Related papers: Limits in Modified Categories of Interest
In this paper using split extensions of group-groupoids we obtain the notion of crossed modules over group-grouoids which are also called 2-groups and we prove a categorical equivalence of these types of crossed modules and double…
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…
This is the second 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 -…
The notion of crossed modules for Lie 2-algebras is introduced. We show that, associated to such a crossed module, there is a strict Lie 3-algebra structure on its mapping cone complex and a strict Lie 2-algebra structure on its…
For any cssc-crossed module a category is constructed, equipped with a structure and proved that this is a coherent categorical group. Together with a result of the previous paper, where to any categorical group the cssc-crossed module is…
The 2-categories of strict 2-groups and crossed modules are introduced and their 2-equivalence is made explicit.
In this paper, we state the notion of morphisms in the category of abelian crossed modules and prove that this category is equivalent to the category of strict Picard categories and regular symmetric monoidal functors. The theory of…
Let $A$ be a finite dimensional algebra and $D^b(A)$ be the bounded derived category of finitely generated left $A$-modules. In this paper we consider lengths of compact exceptional objects in $D^b(A)$, proving a sufficient condition such…
In this paper, we described the GAP implementation of crossed modules of commutative algebras and cat$^{1}$-algebras and their equivalence.
Connections between heaps of modules and (affine) modules over rings are explored. This leads to explicit, often constructive, descriptions of some categorical constructions and properties that are implicit in universal algebra and…
In this paper we explore some categorical results of 2-crossed module of commutative algebras extending work of Porter in [18]. We also show that the forgetful functor from the category of 2-crossed modules to the category of k-algebras,…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
The aim of this paper is to define the notion of lifting of a crossed module via a group morphism and give some properties of this type of the lifting. Further we obtain a criterion for a crossed module to have a lifting of crossed module.…
We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be…
We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining exact closed tensor categories of…
The purpose of this note is to observe that a homomorphism of discrete groups $f:\Gamma\to G$ arises as the induced map $\pi_0(\mathfrak{M})\to \pi_0(\mathfrak{X})$ on path components of some closed normal inclusion of topological groups…
We classify the module categories over the double (possibly twisted) of a finite group.
We prove that the category of preordered groups contains two full reflective subcategories that give rise to some interesting Galois theories. The first one is the category of the so-called commutative objects, which are precisely the…
For a finite braided tensor category we introduce its Picard crossed module consisting of the group of invertible module categories and the group of braided tensor autoequivalences. We describe the Picard crossed module in terms of braided…
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.