Tannaka Reconstruction of Weak Hopf Algebras in Arbitrary Monoidal Categories

Broadly speaking, Tannaka duality describes the relationship between algebraic objects and their representations, for an excellent introduction, see [7]. On the one hand, given an algebraic object H in a monoidal category B (for instance, a hopf object in the category Vec k of vector spaces over a field k), one can consider the functor which takes algebraic objects of the given type to their category of representations, rep B H, for which there is a canonical forgetful functor back to B. This process is representation and it can be defined in a great variety of situations, with very mild assumptions on B. For instance:

• If J is a bialgebra in a braided linearly distributive category B, then rep B J is linearly distributive [2].

• If H is a Hopf algebra in a braided star-autonomous category B, then rep B H is star-autonomous [2]. Note that the first example includes the familiar notion of a bialgebra in a monoidal category giving rise to a monoidal category of representations, since every monoidal category is degenerately linearly distributive by taking both of the monoidal products to be the same.

On the other hand, given a suitable functor F : A -→ B, we can try to use the properties of F (which of course include those of A and B) to build an algebraic object in B; this is called (Tannaka) reconstruction, since historically the algebraic objects have been considered primitive. We denote the reconstructed object as E F , following [12]. This requires making more stringent assumption on B; certainly it must be braided; it must be autonomous, and it is assumed that B admits certain ends or coends which cohere with the monoidal product. For instance, under these assumptions:

• If A is autonomous and F is a strong monoidal functor, then E F is a Hopf algebra in B ( [11], [8]).

To this we add the following:

• If A is autonomous and F is a separable Frobenius functor, then E F is a weak Hopf algebra in B. Note that the notion of weak Hopf algebra considered by Haring-Oldberg [6] is a different notion, the sense of weak we use here is that of [1]. It should be noted that a special case of the weak Hopf algebra result has been obtained by Pfeiffer [10], where A is taken to be a modular category and B is taken to be Vec k ; see also [13].

In favourable circumstances, reconstruction is left adjoint to representation, for instance:

In the weak cases which I discuss in the sequel, however, we do not have these adjunctions.

Before we discuss the reconstruction itself, we discuss notations 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 [2], and has recently been revived and popularized by Mellies [9] 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, F x ⊗ F y -→ F (x ⊗ y) and e -→ F e, which we notate as follows:

Similarly, for a comonoidal F , we have maps F (x ⊗ y) -→ F x ⊗ F y and F e -→ 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.

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 Day and Pastro [4]; moreover, it is the notion which results from considering linear functors in the sense of [2] between degenerate linearly distributive categories. They are more common than strong monoidal functors yet still share the key property of preserving duals.

Let F : A -→ B be a (mere) functor between monoidal categories, where B is assumed to be left closed. Then define

where I assume that A and B are such that the indicated end exists. As Richard Garner imimitably asked at the 2006 PSSL in Nice, “Have you considered enriching everything?”. I do not discuss the matter here, but it has been considered by Brian Day [3].

There is a canonical action of E F on F x for each object x in A, which we denote as α = α x : E F ⊗F x -→ F x. This is defined as:

using the x’th projection from the end followed by the evaluation of the monoidal closed structure of B. The dinaturality of the end in a gives rise to the naturality of the action on F a in a, which we notate as:

Let us now assume that the closed structure of B is given by left duals, that is, [a, b] = b ⊗ La.

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