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Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules over $A$ are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$ of…
We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before…
Crossed squares and 2-crossed modules are both algebraic models for 3-types. This paper explores the interrelationships between these two models.
The notion of crossed product by a coquasi-bialgebra H is introduced and studied. The resulting crossed product is an algebra in the monoidal category of right H-comodules. We give an interpretation of the crossed product as an action of a…
We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.
In this paper, we define the notion of crossed modules of groups with action and investigate related structures. Functions for computing of these structures have been written using the GAP computational discrete algebra programming…
We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph…
We prove that if a totally disconnected locally compact group admits a topologically free boundary, then the reduced crossed product of continuous functions on its Furstenberg boundary by the group is simple. We also prove a partial…
The category $\mathbf{XSq}$ of crossed squares is equivalent to the category $\mathbf{Cat2}$ of cat$^2$-groups. Functions for computing with these structures have been developed in the package $\mathsf{XMod}$ written using the…
We introduce the notion of Whitehead sequence which is defined for a base category together with a system of abstract actions over it. In the classical case of groups and group actions the Whitehead sequences are precisely the…
We give an introduction into the ideal structure and representation theory of crossed products by actions of locally compact groups on C*-algebras. In particular, we discuss the Mackey-Rieffel-Green theory of induced representations of…
The graph C*-algebra of a directed graph E is the universal C*-algebra generated by a family of partial isometries satisfying relations which reflect the path structure of E. In the first part of this paper we consider coverings of directed…
A higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This index-free view allows for a better understanding of the underlying algebraic structures, among which are generalizations of…
We define a homology theory for pre-crossed modules that specifies to rack homology in the case when the pre-crossed module is freely generated by a rack.
We explain how the computation of induced crossed modules allows the computation of certain homotopy 2-types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some…
We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the…
Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a…
In this paper we state some applications of Gr-category theory on the classification of crossed modules and on the classification of extensions of groups of the type of a crossed module.
It is a well known result in the covering groups that a subgroup $G$ of the fundamental group at the identity of a semi-locally simply connected topological group determines a covering morphism of topological groups with characteristic…
We give a simplified approach to the cubulation of small-cancellation quotients of free products of cubulated groups. We construct fundamental groups of compact nonpositively curved cube complexes that do not virtually split.