Ronald Brown
We fill a lacuna in the literature by giving a version in dimension 1 of the Relative Hurewicz Theorem, and relate this to abelianisations of groupoids, covering spaces and covering morphisms of groupoids, and Crowell's notion of derived modules.
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Extension of topological groupoids and Serre, Hurewicz morphisms
该研究提出拓扑群胚扩张概念，将其关联于拓扑栈上的 gerbe，并考察 Serre、Hurewicz 栈上 gerbe 的性质。
Covering morphisms of crossed complexes and of cubical omega-groupoids with connection are closed under tensor product
目的是证明标题中的定理及其推论：群或群胚的两个自由交叉分解的张量积也是乘积群或群胚的自由交叉分解。通向这个推论的途径是通过交叉复形范畴与三次Ω-广群范畴的等价性，其中三次Ω-广群的联络是张量积的初始定义所在。也正是在后一个范畴中，我们能够应用稠密子范畴的技巧来将覆盖态射的张量积识别为覆盖态射。
We summarize several results about the regular coverings and the fundamental groupoids of Alexandroff spaces. In particular, we show that the fundamental groupoid of an Alexandroff space $X$ is naturally isomorphic to the localization, at its set of morphisms, of the thin category associated to the set $X$ considered as a preordered set with the specialization preorder. We also show that the regular coverings of an Alexandroff space $X$ are represented by certain morphism-inverting functors with domain $X$, extending a result of E. Minian and J. Barmak about the regular coverings of locally fi…
We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of weights at most $m-1$ and $RM$ is of weights at least $n+1$ for some $m\le n$, is determined by $M$ if exists. This happens if and only if the weight complex $t(M)\in Obj K(Hw)$ ($Hw$ is the heart of $w$) is homotopy equivalent to a complex with zero terms in degrees $-n,\dots, -m$; hence this condition can be "detected" via pure functors. One can also take $m=-\…
In this paper, for given an algebraic theory $T$ whose category $C$ of models is semi-abelian, we consider the topological models of $T$ called topological $T$-algebras and obtain some results related to the fundamental groups of topological $T$-algebras. We also deal with the internal groupoid structure in the category of models providing that the fundamental groupoid deduces a functor from topological $T$-algebras to the internal groupoids in $C$ and prove a criterion for the lifting of such an internal groupoid structure to the covering 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.
This work represents the concept of an n-groupoid $ Γ^n $ and n-characters $ χ_n $ on n-groupoids as complex-valued maps from spaces of different classes of morphisms satisfying the condition $ χ_n (ψ\circ_k φ) = χ_n (ψ) + χ_n (φ) $ for any possible compositions. A sequence of spaces of n-characters and morphisms between them is constructed and its accuracy is shown. This construction has important application for describing the derivations in a group algebras. In particular, this approach allows us to study the algebra of external derivations from a new point of view, and also to construct so…
In this paper we prove some results on the covering morphisms of internal groupoids. We also give a result on the coverings of the crossed modules of groups with operations.
Coverings and Actions of Structured Lie Groupoids I
本文研究了李群-群胚作为一类结构李群胚的覆盖和作用。首先，我们定义了李群-群胚在某些李群上的作用以及李群-群胚的光滑覆盖。然后，我们证明了李群-群胚在李群上的光滑作用范畴与李群-群胚的光滑覆盖范畴的等价性。进一步证明了如何从李群-群胚得到覆盖李群-群胚和李群-群胚的光滑覆盖射的定理。
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 also prove that the liftings of a certain crossed module constitute a category; and that this category is equivalent to the category of covers of that crossed module and hence to the category of group-groupoid actions of the corresponding groupoid to that crossed module.
In this paper we define the notion of pullback lifting of a lifting crossed module over a crossed module morphism and interpret this notion in the category of group-groupoid actions as pullback action. Moreover, we give a criterion for the lifting of homotopic crossed module morphisms to be homotopic, which will be called homotopy lifting property for crossed module morphisms. Finally, we investigate some properties of derivations of lifting crossed modules according to base crossed module derivations.
Covering groups of non-connected topological groups revisited
一般地，非连通拓扑群的泛覆盖不需要允许拓扑群结构使得覆叠映射是拓扑群的态射。这一结果是由于R.L. Taylor（1953）.我们概括了这一结果，并将其与理论的障碍群扩展。方法用途：X的覆盖映射与X的基本广群的覆盖广群之间的等价性;群广群与交叉模之间的等价性;以及交叉复形上同调的描述。
We address the (pointed) homotopy of crossed module morphisms in modified categories of interest; which generalizes the groups and various algebraic structures. We prove that, the homotopy relation gives rise to an equivalence relation; furthermore a groupoid structure, without any restriction on neither domain nor co-domain of the crossed module morphism. Additionally, we consider the particular cases such as associative algebras, Leibniz algebras, Lie algebras and dialgebras of crossed modules of this generalized homotopy definition. Then as the main part of the paper, we prove that the func…
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