A perspective on non-commutative frame theory

Abstract

This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over byétale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We prove several main results. To start with, we establish a duality between the category of complete restriction monoids and the category ofétale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups andétale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. A wider class of quantales, called multiplicative Ehresmann quantal frames, is put into a correspondence with those localic categories where the multiplication structure map is semiopen, and all the other structure maps are open. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction betweenétale localic categories andétale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectralétale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work

Invariant means on Boolean inverse monoids

“…An attempt to dispell this was made by Wallis in his thesis [35] who developed a coordinate-free module-type theory which led naturally to rook matrices in a way analogous to that in which linear transformations on a vector space lead to (classical) matrices. Another approach to showing that the definition is a natural one uses some ideas from [25] and [14] where we refer the reader for any undefined terms. We in fact generalize an argument to be found in [10].…”

“…We believe that there is something deeper going on. Recent work on noncommutative Stone dualities [14,19,25] has linked classes of inverse semigroups and classes ofétale topological groupoids and, as the word duality suggests, this is a two-way relationship. The rôle ofétale topological groupoids within mathematics as a whole is well-established, particularly within the theory of C * -algebras [24,22].…”

“…If S is a semigroup and e is an idempotent then eSe is a subsemigroup that is a monoid with respect to the identity e. We call such subsemigroups local monoids of S. If S is a Boolean semigroup then for any idempotent e ∈ S the local monoid eSe is a Boolean monoid. Classical Stone duality may be generalized to yield a duality between Boolean monoids and a class ofétale topological groupoids called Boolean groupoids [19,25,14].…”

General non-commutative locally compact locally Hausdorff Stone duality

“…A number of recent papers, notably by Kudryavtseva, Lawson, Lenz and Resende, have generalized these to various settings. For example [Law12] and [KL16] extend the classical Stone duality between generalized Boolean algebras and zero-dimensional locally compact Hausdorff topological spaces, while [Res07], [LL13] and [KL17] extend the duality between spatial frames/locales and sober topological spaces. However, even in the classical commutative cases, both these dualities have their drawbacks.…”

Effective equivalence relations and principal quantales

“…This is proved in [17], where the relations toétale groupoids, both topological and localic, are established precisely and include a bijection, up to isomorphisms, between the class of localicétale groupoids and that of inverse quantal frames. A consequence of these is that the role played by inverse semigroups in relation to topologicalétale groupoids (see, e.g., [3,12,15]) is subsumed by inverse quantal frames (this is further generalized for non-involutive quantales andétale categories in [4]), and in [7] it has been shown that this generalizes classical topological correspondences between inverse semigroups and germ groupoids (see also [5]).…”