We show that the fundamental groupoid~\(Π_1(X)\) of a locally path connected semilocally simply connected space~\(X\) can be equipped with a \emph{natural} topology so that it becomes a topological groupoid; we also justify the necessity and minimality of these two hypotheses on~\(X\) in order to topologise the fundamental groupoid. We find that contrary to a belief -- especially among the Operator Algebraists -- the fundamental groupoid is not {\etale}. Further, we prove that the fundamental groupoid of a topological group, in particular a Lie group, is a \emph{transformation groupoid}; again, this result disproves a standard belief that the fundamental groupoids are \emph{far} away from being transformation groupoids. We also discuss the point-set topology on the fundamental groupoid with the intention of making it a locally compact groupoid.
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.
We construct a 2-category version of tom Dieck's equivariant fundamental groupoid for representable orbifolds and show that the discrete fundamental groupoid is Morita invariant; hence an orbifold invariant for representable orbifolds.
Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid $Π_1(X)$ has well-defined dense products if and …
Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. We prove that the well-definedness of products indexed by a scattered linear order in the fundamental groupoid of a first countable space is equivalent to the homotopically Hausdorff property. To prove this characterization, we employ the machinery of closure operators, on the $π_1$-subgroup lattice, defined in terms of test maps from one-dimensional domains.
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 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.
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.
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.
For a finitely presented discrete group $Γ$, we introduce two generalizations of the orbifold Euler characteristic and $Γ$-orbifold Euler characteristic to a class of proper topological groupoids large enough to include all cocompact proper Lie groupoids. The $Γ$-Euler characteristic is defined as an integral with respect to the Euler characteristic over the orbit space of the groupoid, and the $Γ$-inertia Euler characteristic is the usual Euler characteristic of the $Γ$-inertia space associated to the groupoid. A key ingredient is the application of o-minimal structures to study orbit spaces …
The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action
The paper contains an application of van Kampen theorem for groupoids for computation of homotopy types of certain class of non-compact foliated surfaces obtained by gluing at most countably many strips $\mathbb{R}\times(0,1)$ with boundary intervals in $\mathbb{R}\times\{\pm1\}$ along some of those intervals.
The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the application of a term/function to any other term/function, including itself. The calculus can be seen as a formal theory with certain pre-established axioms and inference rules, which can be interpreted by models. Dana Scott proposed the first non-trivial model of the extensional lambda calculus, known as $ D_\infty$, to represent the $λ$-terms as the typical functio…
