scholarly journals Transgression maps for crossed modules of groupoids

2021 ◽  
pp. 2150061
Author(s):  
Xiongwei Cai
Keyword(s):  
Crossed Product ◽  
Crossed Module ◽  
Natural Homomorphism ◽  
Singular Cohomology ◽  
Crossed Modules

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

Internal Crossed Modules

2003 ◽  
Vol 10 (1) ◽  
pp. 99-114 ◽  
Author(s):  
G. Janelidze
Keyword(s):  
Crossed Module ◽  
Crossed Modules

Abstract We introduce the notion of (pre)crossed module in a semiabelian category, and establish equivalences between internal reflexive graphs and precrossed modules, and between internal categories and crossed modules.

scholarly journals Group cohomology with coefficients in a crossed module

2010 ◽  
Vol 10 (2) ◽  
pp. 359-404 ◽  
Author(s):  
Behrang Noohi
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Geometric Approach ◽  
Group Cohomology ◽  
Theoretic Approach ◽  
Crossed Module ◽  
Explicit Approach ◽  
Crossed Modules ◽  
Cohomology Sequence

AbstractWe 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 the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

SEMI-COMPLETE CROSSED MODULES OF LIE ALGEBRAS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250096
Author(s):  
A. AYTEKIN ◽  
J. M. CASAS ◽  
E. Ö. USLU
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Crossed Module ◽  
Crossed Modules ◽  
Sufficient And Necessary Conditions

We investigate some sufficient and necessary conditions for (semi)-completeness of crossed modules in Lie algebras and we establish its relationships with the holomorphy of a crossed module. When we consider Lie algebras as crossed modules, then we recover the corresponding classical results for complete Lie algebras.

scholarly journals Quasirationality and prounipotent crossed modules

2019 ◽  
Vol 28 (13) ◽  
pp. 1940012
Author(s):  
A. M. Mikhovich
Keyword(s):  
Crossed Module ◽  
Defining Relation ◽  
Crossed Modules

We study quasirational (QR) presentations of (pro-[Formula: see text])groups, which contain aspherical presentations and their subpresentations, and also still mysterious pro-[Formula: see text]-groups with a single defining relation. Using schematization of QR-presentations and embedding of the rationalized module of relations into a diagram related to a certain prounipotent crossed module, we derive cohomological properties of pro-[Formula: see text]-groups with a single defining relation.

scholarly journals Normality and quotient in the category of crossed modules within the category of groups with operations

2019 ◽  
Vol 38 (7) ◽  
pp. 169-179
Author(s):  
Tunçar Şahan ◽  
Osman Mucuk
Keyword(s):  
Crossed Module ◽  
Crossed Modules

In this paper we define the notions of normal subcrossed module and quotient crossed module within groups with operations; and then give same properties of such crossed modules in groups with operations.

scholarly journals Liftings of crossed modules in the category of groups with operations

2019 ◽  
Vol 38 (7) ◽  
pp. 181-193
Author(s):  
H. Fulya Akız ◽  
Osman Mucuk ◽  
Tunçar Şahan
Keyword(s):  
Crossed Module ◽  
Crossed Modules ◽  
Internal Groupoid

In this paper we define the notion of lifting of a crossed module via the morphism in groups with operations and give some properties of this type of liftings. Further we prove that the lifting crossed modules of a certain crossed module are categorically equivalent to the internal groupoid actions on groups with operations, where the internal groupoid corresponds to the crossed module.

CHARACTERIZING n-ISOCLINIC CLASSES OF CROSSED MODULES

2018 ◽  
Vol 61 (03) ◽  
pp. 637-656
Author(s):  
HAJAR RAVANBOD ◽  
ALI REZA SALEMKAR ◽  
SAJEDEH TALEBTASH
Keyword(s):  
Equivalence Relation ◽  
Crossed Module ◽  
Basic Properties ◽  
Crossed Modules

AbstractIn this paper, we introduce the notion of the equivalence relation, called n-isoclinism, between crossed modules of groups, and give some basic properties of this notion. In particular, we obtain some criteria under which crossed modules are n-isoclinic. Also, we present the notion of n-stem crossed module and, under some conditions, determine them within an n-isoclinism class.

scholarly journals Group-groupoid actions and liftings of crossed modules

2019 ◽  
Vol 26 (3) ◽  
pp. 437-447 ◽  
Author(s):  
Osman Mucuk ◽  
Tunçar Şahan
Keyword(s):  
Crossed Module ◽  
Crossed Modules ◽  
Group Morphism

Abstract The aim of this paper is to define the notion of lifting via a group morphism for a crossed module and give some properties of this type of liftings. Further, we obtain a criterion for a crossed module to have a lifting crossed module. We also prove that the category of the lifting crossed modules of a certain crossed module is equivalent to the category of group-groupoid actions on groups, where the group-groupoid corresponds to the crossed module.

ENUMERATION OF WHITEHEAD GROUPS OF LOW ORDER

2002 ◽  
Vol 12 (05) ◽  
pp. 645-658
Author(s):  
MURAT ALP
Keyword(s):  
Group Theory ◽  
Programming Language ◽  
Crossed Module ◽  
Whitehead Group ◽  
Group Of Automorphisms ◽  
Isomorphism Classes ◽  
Crossed Modules ◽  
Whitehead Groups ◽  
A Share ◽  
Low Order

In this paper we describe a share package of functions for computing with finite, permutation crossed modules, cat1-groups and their morphisms, written using the [Formula: see text] group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat1-groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module [Formula: see text], and the corresponding monoid of sections of a cat1-group [Formula: see text], are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of X, are used to construct the actor crossed module of X which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat1-structures on groups of order at most 30 in [2].

scholarly journals From simplicial homotopy to crossed module homotopy in modified categories of interest

2020 ◽  
Vol 27 (4) ◽  
pp. 541-556
Author(s):  
Kadir Emir ◽  
Selim Çetin
Keyword(s):  
Lie Algebras ◽  
Equivalence Relation ◽  
Crossed Module ◽  
Algebraic Structures ◽  
Homotopy Equivalence ◽  
Associative Algebras ◽  
Leibniz Algebras ◽  
Crossed Modules ◽  
Simplicial Objects ◽  
Simplicial Homotopy

AbstractWe address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.

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