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Spheres as Frobenius objects

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Djordje Baralic, Zoran Petric and Sonja Telebakovic

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:
symmetric monoidal category, commutative Frobenius object, oriented
manifold, cobordism, normal form, coherence,
topological quantum field theory, Brauerian representation

2010 MSC:
18D35, 57R56

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 24, pp 691-726.

Published 2018-07-18.

http://www.tac.mta.ca/tac/volumes/33/24/33-24.pdf

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