  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.
