In this paper, we extend the classical theory of crossed $G$-sets and the crossed Burnside ring from a finite group $G$ to a finite groupoid $\mathcal{G}$. We introduce a natural monoidal structure on the category of crossed $\mathcal{G}$-sets over a $\mathcal{G}$-monoid and construct the corresponding crossed Burnside ring of a $\mathcal{G}$-monoid. Finally, we prove a decomposition theorem that expresses the crossed Burnside ring of a groupoid as a product of crossed Burnside rings of groups.
Several models for the Burnside bicategory of groupoids are described and shown to be equivalent. As observed by the late Gaunce Lewis, the corresponding Burnside category is additive.
We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between two categorified group-sets is given by a particular equivalence between the underlying categories. We also exhibit some of the differences between the old Burnside theory and the new one. Lastly, we briefly explain how to extend these new techniques and concepts to the context of groupoids, employing the categories of (right) groupoid-sets, aiming by this to…
We explore the concept of conjugation between subgroupoids, providing several characterizations of the conjugacy relation (Theorem A in §1.2). We show that two finite groupoid-sets, over a locally strongly finite groupoid, are isomorphic, if and only if, they have the same number of fixed points with respect to any subgroupoid with a single object (Theorem B in §1.2). Lastly, we examine the ghost map of a finite groupoid and the idempotents elements of its Burnside algebra. The exposition includes an Appendix where we gather the main general technical notions that are needed along the paper.
In this work we study the notion of Whitehead sequence in the category of crossed modules and actions of crossed modules. As expected, Whitehead sequences in that context are the same as crossed squares. We investigate under which conditions a Whitehead sequence of crossed modules gives rise to an internal groupoid in the category of crossed modules. In other words, we explicitly investigate the so called "Smith is Huq" condition in the category of crossed modules.
Balmer and Dell'Ambrogio introduced the pseudo-functor $P$ from the bicategory of $k$-linear Mackey $2$-motives to the bicategory of $k$-linear cohomological Mackey $2$-motives over a commutative ring $k$. They showed that $P$ maps the general Mackey $2$-motives to the cohomological Mackey $2$-motives by using the ring homomorphism from the crossed Burnside ring of a finite group $G$ over $k$ to the center $ ZkG$ of group algebra $kG$ ([BD21, Theorem 5.3]). We study the behavior of motivic decomposition of cohomological Mackey $2$-motives as images by $P$ of motivic decomposition of Mackey $2$…
This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author's knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2-groupoids introduced by Moerdijk and Svensson.
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.
We consider the categorical equivalence between crossed modules over groupoids and double groupoids with thin structures; and by this equivalence, we prove how normality and quotient concepts are related in these two categories and give some examples of these objects.
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.
We give a detailed description of the structure of the actor 2-crossed module related to the automorphisms of a crossed module of groupoids. This generalises work of Brown and Gilbert for the case of crossed modules of groups, and part of this is needed for work on 2-dimensional holonomy to be developed elsewhere (see math.DG/0009082).
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 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.
We outline the main features of the definitions and applications of crossed complexes and cubical $ω$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types.
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 …
We use the n-globe with its skeletal filtration to define the fundamental globular omega--groupoid of a filtered space; the proofs use an analogous fundamental cubical omega--groupoid due to the author and Philip Higgins. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free globular omega-groupoid on one element of dimension n is the fundamental crossed complex of the n-globe.
We define the thin fundamental Gray 3-groupoid $S_3(M)$ of a smooth manifold $M$ and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps $S_3(M) \to C(H)$, where $H$ is a 2-crossed module of Lie groups and $C(H)$ is the Gray 3-groupoid naturally constructed from $H$. As an application, we define Wilson 3-sphere observables.
For an arbitrary group $G$, a (semi-)Mackey functor is a pair of covariant and contravariant functors from the category of $G$-sets, and is regarded as a $G$-bivariant analog of a commutative (semi-)group. In this view, a $G$-bivariant analog of a (semi-)ring should be a (semi-)Tambara functor. A Tambara functor is firstly defined by Tambara, which he called a TNR-functor, when $G$ is finite. As shown by Brun, a Tambara functor plays a natural role in the Witt-Burnside construction. It will be a natural question if there exist sufficiently many examples of Tambara functors, compared to the wid…
Suppose that $G$ is a groupoid acting on a small category $H$ in the sense of \cite[Definition 4]{NOT} and $H\times_αG$ is the resulting semi-direct product category (as in \cite[Proposition 8]{NOT}). We show that there exists a subcategory $H_r \subseteq H$ satisfying some nice property called ``regularity'' such that $H_r \times_αG = H\times_αG$. Moreover, we show that there exists a so-called ``quasi action'' (see Definition \ref{quasi}) $β$ of $G$ on $C^*(H_r)$ (where $C^*(H_r)$ is the semigroupoid $C^*$-algebra as defined in \cite{EXE}) such that $C^*(H_r\times_αG) = C^*(H_r)\times_βG$ (w…
We introduce the notion of Whitehead sequence which is defined for a base category together with a system of abstract actions over it. In the classical case of groups and group actions the Whitehead sequences are precisely the crossed-modules of groups. For a general setting we give sufficient conditions for the existence of a categorical equivalence between internal groupoids and Whitehead sequences.
Previous work (Pradines, 1966, Aof and Brown, 1992) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involv…
By regarding the classical non abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed modules are 2-groupoids and, cocycles are lax 2-functors and the cocycle conditions are precisely the coherence axioms for lax 2-functors, and, on the other hand group extensions are fibrations of categor…
A crossed module is (A,H,d,\la) where d:A\to H is a homomorphism of groups and H acts on A, with conditions leading to a groupoid A\lcross H{\to\atop \to}H as an example of a strict 2-group. We give the corresponding notion of a quantum 2-group where we replace the above by Hopf algebras and introduce a new version of quantum groupoid. The work also suggests a natural notion of braided crossed module where A a braided-Hopf algebra in the braided category Z({}_H\CM) of crossed H-modules, although without the full groupoid picture in this more general case.
In this study, using the Brown-Spencer theorem and in the ligth of the works of Norrie, in the category of internal categories within groups, also called group-groupoids, we interpret the notion of actor of a crossed module over groups. Further, we construct the action of a group-groupoid on a group-groupoid. Moreover, we give the explicit construction of semi-direct product of two group-groupoids and of holomorph of group-groupoids.
In general a universal covering of a non connected topological group need not admit a topological group structure such that the covering map is a morphism of topological groups. This result is due to R.L. Taylor (1953). We generalise this result and relate it to the theory of obstructions to group extensions. The methods use: the equivalence between covering maps of X and covering groupoids of the fundamental groupoid of X; the equivalence between group groupoids and crossed modules; and descriptions of cohomology in terms of crossed complexes.
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.
