A Categorical approach to Tannaka Duality

Abstract

The aim of this work is to study, in a categorical context, similar results to the Tannaka-Krein duality theorem. We prove a reconstruction theorem which allows one to recover a given group G as the monoid of natural endomorphisms of the forgetful functor from its category of permutation representations to Set. The proof of this result follows as a consequence of the Yoneda lemma. After introducing some necessary categorical concepts and results, and introducing the context of enriched category theory and the enriched version of the Yoneda lemma, we prove the enriched version of the reconstruction theorem for a general monoid A in a symmetric monoidal closed category V, which is recovered as the monoid in V of enriched natural endomorphisms of the forgetful V-functor from the enriched category of representations of A to V.