Coends and categorical Hopf algebras

K. Shimizu has proved that, in a braided finite tensor category over an algebraically closed field, the triviality of the M¨uger centre implies that a certain Hopf pairing is non-degenerate. It is an open question whether the hypothesis that the base field is algebraically closed is necessary. In this thesis, we show, following some unpublished notes of Y. Sommerh¨auser and his coauthors, that this hypothesis is indeed not necessary in the case of the category of finite-dimensional modules over a finite-dimensional quasitriangular ribbon Hopf algebra H. In this category, the coend can be constructed as the dual space of H. We first review some basics of category theory, the construction of a coend as a categorical Hopf algebra, and duals and homomorphic images of categorical Hopf algebras. We then prove the result mentioned above. We conclude by constructing an example of a similar category where the dual space fails to be a coend.