Related papers: Diffeological Coarse Moduli Spaces of Stacks over …

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Fr\"olicher spaces, another generalisation of smooth structures. We then…

Let $\mathcal{G}$ be a Lie groupoid. The category $B\mathcal{G}$ of principal $\mathcal{G}$-bundles defines a differentiable stack. On the other hand, given a differentiable stack $\mathcal{D}$, there exists a Lie groupoid $\mathcal{H}$…

In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375],…

We construct a moduli space of stable pairs over a smooth projective variety, parametrizing morphisms from a fixed coherent sheaf to a varying sheaf of fixed topological type, subject to a stability condition. This generalizes the notion…

In this article, we derive many properties of \'etale stacks in various contexts, and prove that \'etale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local…

This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for stacks. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids.…

The purpose of this note is to define sheaves for diffeological spaces and give a construction of their \v{C}ech cohomology. As an application, we prove that the first degree \v{C}ech cohomology classes for the sheaf of smooth functions to…

This paper aims to describe the behavior of diffeological differential forms under the operation of gluing of diffeological spaces along a smooth map. In the diffeological context, two ways of looking at diffeological forms are available,…

A diffeological space is a set equipped with a smooth structure, known as a diffeology, which allows us to extend certain notions from manifolds to these more general spaces. We study a generalized notion of tangent space to a point of a…

In this paper, we study the geometric invariant theory on algebraic spaces, and construct te moduli spaces of $\mathcal{H}$-semistable sheaves on projective Deligne-Mumford stacks over algebraic spaces $S$. We prove that this moduli space…

Diffeological spaces are generalizations of smooth manifolds. In this paper, we study the homotopy theory of diffeological spaces. We begin by proving basic properties of the smooth homotopy groups that we will need later. Then we introduce…

In this paper, we introduce a Grothendieck topology on the category of totally bounded metric spaces and develop a theory of stacks with respect to this topology. We further define the fine moduli stack of compact metric spaces and prove…

In the first part of the paper, we prove that the category of diffeological spaces does not admit a model structure transferred via the smooth singular complex functor from simplicial sets, resolving in the negative a conjecture of…

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate…

We show that the inertia stack of a topological stack is again a topological stack. We further observe that the inertia stack of an orbispace is again an orbispace. We show how a U(1)-banded gerbe over an orbispace gives rise to a flat line…

We construct moduli stacks of stable sheaves for surfaces fibered over marked nodal curves by using expanded degenerations. These moduli stacks carry a virtual class and therefore give rise to enumerative invariants. In the case of a…

We study fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. Then use this additional structure to analyze geometric/ topological properties of stacks.…

We construct a weak categorification of the quantum toroidal algebra action on the Grothendieck group of moduli space of stable (or framed) sheaves over an algebraic surface, which is constructed by Schiffmann-Vasserot and Negu\c{t}. The…

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…

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) $F$-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected…