Reconstruction Notes

We discuss Tannaka reconstruction in general categories, not necessarily vector spaces. For an excellent introduction, see [4]. Before that, though, an introduction to my notation for monoidal and comonoidal functors. The original notion for graphically depicting monoidal functors as transparent boxes in string diagrams is due to Cockett and Seely[1], and has recently been revived and popularized by Mellies[5] with prettier graphics and an excellent pair of example calculations which nicely show the worth of the notation. However, a small modification improves the notation considerably. For a monoidal functor F: A − → B, we have a pair of maps, Fx ⊗ Fy − → F (x ⊗ y) and e − → Fe, which we notate as follows: Similarly, for a comonoidal F, we have maps F (x ⊗ y) − → Fx ⊗ Fy and Fe − → e which we notate in the obvious dual way, as follows: Graphically, the axioms for a monoidal functor are depicted as follows: where, once again, the similar constraints for a comonoidal functor are exactly the above with composition read right-to-left instead of left-to-right. 1 It is curious and pleasing that the two unit axioms bear a superficial resemblence to triangle-identies being applied along the boundary of the “F-region”. The above axioms seem to indicate some sort of “invariance under continuous deformation of F-regions”. For a functor which is both monoidal and comonoidal, pursuing this line of thinking leads one to consider the following pair of axioms: A functor with these properties has been called a Frobenius monoidal functor by Brian and Craig. Let F: A − → B be a (mere) functor between monoidal categories, where B is assumed to be left closed. Then define EF = a∈