Abstract It is often useful to be able deal with locales in terms of presentations their underlying frames, or equivalently, the geometric theories which they classify. Given a presentation for locale, its sublocales can obtained by simply appending additional relations, but case quotient more subtle. We provide simple procedures obtaining open quotients, proper quotients general triquotients f...

Let (M,ω) be a connected, compact symplectic manifold equipped with a Hamiltonian S action. We prove that, as fundamental groups of topological spaces, π1(M) = π1(minimum) = π1(maximum) = π1(Mred), where Mred is the symplectic quotient at any value in the image of the moment map φ. Let (M,ω) be a connected, compact symplectic manifold equipped with a circle action. If the action is Hamiltonian,...