#
Graphical Methods for
Tannaka duality of weak bialgebras and weak Hopf algebras

##
Micah Blake McCurdy

Tannaka duality describes the relationship between algebraic objects in a
given category and functors into that category; an important case is that
of Hopf algebras and their categories of representations; these have
strong monoidal forgetful ``fibre functors'' to the category of vector
spaces. We simultaneously generalize the theory of Tannaka duality in two
ways: first, we replace Hopf algebras with *weak Hopf algebras* and
strong monoidal functors with *separable Frobenius monoidal
functors*; second, we replace the category of vector spaces with an
arbitrary braided monoidal category. To accomplish this goal, we make use
of a graphical notation for functors between monoidal categories, using
string diagrams with *coloured regions*. Not only does this notation
extend our capacity to give simple proofs of complicated calculations, it
makes plain some of the connections between Frobenius monoidal or
separable Frobenius monoidal functors and the topology of the axioms
defining certain algebraic structures. Finally, having generalized Tannaka
duality to an arbitrary base category, we briefly discuss the
functoriality of the construction as this base is varied.

Keywords:
Tannaka duality, Tannaka reconstruction, bialgebras, Hopf
algebras, weak bialgebras, weak Hopf algebras, separable Frobenius
monoidal functors, graphical methods

2000 MSC:
18D10

*Theory and Applications of Categories,*
Vol. 26, 2012,
No. 9, pp 233-280.

Published 2012-05-28.

http://www.tac.mta.ca/tac/volumes/26/9/26-09.pdf

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