutative C*-algebra in the internal language of the functor topos. Using a constructive version of Gelfand duality we calculate the spectrum of this internal commutative C*-algebra.