In this paper, projective modules over a quantale are characterized by distributivity, continuity, and adjointness conditions. It is then shown that a morphism Q --> A of commutative quantales is coexponentiable if and only if the corresponding Q-module is projective, and hence, satisfies these equivalent conditions.
Keywords: projective, flat, completely distributive, totally continuous, quantale module
2010 MSC: 06F07,18A40,18D10,18D20
Theory and Applications of Categories, Vol. 31, 2016, No. 30, pp 839-851.