Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of Etingof-Kazhdan on quantization of Lie bialgebras.


Briefly, the coproduct of a Hopf algebra $H$ (say, in vector spaces $Vect\_{\mathbb{K}}$) defines a symmetric monoidal structure on its category of modules $Mod\_H$. We have a forgetful functor $U:Mod\_H\rightarrow Vect\_{\mathbb{K}}$ called the fiber functor,
so that if $U$ is equipped with a symmetric monoidal structure, then one can recover $H$ via an isomorphism $H\cong End(U)$ (the linear endomorphisms of $U$).


My question is the following: is there a Tannaka duality for topological Hopf algebras, e.g. Hopf algebras in Fréchet spaces, Banach spaces, etc...(equipped with the appropriate tensor product) ? If so, what are the main results and good references about this ?



Tannakian formalism for topological Hopf algebras
I believe the best answer to your question at the moment is in this paper (which only treats the case of bialgebras)
<http://adsabs.harvard.edu/abs/2014arXiv1411.3183L>


Indeed, in topological setting one has different tensor products. But if one looks carefully on the proofs on Tannaka duality in algebraic setting - they use universal properties of objects of the form $X\otimes X^\wedge$, that are dual to the universal properties of endomorphism objects. 
Instead of requiring the structure of rigid category (that fails in topological setting due to the necessity to use different tensor products), one can only require the existence of the above-mentioned objects, that are called *coendomorphisms* in the paper, and it appears that all the proofs can be carried out in this new setting (although it is probably not entirely new). At least for topological vector spaces over $p$-adic fields, the coendomorphisms are precisely the inductive tensor products of the corresponding space and it's dual, which fits into the pattern of the construction.


