arXiv research

The paper solves a problem in constructing a bicategory of algebra bundles.

Equivalent bicategories constructed from action Lie groupoids.

A bicategory approach to differential cohomology is presented. Based on the axioms of Bunke-Schick, a symmetric monoidal groupoid is associated to differential refinements of cohomology theories. It is proven that such differential refinements are unique up to equivalence of the corresponding symmetric monoidal groupoi…

A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity conditions. We prove that the oriented 3-dimensional bordism bicategory of 1-, 2-…

Defines extended TQFTs using handle attachments.

We describe a bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a …

We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans. This functor simultaneously extends the Burau representation of the braid groups, its generalization to tangles due to Turaev and the first-named author, and the Alexander module of 1 and 2-dimensional links.

Theory of 2-vector bundles for smooth manifolds developed.

Quasifold groupoids and diffeological quasifolds are studied, showing an equivalence of categories.

Solves open problem on Lie groupoids equivalence.

Extends 4D cornered skein theory to surfaces, proving gluing formulas.

New construction of Turaev-Viro invariants invariant under Morita equivalence.

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory $\mathbf{C}$ and endobifunctor $Σ\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $Σ_q$ such th…

We study (pre-)sheaves in bicategories on geometric categories: smooth manifolds, manifolds with a Lie group action and Lie groupoids. We present three main results: we describe equivariant descent, we generalize the plus construction to our setting and show that the plus construction yields a 2-stackification for 2-pr…

This thesis is divided into four chapters. The first chapter discusses the relationship between stacks on a site and groupoids internal to the site. It includes a rigorous proof of the folklore result that there is an equivalence between the bicategory of internal groupoids and the bicategory of geometric stacks. The s…

We concretely construct a 2-categorically extended TQFT that extends the Reshetikhin-Turaev TQFT to cobordisms with corners. The source category will be a well chosen 2-category of decorated cobordisms with corners and the target bicategory will be the Kapranov-Voevodsky 2-vector spaces.

The paper discusses strictification and non-strictification of monoidal categories.

This paper categorifies Quinn's TQFTs and computes them for specific omega-groupoids.

New Morita equivalence for diffeological groupoids defined.

Constructs 2-vector bundles and 2K-theory for Lie groupoids and 2-equivariant settings.

We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit construction of a geometric cobordism bicategory which incorporates general backgro…

We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial constructi…

Constructs a representation of the string 2-group on a von Neumann algebra.

We give another definition of two-dimensional extended homotopy field theories (E-HFTs) with aspherical targets and classify them. When the target of E-HFT is chosen to be a $K(G,1)$-space, we classify E-HFTs taking values in the symmetric monoidal bicategory of algebras, bimodules, and bimodule maps by certain Frobeni…

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 results to obtain a correspondence between the category of representations of an et…

We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2-category of pointed ho…

Develops a new framework for anomaly description in quantum field theories.

We describe discrete symmetries of two-dimensional Yang-Mills theory with gauge group $G$ associated to outer automorphisms of $G$, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted $G$-bundles, and calculate it ex…

In this article, we derive many properties of étale stacks in various contexts, and prove that étale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms. Moreover, we show that the bicategory of étale differentiable stacks and loca…