非交换 $H^2$ 与在群中的系数及在交叉模中的系数

The third cohomology group classifies crossed module extensions

We give an elementary proof of the well-known fact that the third cohomology group H^3(G, M) of a group G with coefficients in an abelian G-module M is in bijection to the set Ext^2(G, M) of equivalence classes of crossed module extensions of G with M.

Isoclinism of crossed modules

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 library.

On finite induced crossed modules and the homotopy 2-type of mapping cones

Results on the finiteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2-types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of some finite crossed modules are given, using crossed complex methods.

Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups

We define and study the notion of a crossed module over an inverse semigroup and the corresponding $4$-term exact sequences, called crossed module extensions. For a crossed module $A$ over an $F$-inverse monoid $T$, we show that equivalence classes of admissible crossed module extensions of $A$ by $T$ are in a one-to-one correspondence with the elements of the cohomology group $H^3_\le(T^1,A^1)$.

Braids and crossed modules

We describe Artin's braid group on a (fixed) finite number of strings as a crossed module over itself. In particular, we interpret the braid relations as crossed module structure relations.

交叉模

Crossed modules

本文综述了交叉模的概念，指出其由群、自同构群及典范同态构成，并源于群展示关系、扩张问题、低维拓扑及非交换代数推广，且对Lie代数同样适用。

群在交叉模系数下的非阿贝尔上同调与伽罗瓦上同调

Non-abelian hypercohomology of a group with coefficients in a crossed module, and Galois cohomology

发展了群在交叉模系数下的上同调理论，并应用于定义约化代数群伽罗瓦上同调的阿贝尔化映射。

Free crossed resolutions of groups and presentations of modules of identities among relations

We give formulae for a module presentation of the module of identities among relations for a presentation of a group, in terms of information on 0- and 1-combings of the Cayley graph. This is seen as a special case of extending a partial free crossed resolution of a group given a partial contracting homotopy of its universal cover.

Quasirationality and prounipotent crossed modules

Quasirational presentations ($QR$-presentations) of (pro-$p$)groups are studied. Such presentations include, in particular, aspherical presentations of discrete groups and their subpresentations and pro-$p$-groups with a single defining relation. Using schematization of $QR$-presentations we answer a variation of J.-P. Serre's question on one-relator pro-$p$-groups.

Crossed modules as maps between connected components of topological groups

The purpose of this note is to observe that a homomorphism of discrete groups $f:Γ\to G$ arises as the induced map $π_0(\mathfrak{M})\to π_0(\mathfrak{X})$ on path components of some closed normal inclusion of topological groups $\mathfrak{M}\subseteq \mathfrak{X},$ if and only if the map $f$ can be equipped with a crossed module structure. In that case an essentially unique realization $\mathfrak{M}\subseteq \mathfrak{X}$ exists by homotopically discrete topological groups.

Crossed extensions and equivalences of topological 2-groupoids

We provide concrete models for generalized morphisms and Morita equivalences of topological 2-groupoids by introducing the notions of crossings and crossed extensions of groupoid crossed modules. A systematic study of these objects is elaborated and an explicit description of how they do yield a groupoid and geometric picture of weak 2-groupoid morphisms is presented. Specifically, we construct a weak 3-category whose objects are crossed modules of topological groupoids and in which weak 1-isomorphisms correspond to Morita equivalences in the "category" of topological 2-groupoids.

李群交叉模上的Rota-Baxter算子与Yang-Baxter方程的范畴解

Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation

本文定义了李群交叉模上的Rota-Baxter算子并确立其性质，利用该算子构造了Yang-Baxter方程的范畴解，并进一步研究了李代数交叉模上算子的无穷小量、积分及微分性质。

Proalgebraic crossed modules of quasirational presentations

We introduce the concept of quasirational relation modules for discrete and pro-$p$ presentations of discrete and pro-$p$ groups and show that aspherical presentations and their subpresentations are quasirational. In the pro-$p$-case quasirationality of pro-$p$-groups with a single defining relation holds. For every quasirational (pro-$p$)relation module we construct the so called $p$-adic rationalization, which is a pro-fd-module $\overline{R}\widehat{\otimes}\mathbb{Q}_p= \varprojlim R/[R,R\mathcal{M}_n]\otimes\mathbb{Q}_p$. We provide the isomorphisms $\overline{R^{\wedge}_w}(\mathbb{Q}_p)=…

On Hopf 2-algebras

Our main goal in this paper is to translate the diagram relating groups, Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to categorify it. This is done interpreting 2-objects as crossed modules and showing the compatibility of the standard functors linking groups, Lie algebras and Hopf algebras with the concept of a crossed module. One outcome is the construction of an enveloping algebra of the string Lie algebra of Baez-Crans, another is the clarification of the passage from crossed modules of Hopf algebras to Hopf 2-algebras.

Twisted Weyl groups of Lie groups and nonabelian cohomology

For a cyclic group $A$ and a connected Lie group $G$ with an $A$-module structure (with the additional conditions that $G$ is compact and the $A$-module structure on $G$ is 1-semisimple if $A\cong\ZZ$), we define the twisted Weyl group $W=W(G,A,T)$, which acts on $T$ and $H^1(A,T)$, where $T$ is a maximal compact torus of $G_0^A$, the identity component of the group of invariants $G^A$. We then prove that the natural map $W\backslash H^1(A,T)\to H^1(A,G)$ is a bijection, reducing the calculation of $H^1(A,G)$ to the calculation of the action of $W$ on $T$. We also prove some properties of the …

极小极大幂零线性群交叉积上模的交换不变量

On commutative invariants for modules over crossed products of minimax nilpotent linear groups

本文针对有限秩可解群$G$的极小极大幂零无挠正规子群$N$与有限生成交换整环$R$的交叉积$R*N$上的模$W$，构建其与特定素理想等价类集$M$的对应关系，证明$G$对$N$的共轭作用可延拓至$M$，为用交换代数方法研究$W$提供基础。

On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras

The aim here is to sketch the development of ideas related to brackets and similar concepts: Some purely group theoretical combinatorics due to Ph. Hall led to a proof of the Jacobi identity for the Whitehead product in homotopy theory. Whitehead introduced crossed modules to characterize a second relative homotopy group; guided by combinatorial group theory considerations, Reidemeister and Peiffer explored this kind of structure to develop normal forms for the decomposition of a 3-manifold; but crossed modules are also lurking behind a forgotten approach of Turing to the extension problem for…