12 September 2018, Algebra|Coalgebra Seminar, Ganna Kudryavtseva

We discuss an extension of fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by etale localic categories. These categories are put in a duality with complete and infinitely distributive restriction monoids (restriction monoids being a well-established class of non-regular generalizations of inverse monoids). As a special case this includes the duality between etale localic groupoids and pseudogroups (defined as complete and infinitely distributive inverse monoids). The relationship between categories and monoids is mediated by a class of quantales called restriction quantal frames. Projecting down to topological setting, we extend the classical adjunction between locales and topological spaces to an adjunction between etale localic categories and etale topological categories. As a consequence, we deduce a duality between distributive restriction semigroups and spectral etale topological categories. Our work unifies and upgrades the earlier work by Pedro Resende, and also by Mark V. Lawson and Daniel H. Lenz.

The talk is based on a joint work with Mark V. Lawson.