arXiv subjects
Publications and source records attributed to Osman Mucuk.
In [8] we proved that any categorical group defines a c-crossed module, which is a cssc-crossed module defined in the same paper. In [9] we constructed a categorical group for any cssc-crossed module. In the presented paper we prove that these correspondences define functors between the corresponding categories, which realize an equivalence of the categories of categorical groups and cssc-crossed modules.
For any cssc-crossed module a category is constructed, equipped with a structure and proved that this is a coherent categorical group. Together with a result of the previous paper, where to any categorical group the cssc-crossed module is associated, this construction will enable us to prove an equivalence between the categories of categorical groups and of cssc-crossed modules in the sequel to this paper.
It is well known that for a Hausdorff topological group $X$, the limits of convergent sequences in $X$ define a function denoted by $\lim$ from the set of all convergent sequences in $X$ to $X$. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing $\lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Recently some authors have extended the concept to the topological group setting and introduced the concepts of $G$-continuity, $G$-compactness and $G$-connectedness. In this paper we prove some results on different types of $G$-compactness for topological group with operations which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.
We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to c-groups and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.
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.
In this paper we develop star topological and topological group-groupoid structures of monodromy groupoid and prove that the monodromy groupoid of a topological group-groupoid is also a topological group-groupoid.
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.
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 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 coverings of cat$^{1}$-groups with operations.
If $X$ is a topological group, then its fundamental groupoid $π_1X$ is a group-groupoid which is a group object in the category of groupoids. Further if $X$ is a path connected topological group which has a simply connected cover, then the category of covering spaces of $X$ and the category of covering groupoids of $π_1X$ are equivalent. In this paper we prove that if $(X,x_0)$ is an $H$-group, then the fundamental groupoid $π_1X$ is a categorical group. This enable us to prove that the category of the covering spaces of an $H$-group $(X,x_0)$ is equivalent to the category of covering groupoid of the categorical group $π_1X$.
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.
It is a well known result in the covering groups that a subgroup $G$ of the fundamental group at the identity of a semi-locally simply connected topological group determines a covering morphism of topological groups with characteristic group $G$. In this paper we generalize this result to a large class of algebraic objects called topological groups with operations, including topological groups. We also give the cover of crossed modules within topological groups with operations.
This paper gives an introduction to some results on monodromy groupoids and the monodromy principle, and then develops the notion of monodromy groupoid for group groupoids.
Recently, Cakalli has introduced a concept of $G$-sequential connectedness in the sense that a non-empty subset $A$ of a Hausdorff topological group $X$ is $G$-sequentially connected if there are no non-empty, disjoint $G$-sequentially closed subsets $U$ and $V$ meeting $A$ such that $A\subseteq U\bigcup V$. In this paper we investigate further properties of $G$-sequential connectedness and prove some interesting theorems.
Let $X$ be a first countable Hausdorff topological group. The limit of a sequence in $X$ defines a function denoted by $lim$ from the set of all convergence sequences to $X$. This definition was modified by Connor and Grosse-Erdmann for real functions by replacing $lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Çakallı extended the concept to topological group setting and introduced the concept of $G$-sequential compactness and investigated $G$-sequential continuity and $G$-sequential compactness in topological groups. In this paper we give a further investigation of $G$-sequential continuity in topological groups most of which are also new for the real case.
We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.
The notion of local subgroupoids as generalition of a local equivalence relations was defined by the first author and R.Brown. Here we investigate some relations between transitive components and coherence properties of the local subgroupoids.