TITLE: Tannaka-Krein duality in Chari-Pressley's book
QUESTION [8 upvotes]: I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here.
V.Chari and A.N.Pressley in their "Guide to Quantum Groups" formulate the following version of Tannaka-Krein's duality (Theorem 5.1.11):

Let $R$ be a commutative ring, let $C$ be an essentially small, $R$-linear, rigid, abelian, monoidal category, and let $\varPhi:C\to\operatorname{mod}_R$  be a $R$-linear exact faithful monoidal functor. Then there exists a Hopf algebra $A$ over $R$ and an equivalence of  $R$-linear categories $C\to\operatorname{corep}_A$ whose composite with the forgetful functor $\operatorname{corep}_A\to\operatorname{mod}_R$ is $\varPhi$.

They give an outline of the proof, but I must say that I don't understand some of its details. I think that there must exist an accurate text with complete proof. Can anybody give me a reference to it?  
Also if somebody could cast some light on possible generalizations of this proposition (to the case of Hopf algebras in braided monoidal categories), I would appreciate this very much.

REPLY [2 votes]: In the case where $k$ is a field this should be worked out in full generality (for Hopf algebras) in Chapter 5 of 'Tensor Categories' by Etingof, Gelaki, Nikshych and Ostrik. This includes the infinite dimensional case.
For the finite dimensional case this can be generalized even further to weak Hopf algebras, so that every Fusion category is monoidally equivalent to the category of representations for a weak Hopf algebra. This is stated in 'On Fusion Categories.' (Corollary 2.22) where it is there referenced as due to Hayashi in 'A canonical Tannaka duality for finite seimisimple tensor categories
' and Szlach´anyi in 'Finite quantum groupoids and inclusions of finite type.' A proof should also be provided in Ostrik's 'Module categories, weak Hopf algebras and modular invariants.' An additional reference is section 7.23 of 1.
As a note, this is referenced in the nlab article, under the table 'Tannaka duality for categories of modules over monoids/associative algebras' for 'Hopf algebras.' Following the Hopf algebras link takes you to the Hopf algebras article which includes a section on 'Tannakian Duality' and includes a link to Bakke's 'Hopf algebras and monoidal categories' and Vercruysse's 'Hopf algebras---Variant notions and reconstruction theorems'. For weak hopf algebras following the links leads to 3 and 5.
Finally, a gold standard reference for understanding Tannakian duality is Pierre Delign's 'Catégories Tannakiennes' and its related generalizations, linked in the nLab article on said paper.