Fundamental Groupoid Research Articles

Groupoidal realizability for intensional type theory

Abstract We develop realizability models of intensional type theory, based on groupoids, wherein realizers themselves carry non-trivial (non-discrete) homotopical structure. In the spirit of realizability, this is intended to formalize a homotopical BHK interpretation, whereby evidence for an identification is a path. Specifically, we study partitioned groupoidal assemblies. Categories of such are parameterized by “realizer categories” (instead of the usual partial combinatory algebras) that come equipped with an interval qua internal cogroupoid. The interval furnishes a notion of homotopy as well as a fundamental groupoid construction. Objects in a base groupoid are realized by points in the fundamental groupoid of some object from the realizer category; isomorphisms in the base groupoid are realized by paths in said fundamental groupoid. The main result is that, under mild conditions on the realizer category, the ensuing category of partitioned groupoidal assemblies models intensional (1-truncated) type theory without function extensionality. Moreover, when the underlying realizer category is “untyped,” there exists an impredicative universe of 1-types (the modest fibrations). This is a groupoidal analog of the traditional situation.

Topological fundamental groupoids: Brown's topology

In this paper, we generalize the Brown$^{^,}$s topology on the fundamental groupoids. For a locally path connected space $X$ and a totally disconnected normal subgroupoid $M$ of $\pi X$, we define a topology on the quotient groupoid $\dfrac{\pi X}{M}$ which is a generalization of what introduced by Brown for locally path connected and semilocally simply connected spaces. We prove that $\dfrac{\pi X}{M}$ equipped with this topology is a topological groupoid. Also, we will find a class of subgroupoids of topological groupoids whose their related quotient groupoids will be topological groupoids. By using this, we show that our topology on $\dfrac{\pi X}{M}$ is equivalent to the quotient of the Lasso topology on the topological fundamental groupoids, $\dfrac{\pi^L X}{M}$.

Local constancy of pro‐unipotent Kummer maps

AbstractIt is a theorem of Kim–Tamagawa that the ‐pro‐unipotent Kummer map associated to a smooth projective curve over a finite extension of is locally constant when . This paper establishes two generalisations of this result. First, we extend the Kim–Tamagawa theorem to the case that is a smooth variety of any dimension. Second, we formulate and prove the analogue of the Kim–Tamagawa theorem in the case , again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale–de Rham comparison theorem for pro‐unipotent fundamental groupoids using methods of Scholze and Diao–Lan–Liu–Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids.

A kind of characterization of homeomorphism and homeomorphic spaces by Core fundamental groupoid: a good invariant

In this paper, we give a new topological invariant and a kind of characterization to homeomorphisms. We introduce this topological invariant by constructing an algebraic kind of groupoid structure to each topological space, which is an extension of the concept of fundamental groups. This construction we call it “Core fundamental groupoid” and it is different from the fundamental groupoid as in the book and articles of Ronald Brown [30, 31]. Moreover, both groupoid notions are not equivalent, but, it will be a wide subgroupoid of the fundamental groupoid in category-theoretic. In the entire paper, we consider groupoid in the algebraic sense. First, we discuss basic properties of the Core fundamental groupoid and their significant importance in topological invariants. Further, we separately present the Core fundamental groupoid as an algebraic structure and a topo-algebraic structure and investigate their properties separately. We have an explicit description of both the algebraic structure of the groupoid and a unique topological structure of Core fundamental groupoid. Besides, we give a kind of characterization of homeomorphisms in terms of an invariant that we have obtained, i.e., “f : M → N is a homeomorphism if and only if f# 1M → 1N is a topological groupoid isomorphism”. This result addresses the open problem stated in [1, 19, 24, 26] and also gives a kind of characterization for homeomorphic spaces, but, computationally it seems difficult and it becomes sometimes trivial when intact with topology. Induced groupoid homomorphism, induced base map, topological and smooth structure on the Core fundamental groupoid, homotopic properties of induced groupoid homomorphisms of continuous maps and topological groupoid homomorphism-related properties of induced groupoid homomorphisms are discussed. We also present the relation of homotopy type on Core fundamental groupoid.

On a random entanglement problem

Abstract We study a model for the entanglement of a two-dimensional reflecting Brownian motion in a bounded region divided into two halves by a wall with three or more small windows. We map the Brownian motion into a Markov Chain on the fundamental groupoid of the region. We quantify entanglement of the path with the length of the appropriate element in this groupoid. Our main results are a law of large numbers and a central limit theorem for this quantity. The constants appearing in the limit theorems are expressed in terms of a coupled system of quadratic equations.

Isotropy quotients of Hopf algebroids and the fundamental groupoid of digraphs

We build on our construction of Hopf algebroids from noncommutative calculi under the further assumption of surjectivity for the calculus. We also introduce the notions of Hopf ideals and isotropy quotients for arbitrary Hopf algebroids. Using these ingredients, we prove a Riemann-Hilbert correspondence for digraphs, by showing that the groupoid algebra of the fundamental groupoid of a digraph is isomorphic to the isotropy quotient of the Hopf algebroid corresponding to flat connections over the digraph.

Quantum difference equation for Nakajima varieties

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in $${{\,\mathrm{Pic}\,}}(X)\otimes {\mathbb {C}}$$ . We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.

Core Fundamental Groupoid Bundle, Its Sections and Relatedness of Homeomorphisms and Sections

In this paper, we introduce bundle, fibre bundle and principal G-bundle structures on the Core fundamental groupoid keeping its standard projections and quotient topology intact. We give an explicit description of Core fundamental groupoids as such bundles including for the uniquely geodesic spaces and formulate some results on bundle maps. Further, we introduce sections on the Core fundamental groupoid bundle, and also, present some basic properties including composition and inverse with the help of the induced groupoid homomorphisms on the Core fundamental groupoids. With a group structure on the set of all continuous sections of the Core fundamental groupoid bundle, a group action has been built on the Core fundamental groupoid. A notion of relatedness of homeomorphism and section are defined and discussed the pushforward, pullback of sections and their properties. Finally, we investigate more about relatedness notions relΓ0(¯π1M)(f), relHomeo(M)(X) and same on the subsets based on both section related homeomorphisms and homeomorphism related sections. Further, some consequences based on an algebraic structure on the new class of relΓ0(¯π1M)(f), relHomeo(M)(X), etc. have been placed. We present an interrelationship between subsets of Homeo(M) and Γ 0 (¯π1M), which have nice applications in the leftinvariant sections and topological group

The C2-equivariant cohomology of complex projective spaces

We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of C2, using ordinary cohomology graded on representations of the fundamental groupoid, with coefficients in the Burnside ring Mackey functor. This extension of the RO(C2)-graded theory allows for the definition of Euler classes, which are used as generators of the cohomology of the projective spaces. As an application, we give an equivariant version of Bézout's theorem.

Fundamental Groupoids for Graphs

In this paper, we develop a $\times$-homotopy fundamental groupoid for graphs, and show a functorial relationship to the 2-category of graphs. We further explore the fundamental groupoid of graph products and develop a groupoid product which respects the graph product. A van Kampen Theorem for these groupoids is provided. Finally, we generalize previous work on a fundamental group for graphs, developing a looped walk groupoid and showing a connection to the polyhedral complex of graph morphisms.

The fundamental groupoid as a topological groupoid: Lasso topology

This paper is devoted to study of a topology on the fundamental groupoid which makes it a topological groupoid and give up a functor from category of topological spaces to the category of topological groupoids. The resulting topological fundamental groupoid, denoted by πlX for a given space X is a generalization of the universal path space and also is a generalization of the topological fundamental group equipped by Lasso topology such that πlX(x)=π1l(X,x), for x∈X.

Integration of Singular Foliations via Paths

Abstract We give a new construction of the holonomy and fundamental groupoids of a singular foliation. In contrast with the existing construction of Androulidakis and Skandalis, our method proceeds by taking a quotient of an infinite-dimensional space of paths. This strategy is a direct extension of the classical construction for regular foliations and mirrors the integration of Lie algebroids via paths (per Crainic and Fernandes). In this way, we obtain a characterization of the holonomy and fundamental groupoids of a singular foliation that more clearly reflects the homotopic character of these invariants. As an application of our work, we prove that the constructions of the fundamental and holonomy groupoid of a foliation have functorial properties.

Uniconnected solutions to the Yang–Baxter equation arising from self-maps of groups

AbstractSet-theoretic solutions to the Yang–Baxter equation can be classified by their universal coverings and their fundamental groupoids. Extending previous results, universal coverings of irreducible involutive solutions are classified in the degenerate case. These solutions are described in terms of a group with a distinguished self-map. The classification in the nondegenerate case is simplified and compared with the description in the degenerate case.

Groupoids and the algebra of rewriting in group presentations

Presentations of groups by rewriting systems (that is, by monoid presentations), have been fruitfully studied by encoding the rewriting system in a 2-complex—the Squier complex—whose fundamental groupoid then describes the derivation of consequences of the rewrite rules. We describe a reduced form of the Squier complex, investigate the structure of its fundamental groupoid, and show that key properties of the presentation are still encoded in the reduced form.

Pro-unipotent harmonic actions and dynamical properties of p-adic cyclotomic multiple zeta values

$p$-adic cyclotomic multiple zeta values depend on the choice of a number of iterations of the crystalline Frobenius of the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$. In this paper we study how the iterated Frobenius depends on the number of iterations, in relation with the computation of $p$-adic cyclotomic multiple zeta values in terms of cyclotomic multiple harmonic sums. This provides new results on that computation and the definition of a new pro-unipotent harmonic action.

Transfinite product reduction in fundamental groupoids

Infinite products, indexed by countably infinite linear orders, arise naturally in the context of fundamental groupoids. Such products are called “transfinite” if the index orders are permitted to contain a dense suborder and are called “scattered” otherwise. In this paper, we prove several technical lemmas related to the reduction (i.e., combining of factors) of transfinite products in fundamental groupoids. Applying these results, we show that if the transfinite fundamental group operations are well-defined in a space X with a scattered algebraic 1-wild set $$\mathbf {aw}(X)$$ , then all transfinite fundamental groupoid operations are also well-defined.

Covering morphisms of topological internal groupoids

Let $X$ be a topological group with operations whose underlying space has a universal cover. Then the fundamental groupoid $\pi X$ becomes a topological internal groupoid, i.e., an internal groupoid in the category of topological groups. In this paper, we prove that the slice category $\text{Cov}_{sTC}/X$ of covering morphisms $p:\tilde{X}\rightarrow X$ of topological groups with operations in which $\tilde{X}$ has also a universal cover and the category $\text{Cov}_{Gpd(TC)}/\pi X$ of covering morphisms $q:\tilde{G}\rightarrow \pi X $ of topological internal groupoids based on $\pi X$ are equivalent. We also prove that for a topological internal groupoid $G$, the category $\text{Cov}_{Gpd(TC)}/G$ of covering morphisms of topological internal groupoids based on $G$ and the category $\text{ACT}_{Gpd(TC)}/G$ of topological internal groupoid actions of $G$ on topological groups with operations are equivalent.

Combinatorial constructions of derived equivalences

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more complicated wall crossings than the balanced wall crossings studied in recent work on derived categories and variation of GIT quotients. Our construction is algorithmic and quite explicit, allowing us to: 1) describe a tilting vector bundle which generates the derived category of such a GIT quotient, 2) provide a combinatorial basis for the K-theory of the GIT quotient in terms of the representation theory of G, and 3) show that our derived equivalences satisfy certain relations, leading to a representation of the fundamental groupoid of a "K\"ahler moduli space" on the derived category of such a GIT quotient. Finally, we use graded categories of singularities to construct derived equivalences between all Deligne-Mumford hyperk\"ahler quotients of a symplectic linear representation of a reductive group (at the zero fiber of the algebraic moment map and subject to a certain genericity hypothesis on the representation), and we likewise construct actions of the fundamental groupoid of the corresponding K\"ahler moduli space.

Mapping class groups of covers with boundary and braid group embeddings

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of non-geometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman-Hilden theorem and the action of the mapping class group on a particular fundamental groupoid of the surface.

The algebra of rewriting for presentations of inverse monoids

We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an example of which now arises as the fundamental groupoid of our version of the Squier complex. A further key ingredient is the factorization of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. The relation module of a presentation is then defined as the abelianized kernel of this idempotent separating map. We then use the properties of idempotent separating maps to derive a free presentation of the relation module. The construction of its kernel – the module of identities – uses further facts about pseudoregular groupoids.