Quotients and colimits of κ-quantales

Let κ Qnt be the category of κ-quantales, quantales closed under κ-joins in which the monoid identity is the largest element. (κ is an infinite regular cardinal.) Although the lack of lattice completeness in this setting would seem to mitigate against the techniques which lend themselves so readily to the calculation of frame quotients, we show how to easily compute κ Qnt quotients by applying generalizations of the frame techniques to suitable extensions of this category. cond major tool in the analysis is the free κ-quantale over a λ-quantale, κ ⩾ λ . Surprisingly, these can be characterized intrinsically, and the generating sub-κ-quantale can even be identified. The result that the λ-free κ-quantales coincide with the λ-coherent κ-quantales directly generalizes Maddenʼs corresponding result for κ-frames. tools permit a direct and intuitive construction of κ Qnt colimits. We provide two applications: an intrinsic characterization of κ Qnt colimits, and of free (over sets) κ-quantales. The latter is a direct generalization of Whitmanʼs condition for distributive lattices.