Remarks on Tannaka recovery of coalgebras
Let \({\mathcal C}\) be a small category and \({\mathcal V}\) a tensor category co-complete over \({\mathcal C}\) such that tensor preserves colimits in each variable. Suppose \(F:{\mathcal C}\to{\mathcal V}\) is a functor such that for each \(X\in{\mathcal C}\), the dual \((FX)^*\) of \(FX\) exists. Then one can form the coend, \[ \operatorname {coend} (F)= \int^X F(X)^*\otimes F(X). \] This has a natural \({\mathcal V}\)-coalgebra structure and so we can consider the category \({\mathcal C}omod_{\mathcal V}(\operatorname {coend}(F))\). An important case of Tanaka-Krein duality is the characterization of those \(F:{\mathcal C}\to{\mathcal V}\) equivalent to the forgetful functor \(U_H: {\mathcal C}omod_{\mathcal V}(H)_c\to{\mathcal V}\) for some Hopf algebra \(H\). (The suffix \(c\) is used to indicate the subcategory of dualisable objects.) A more general question is whether \(\operatorname {coend} (F)\cong C\) when \(F= U_C\) for a coalgebra \(C\). This is known if \({\mathcal C}\) is a coalgebra over a field and \(U: {\mathcal C}omod (C)_c\to {\mathcal V}ect\) is the forgetful functor. The author here generalizes this to investigate the case when \({\mathcal V}\) is a module category over an associative ring. The investigation shows that versions of this coalgebra recovery theorem do hold for important classes of rings.
scientific article; zbMATH DE number 1384160