Authors: Baralić, Djordje 
Petrić, Zoran 
Telebaković, Sonja
Title: Spheres as frobenius objects
Journal: Theory and Applications of Categories
Volume: 33
First page: 691
Last page: 726
Issue Date: 18-Jul-2018
Rank: M23
ISSN: 1201-561X
URL: http://www.tac.mta.ca/tac/volumes/33/24/33-24.pdf
Abstract: 
Following the pattern of the Frobenius structure usually assigned to the 1-dimensional sphere, we investigate the Frobenius structures of spheres in all other dimensions. Starting from dimension d = 1, all the spheres are commutative Frobenius objects in categories whose arrows are (d + 1)-dimensional cobordisms. With respect to the language of Frobenius objects, there is no distinction between these spheres–they are all free of additional equations formulated in this language. The corresponding structure makes out of the 0-dimensional sphere not a commutative but a symmetric Frobenius object. This sphere is mapped to a matrix Frobenius algebra by a 1-dimensional topological quantum field theory, which corresponds to the representation of a class of diagrammatic algebras given by Richard Brauer.
Keywords: Brauerian representation | Cobordism | Coherence | Commutative Frobenius object | Normal form | Oriented manifold | Symmetric monoidal category | Topological quantum field theory
Publisher: Mount Allison University
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Representations of logical structures and formal languages and their application in computing 
Analysis and algebra with applications 

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