Given a crossed module of groupoids [Formula: see text], we construct (1) natural homomorphism from the product groupoid text] to and (2) transgression map singular cohomology nerve text]. The latter turns out be identical obtained by Tu–Xu in their study equivariant text]-theory.

The aim of this paper is to show one more facet of the role of crossed complexes as generalisations of both groups (or groupoids) and of chain complexes. We do this by defining and establishing the main properties of a classifying space functor B: mia -> SToft from the category of crossed complexes to the category of spaces. The basic example of a crossed complex is the fundamental crossed comp...

This thesis contains some results concerning groupoid dynamical systems and crossed products. We introduce the notion of a proper groupoid dynamical system and of its generalized fixed point algebra. We show that our notion of proper groupoid dynamical system extends both the notion of proper actions of groups on topological spaces and the notion of the proper group dynamical systems introduced...

In this paper we study invariant differential operators on manifolds with a given parabolic structure. The model for the parabolic geometry is the quotient of the orthogonal group by a maximal parabolic subgroup corresponding to crossing of the k-th simple root of the Dynkin diagram. In particular, invariant differential operators discussed in the paper correspond (in a flat model) to the Dirac...

Let M be a right R-module. We call M Rad-H-supplemented iffor each Y M there exists a direct summand D of M such that(Y + D)/D (Rad(M) + D)/D and (Y + D)/Y (Rad(M) + Y )/Y .It is shown that:(1) Let M = M1M2, where M1 is a fully invariant submodule of M.If M is Rad-H-supplemented, thenM1 andM2 are Rad-H-supplemented.(2) Let M = M1 M2 be a duo module and Rad--supplemented. IfM1 is radical M2-...

In [BH5] it is shown how the Relative Hurewicz Theorem follows from a Generalised Van Kampen Theorem (GVKT) for the fundamental crossed complex of a filtered space, and in [BL3] it is shown how a new multirelative Hurewicz Theorem follows from a GVKT for the fundamental cat-group of an n-cube of spaces. The purpose of this paper is to advertise and explain some implications and special cases of...

Grothendieck Toposes are often considered as generalized spaces; indeed, every space gives rise to a topos of sheaves, and various invariants and constructions from (algebraic) topology can be generalized to the level of toposes. In this talk, I will introduce a newly discovered invariant called the isotropy group of a topos and illustrate by considering special cases such as continuous group a...

The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfeld modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfeld modules over a braided system...