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We construct a category $\mathrm{HomCob}$ whose objects are {\it homotopically 1-finitely generated} topological spaces, and whose morphisms are {\it cofibrant cospans}. Given a manifold submanifold pair $(M,A)$, we prove that there exists…

Mathematical Physics · Physics 2022-09-01 Fiona Torzewska

Following the theory of principal $\infty$-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of…

Quantum Algebra · Mathematics 2026-01-23 Hank Chen , Florian Girelli

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries…

Differential Geometry · Mathematics 2016-08-29 Virginie Charette , Todd A. Drumm , Youngju Kim

We give a proof of the existence of radial (smooth) parallel sections of vector bundles endowed with a linear connection.

Differential Geometry · Mathematics 2018-01-23 Antonio J. Di Scala

We consider commutative Frobenius pseudomonoids in the bicategory of spans, and we show that they are in correspondence with 2-Segal cosymmetric sets. Such a structure can be interpreted as a coherent 2-dimensional topological quantum field…

Algebraic Topology · Mathematics 2026-01-01 Sophia E Marx , Rajan Amit Mehta

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…

Geometric Topology · Mathematics 2016-11-29 David Cimasoni , Anthony Conway

Drawing parallels with hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\mathcal{A}$ of locally flat, codimension-1…

Algebraic Topology · Mathematics 2013-06-13 Priyavrat Deshpande

We show that the complete bornological convolution algebras of Lie groupoids and convolution bimodules of groupoid bibundles define a monoidal functor from the 2-category of differentiable stacks to the Morita 2-category of complete…

Differential Geometry · Mathematics 2026-05-29 David Aretz , Christian Blohmann

The bicategorical point of view provides a natural setting for many concepts in the representation theory of monoidal categories. We show that centers of twisted bimodule categories correspond to categories of 2-dimensional natural…

Category Theory · Mathematics 2023-06-09 Bojana Femić , Sebastian Halbig

In this article we discuss local aspects of 2-functors defined on the path 2-groupoid of a smooth manifold; in particular, local trivializations and descent data. This is a contribution to a project that provides an axiomatic formulation of…

General Topology · Mathematics 2017-02-01 Urs Schreiber , Konrad Waldorf

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a…

Mathematical Physics · Physics 2009-10-31 R. Coquereaux , A. O. Garcia , R. Trinchero

It is well-known that pseudo functors from bicategories of spans are equivalent to Beck-Chevalley bifibrations, and therefore capture the relationships underlying the adjunctions suitable as semantics for existential quantification. This…

Category Theory · Mathematics 2025-09-26 José Siqueira

Here we consider the product of varieties with $n$-blocks collections . We give some cohomological splitting conditions for rank 2 bundles. A cohomological characterization for vector bundles is also provided. The tools are Beilinson's type…

Algebraic Geometry · Mathematics 2008-04-02 Edoardo Ballico , Francesco Malaspina

I discuss possible definitions of categories of vector spaces enriched with a notion of formal infinite linear combination in the likes of the formal infinite linear combinations one has in the context of generalized power series, I call…

Category Theory · Mathematics 2026-03-10 Pietro Freni

We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections…

Algebraic Geometry · Mathematics 2019-02-08 Sandra Di Rocco , Kelly Jabbusch , Gregory G. Smith

This article is dedicated to the study of the normal functor in the category of smooth real vector bundles. Particularly, we focus on a symmetry phenomena which occurs after iterating two times the normal functor on a commutative square of…

Category Theory · Mathematics 2026-04-10 Quentin Karegar Baneh Kohal

In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new…

Differential Geometry · Mathematics 2011-02-23 Florin Dumitrescu

In this paper we study a 2-dimensional version of Quillen's homotopy category construction. Given a category $\mathscr{A}$ and a class of morphisms $\Sigma \subset \mathscr{A}$ containing the identities, we construct a 2-category…

Category Theory · Mathematics 2023-02-28 Eduardo J. Dubuc , Jaqueline Girabel

Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\infty$-)categories of spans (or correspondences). In this paper we study the…

Category Theory · Mathematics 2021-11-30 Elden Elmanto , Rune Haugseng

We improve Ottaviani's splitting criterion for vector bundles on a quadric hypersurface and obtain the equivalent of the result by Rao, Mohan Kumar and Peterson. Then we give the classification of rank 2 bundles without "inner" cohomology…

Algebraic Geometry · Mathematics 2007-05-23 F. Malaspina