Simona Paoli
An algebraic category $\mathcal{C}$ is called balanced if the cotriple cohomology of any object of $\mathcal{C}$ vanishes in positive dimensions on injective coefficient modules. Important examples of balanced and of non-balanced categories occur in the literature. In this paper we prove that the category of crossed modules in groups is non-balanced.
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In this study, internal categories in the category of the crossed modules are characterized and it has been shown that there is a natural equivalence between the category of the crossed modules over crossed modules, i.e. crossed squares, and the category of the internal categories within the category of crossed modules. Finally, we obtain examples of crossed squares using this equivalence.
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 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.
In this study, we interpret the notion of homotopy of morphisms in the category of crossed modules in a category $\mathsf{C}$ of groups with operations using the categorical equivalence between crossed modules and internal categories in $\mathsf{C}$. Further, we characterize the derivations of crossed modules in a category $\mathsf{C}$ of groups with operations and obtain new crossed modules using regular derivations of old one.
In this paper, we define the pullback crossed modules in the category of racks which mainly based on a pullback diagram of rack morphisms with extra crossed module data on some of its arrows. Furthermore we prove that the conjugation functor, which is defined between the category of crossed modules of groups and of racks, preserves the pullback crossed modules.
A Model Categoric Equivalence for Crossed Simplicial Modules
本文在有限交叉单纯群且基域特征为0的条件下，构造了单纯向量空间范畴与交叉单纯群表示范畴之间的模型范畴等价。
We introduce the notion of 3-crossed module, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduché). We show that the category of 3-crossed modules is equivalent to the category of simplicial groups having a Moore complex of length 3. We make explicit the relationship with the cat$^{3}$-groups (Loday) and the 3-hyper-complexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types.
The category $\mathbf{XSq}$ of crossed squares is equivalent to the category $\mathbf{Cat2}$ of cat$^2$-groups. Functions for computing with these structures have been developed in the package $\mathsf{XMod}$ written using the $\mathsf{GAP}$ computational discrete algebra programming language. This paper includes details of the algorithms used. It contains tables listing the $1,000$ isomorphism classes of cat$^2$-groups on groups of order at most $30$.
In this paper we define the notions of normal subcrossed module and quotient crossed module within groups with operations; and using the equivalence of crossed modules over groups with operations and internal groupoids we prove how normality and quotient concepts are related in these two categories. Further we prove an equivalence of crossed modules over groups with operations and cat$^1$-groups with operations for a certain algebraic category; and then by this equivalence we determine normal and quotient objects in the category of cat$^{1}$-groups with operations. Finally we characterize the …
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 group-groupoids which are internal to the category of group-groupoids. This equivalence enables us to produce more examples of double groupoids.
The correspondence between the concept of conditional flatness and admissibility in the sense of Galois appears in the context of localization functors in any semi-abelian category admitting a fiberwise localization. It is then natural to wonder what happens in the category of crossed modules where fiberwise localization is not always available. In this article, we establish an equivalence between conditional flatness and admissibility in the sense of Galois (for the class of regular epimorphisms) for regular-epi localization functors. We use this equivalence to prove that nullification functo…
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.
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.
The category of crossed complexes gives an algebraic model of CW-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNN-extensions of groups, and so obtain computations of higher homotopical syzygies in these cases.
Crossed modules are known to be a model of pointed connected homotopy 2-types; formally, the homotopy category of crossed modules is equivalent to the category of pointed connected homotopy 2-types. In forming the homotopy category of crossed modules, one must resort to computing derived morphisms using non-constructive topological methods, but Behrang Noohi was able to find an algebraic model of these derived morphisms called butterflies. The result is a completely algebraic model of pointed connected homotopy 2-types. Reduced crossed complexes are a generalization of crossed modules and mode…
The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical omega-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.
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 groups, given free crossed resolutions of the individual groups.
We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occuring in topology such as Segal's category Gamma, or the total category of a crossed simplicial group such as Connes' cyclic category Lambda. For any generalized Reedy category R and any cofibrantly generated model category E, the functor category E^R is shown to carry a canonical model structure of Reedy type.
Any simplicial Hopf algebra involves $2n$ different projections between the Hopf algebras $H_n,H_{n-1}$ for each $n \geq 1$. The word projection, here meaning a tuple $\partial \colon H_{n} \to H_{n-1}$ and $i \colon H_{n-1} \to H_{n}$ of Hopf algebra morphisms, such that $\partial \, i = \mathrm{id}$. Given a Hopf algebra projection $(\partial \colon I \to H,i)$ in a braided monoidal category $\mathfrak{C}$, one can obtain a new Hopf algebra structure living in the category of Yetter-Drinfeld modules over $H$, due to Radford's theorem. The underlying set of this Hopf algebra is obtained by an…
In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their…
The Morita $(\infty,2)$-category of a monoidal category as a $2$-complicial set
本文为满足极简条件的幺半范畴构造了Morita $(\infty,2)$-范畴，将其实现为$3$-余骨架的$2$-复合单纯集，并明确编码了幺半群、双模及其映射和复合结构。
The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We give an answer by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes.
3-Crossed modules, Quasi-categories, and the Moore complex
本文提出了一种新的3-交错模定义，用于构建更高阶范畴对应，证明其诱导的simplicial集为准范畴，并展示摩尔复形自然具备该结构。
Categorifying equivariant monoids
本文结合 PROPs、PROBs 与交叉单纯群理论，构造出编码等变幺半群结构的范畴，其代数范畴等价于带群作用的各类幺半群，并扩展至平衡辫张范畴和对合幺半群。
The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel'd modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfe' d modules over a braided system. A systematic construction of braiding structures on such modules is provided. Some general categorical methods of obtaining such generalized Yetter-Drinfel'd (=GYD) modules are described. Among the b…
We propose a generalization of the concept of a Ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the necessary combinatorics is neatly captured in the concept of a crossed simplicial group as introduced, independently, by Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category leads to Ribbon graphs while other crossed simplicial groups naturally yield different notions of structured graphs which model unoriented, N-spin, framed, etc, surfaces. Our main result is that structured graphs provide orb…
Realization of Lie algebras and classifying spaces of crossed modules
本文研究集中于0次度和1次度的完备微分分次李代数的实现，建立其与交叉模范畴的等价，并证明该李代数的实现同构于其关联交叉模的分类空间。
The theorem of the title is deduced from the equivalence between crossed complexes and cubical $ω$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an additive category with kernels.
On the Gottlieb group, Drinfeld centre and the centre of a crossed module
本文引入交叉模的中心概念，利用交叉同态建立其与群上同调、Gottlieb 群、Drinfeld 中心及 Whitehead 中心的联系。
From Lie algebra crossed modules to tensor hierarchies
本文提出Lie-Leibniz三元组（含嵌入张量的莱布尼茨代数与李代数配对结构）可诱导张量层级（微分分次李代数），证明其是李代数交叉模的自然推广，构造更清晰，还暗示存在更多莱布尼茨规范理论。
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