Related papers: Crossed modules, double group-groupoids and crosse…
We compare three different ways of defining group cohomology with coefficients in a crossed-module: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where…
This is the first part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans -…
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded classical and graded strongly classical 2-absorbing second submodules of graded…
We firstly prove the completeness of the category of crossed modules in a modified category of interest. Afterwards, we define pullback crossed modules and pullback cat$^1$-objects that are both obtained by pullback diagrams with extra…
The category of crossed complexes gives an algebraic model of the category of $CW$-complexes and cellular maps. We explain basic results on crossed complexes which allow the computation of free crossed resolutions of graph products of…
We introduce some algebraic structures such as singularity, commutators and central extension in modified categories of interest. Additionally, we introduce the cat$^{1}$-objects with their connection to crossed modules in these categories…
We give an explicit description for the nerve of crossed module of categories.
A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised…
Double groupoids are a type of higher groupoid structure that can arise when one has two distinct groupoid products on the same set of arrows. A particularly important example of such structures is the irrational torus and, more generally,…
In this work we develop some categorical aspects of the double structure of a module.
We decompose the crossed product functor for actions of crossed modules of locally compact groups on C*-algebras into more elementary constructions: taking crossed products by group actions and fibres in C*-algebras over topological spaces.…
Symmetry groups or groupoids of C*-algebras associated to non-Hausdorff spaces are often non-Hausdorff as well. We describe such symmetries using crossed modules of groupoids. We define actions of crossed modules on C*-algebras and crossed…
We introduce the isoclinism of crossed modules. We also give GAP implementations for constructing the isoclinism families of finite crossed modules and consequently give an enumeration about isoclinic crossed modules existing in the GAP…
Given a crossed module $\chi$, we introduce Hopf $\chi$-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of…
In this paper, we study the higher Yang-Mills theory in the framework of higher gauge theory. It was shown that the 2-form electromagnetism can be generalized to the 2-form Yang-Mills theory with the group $U(1)$ replaced by a crossed…
In this paper we define 3-crossed modules for commutative (Lie) algebras and investigate the relation between this construction and the simplicial algebras. Also we define the projective 3-crossed resolution for investigate a higher…
We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for…
This is the last part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed…
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 show that the integrability obstruction of a transitive Lie algebroid coincides with the lifting obstruction of a crossed module of groupoids associated naturally with the given algebroid. Then we extend this result to general extensions…