Noncommutative frames

Abstract

We characterize the left-handed noncommutative frames that arise from sheaves on topological spaces. Further, we show that a general left-handed noncommutative frame A arises from a sheaf on the dissolution locale associated to the commutative shadow of A. Both constructions are made precise in terms of dual equivalences of categories, similar to the duality result for strongly distributive skew lattices in [BCVG + 13].

Noncommutative frames revisited

“…The following is a correction of a result in [5], where the assumption of being join complete was erroneously omitted.…”

“…In [5], the first author introduced noncommutative frames, motivated by a noncommutative topology constructed by Le Bruyn [7] on the points of the Connes-Consani Arithmetic Site [2,3]. The definition of noncommutative frame fits in the general theory of skew lattices, a theory that goes back to Pascual Jordan [6] and is an active research topic starting with a series of papers of the third author [8,9,10].…”

“…In Section 4 we study join completeness in terms of D-classes. In Section 5 we state and prove a correction of Theorem 4.4 of [5], where the assumption of join completeness was erroneously omitted. Theorem 5.1 states that if S is a join complete, strongly distributive skew lattice with 0, then S is a noncommutative frame if and only if its commutative shadow S/D is a frame.…”