Related papers: Sheaf theoretic characterization of etale groupoid…
Etale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under…
We introduce the notion of wide representation of an inverse semigroup and prove that with a suitably defined topology there is a space of germs of such a representation which has the structure of an etale groupoid. This gives an elegant…
We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an \'etale topological or differentiable stack. We then provide a construction analogous to the…
The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these…
Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale Q satisfying mild algebraic properties the sheaves on Q can be defined…
Chevalley's theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian…
We introduce the notion of a "graded topological space": a topological space endowed with a sheaf of abelian groups which we think of as a sheaf of gradings. Any object living on a graded topological space will be graded by this sheaf of…
We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of etale groupoid is subsumed in a natural way by that of quantale. In particular, to each etale groupoid, either localic or…
To an inverse semigroup, we associate an \'etale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that…
In this paper we prove that the etale sheafification of the functor arising from the quotient of an algebraic supergroup by a closed subsupergroup is representable by a smooth superscheme.
In this paper, we introduce a new class of $\ell$-adic sheaves, which we call quadratic $\ell$-adic sheaves, on connected unipotent commutative algebraic groups over finite fields. They are sheaf-theoretic enhancements of quadratic forms on…
Paterson showed how to construct an etale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz's construction can itself be further simplified by using…
A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the…
We define the notion of a sheaf over a complex of groups. As an application, we give a criterion for the developability of a complex of groups. When the developability is witnessed by a morphism to $\mathrm{GL}(V)$ for some $V$, our…
A cosheaf is the dual notion of a sheaf, but we cannot define its homology as the formal dual of sheaf cohomology, in general, because of the lack of the cosheafification. A cellular cosheaf is a contravariant functor from the face poset of…
In this article, the theory of sheaves is studied from a categorical point of view. This perspective vastly generalizes the usual theory of sheaves of sets to a more abstract setting which allows us to investigate the theory of sheaves with…
This is an introduction to the notion of local subgroupoid introduced by the author and R. Brown. It can also serve as an introduction to an application of sheaf theory, and so could be useful to beginners in that theory. The main results…
A compactification of Fell is applied to locally compact non-Hausdorff groupoids and yields locally compact Hausdorff groupoids. In the etale case, this construction provides a geometric picture for the left-regular representations…
In this paper we introduce a description of ordered groupoids as a particular type of double categories. This enables us to turn Lawson's correspondence between ordered groupoids and left-cancellative categories into a biequivalence. We use…
Cosheaves are a dual notion of sheaves. In this paper, we prove existence of a dual of sheafifications, called \textit{cosheafifications}, in the $\infty$-category theory. We also prove that the $\infty$-category of $\infty$-cosheaves is…