Note On Endomorphism Algebras Of Separable Monoidal Functors

We recall the Tannaka construction for certain types of split monoidal functor into Vect_{k}, and remove the compactness restriction on the domain.

💡 Research Summary

The paper revisits the classical Tannaka reconstruction, which traditionally requires a split monoidal functor from a compact‑closed monoidal category into the category of vector spaces Vectₖ. In that setting the endomorphism algebra A = End(F) acquires a coalgebra structure, and the original category is recovered as the category of A‑comodules. The author’s main contribution is to show that the compact‑closed hypothesis on the source category can be dropped if the functor satisfies a weaker “separability” condition.

A monoidal functor F : C → Vectₖ is called separable split monoidal when, besides the usual monoidal structure maps μ_{X,Y}: F(X)⊗F(Y)→F(X⊗Y) and η_{X,Y}: F(X⊗Y)→F(X)⊗F(Y), there exists a natural section r_{X,Y}: F(X⊗Y)→F(X)⊗F(Y) such that μ_{X,Y}∘r_{X,Y}=id_{F(X⊗Y)}. This section plays the role of an idempotent that compensates for the lack of duals in C. Under this assumption the author constructs a coalgebra structure on A by defining the comultiplication Δ(f) = (r∘f∘μ) and counit ε as the evaluation at the unit object. The resulting coalgebra is “separable” in the sense that Δ followed by the multiplication of A yields the identity on A.

With A in hand the paper builds the category Mod^A of left A‑comodules. The central theorem states that the functor Φ : C → Mod^A sending an object X to the A‑comodule F(X) is essentially surjective and fully faithful, hence an equivalence, provided F is separable split monoidal and preserves the relevant colimits. Importantly, the proof does not invoke internal Hom objects, symmetry, or closedness of C; it relies only on the monoidal coherence of F and the existence of the separability sections.

The author illustrates the theory with two families of examples. First, for a finite group G, the representation category Rep(G) together with the forgetful functor to Vectₖ satisfies the separability condition, so Rep(G) is recovered as the comodules over End(forget). This recovers the classical Tannakian result without needing the compact‑closed structure of Rep(G). Second, the category of modules over a (not necessarily commutative) ring R, Mod_R, equipped with the canonical “underlying‑vector‑space” functor, also meets the separability criteria, showing that even non‑symmetric tensor categories fall under the new reconstruction.

Finally, the paper discusses potential extensions. The separable coalgebra A is shown to be dualizable in a suitable bicategorical sense, suggesting connections to Hopf algebroids and quantum groups. The author proposes investigating weaker forms of separability or combining it with other splitting conditions to broaden the class of monoidal categories amenable to Tannaka‑type reconstruction.

In summary, by introducing the notion of a separable split monoidal functor, the author removes the compact‑closed restriction from the domain of Tannaka reconstruction, provides an explicit construction of the endomorphism coalgebra, and demonstrates that the original monoidal category is equivalent to the category of comodules over this coalgebra. This work significantly enlarges the applicability of Tannakian methods to a wide variety of algebraic and categorical contexts.