Quantales as geometric objects: symmetry beyond groupoids?

Modern mathematics has become pervaded by the idea that in order to cater for certain notions of symmetry, in particular of a local nature, one needs to go beyond group theory, replacing groups by groupoids. A nice survey of some implications of this idea in algebra, geometry, and analysis (as of 1996) can be found in [19]. The same idea can be expressed in terms of pseudogroups, which provide another generalization of groups, or, even more generally, by abstract inverse semigroups. See the first chapters of [7] for motivations and a good historical account.