Taking advantage of the quantale-theoretic description \'etale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms modules on inverse quantal frames. The Hilbert module quantale sheaves leads naturally to a formulation bimodules that resemble imprimitivity C*-algebras.

We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a which relates the topological full group an ample groupoid with homology groupoid. Our main result complements in finite case, makes true all covered by assumptions said conjecture. Furthermore, we observe arbitrary graphs, coincides $K$-theory its $C^*$-algebra.

We consider circuits and expressions whose gates carry out multiplication in a non-associative groupoid such as loop. We deene a class we call the polyabelian groupoids, formed by iterated quasidirect products of Abelian groups. We show that a loop can express arbitrary Boolean functions if and only if it is not polyabelian, in which case its Expression Evaluation and Circuit Value problems are...

A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced by the authors. There are two natural operations acting on double pants decompositions: flips and handle-twists. It is shown by the authors that the groupoid generated by flips and handle-twists acts transitively on admissible double pants decompositions, where the class...

In the present paper, based on the previous work (Part I), we present a game semantics for the intensional variant of intuitionistic type theory that refutes the principle of uniqueness of identity proofs and validates the univalence axiom, though we do not interpret non-trivial higher propositional equalities. Specifically, following the historic groupoid interpretation by Hofmann and Streiche...

In the present paper we study some geometric objects and constructions related to groupoids of matrix subalgebras in a fixed matrix algebra. 1. The definition of groupoids In this paper we will freely use the notation from [3]. First recall some definitions and constructions [3]. 1.1. Groupoids Gk, l. Let Mkl(C) be the complex matrix algebra. Unital ∗-subalgebras isomorphic Mk(C) in some unital...

The multiplicative structure of the trivial symplectic groupoid over R associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable condi...

An equational presentation P = h ; V; E i induces a graph G P of terms and equations, while proof gures in P induces an -algebra A P on groupoids. Although the construction of A P is quite syntactic, it is not free over G P . In order to make the meaning of freeness precise, we introduce a notion of -graph, and show that A P is freely generated by the -graph G P induced by P . Incidentally, thi...

The setup of a quantization in a Tangent Groupoid structure is reviewed from the point of view of the action of Butcher Group. In this sense, quantization is implied by renormalization.

Proposed here is a new framework for the analysis of complex systems as a non-explicitly programmed mathematical hierarchy of subsystems using only the fundamental principle of causality, the mathematics of groupoid symmetries, and a basic causal metric needed to support measurement in Physics. The complex system is described as a discrete set S of state variables. Causality is described by an ...