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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 ?

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    $\begingroup$ See ncatlab.org/nlab/show/Tannaka+duality . It works for any monoid in nice enough categories. I do not know enough about the particular categories you want to know if they are well behaved enough for the usual arguments to carry through. $\endgroup$
    – zibadawa timmy
    Commented Sep 26, 2014 at 9:15
  • $\begingroup$ This might not be relevant to the categories you care about, but when people try to look at Hopf von Neumann algebras one has to use a different tensor product for the codomain of the comultiplication. So when you speak of Hopf algebras in the category of Banach spaces with its usual SMC structure, which examples do you have in mind? $\endgroup$
    – Yemon Choi
    Commented Sep 30, 2014 at 0:56
  • $\begingroup$ I was thinking for instance about smooth functions on a compact group, or more generally smooth functions on a compact monoid in the category of smooth manifolds, which form, if I'm not mistaken, a bialgebra in the category of Frechet spaces with the projective tensor product. $\endgroup$
    – Sinan Yalin
    Commented Sep 30, 2014 at 10:18
  • $\begingroup$ Some related stuff can be found in a paper by Joyal and Street maths.mq.edu.au/~street/CT90Como.pdf $\endgroup$
    – Dimitri Chikhladze
    Commented Oct 2, 2014 at 16:23

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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.

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