What is a noncommutative topos?

Abstract

In [1] noncommutative frames were introduced, generalizing the usual notion of frames of open sets of a topological space. In this paper we extend this notion to noncommutative versions of Grothendieck topologies and their associated noncommutative toposes of sheaves of sets.

My journey into noncommutative lattices and their theory

“…Further studies in duality are listed below. Karin has also explored connections to Church algebras (with Antonino Salibra) [29], skew Heyting algebras [18] and noncommutative toposes (with Jens Hemelaer and Lieven Le Bruyn) [21]. Again, see the references near the end.…”

Noncommutative frames

“…The main difference with the case of topological spaces, is that we cannot recover the sheaf E from its étale space E. So here sheaves are more general than étale spaces (one can remedy this by defining local homeomorphisms of locales, but we will not follow this approach here). [CVHLB19] that we can then construct a noncommutative frame H(Y, E) as follows. The elements are pairs (U, s) with U ⊆ Y open (for the original topology on Y ) and s ∈ E(U ).…”

“…In [CVHLB19], noncommutative frames were used to define noncommutative generalizations of toposes, by replacing the subobject classifier Ω with an internal noncommutative frame H having the subobject classifier as commutative shadow. While Ω has a single top element (corresponding to the statement "true" in logic), there are now multiple top elements in H. An example of a noncommutative topos that is not an elementary topos, is the category of complete directed graphs with a 4-coloring of the edges.…”