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.