Duality

The Frobenius equations mentioned for example in the last post were introduced in my paper with Carboni in 1987 to express the fact that an object V of a symmetric monoidal category might be self dual - that is, that  V is adjoint to V - in a strong sense. Each object of the category of sets and relations is self-dual in the strong sense that it has a Frobenius algebra structure (given by the diagonal map; that is, arising from equality). Even the Frobenius structure of wires in electrical circuits mentioned in the last post has an idea of equality associated, in that case the wires are equipotential regions.

However categories are not self-dual objects in the category of categories and bimodules (=profunctors); the natural dual of a category, the opposite category, is not isomorphic to the category. Here I describe a generalization of the notion of Frobenius algebra which implies that an object has a dual, but is not necessarily self-dual, and which applies to the category of categories and bimodules.
(A simpler example you may like to consider is the category of posets with arrows being bi-ideals (a bi-ideal from V to W is a subset of VxW with the property that if (v,w) is in the subset and v'<=v, w<=w' then also (v',w') is in the subset).)

Definition A generalized Frobenius algebra in a symmetric monoidal category (tensor written x) consists of two objects V and W and six arrows (for my ease I have used the same name for last two)
delta:V -> VxV, nabla: WxW->W,
d:W->WxV, n:VxW->V,
!:I->W, !:V->I
satisfying the following equations (writing composition in diagrammatic order):
(i) nabla and ! make W a  monoid; delta and ! make V a  comonoid,
(ii) (Vxd) (nxV) = n delta = (delta x W)(Vxn) : VxW -> VxV,
(iii) (dxW)(Wxn) = nabla d = (Wxd)(nabla xV) ; WxW -> WxV,
(iv) (Vx!)n = 1 : V->V,
(v) d (Wx!) = 1: W->W.

Example A  Frobenius algebra is a special case taking V=W, d=delta, n=nabla.

Example In the category of categories and bimodules consider a category V and its opposite W. Then
delta: V->VxV is the profunctor corresponding to the diagonal functor of V. That is,
delta(v1,(v2,v3))=Hom(v1,v2)xHom(v1,v3).
In this and the following Hom always means Hom in V.  Then nabla:WxW->W is defined by
nabla((v1,v2),v3)=Hom(v3,v1)xHom(v3,v2),
 n;VxW->V by
n((v1,v2),v3)=Hom(v1,v2)xHom(v1,v3),
d;W->WxV by
d(v1,(v2,v3))=Hom(v2,v1)xHom(v2,v3).
Finally !(*,v)={*} and !(v,*)={*}.

It is straightforward to check the equations, using the Yoneda lemma.

I want to prove the simple result, that V is the right adjoint of W. Then since in a symmetric monoidal category this implies that V is also the left adjoint of W,  if we call W the dual of V, then the dual of the dual of V is V.

V is the right adjoint of W.  Define eta to be (! d): I -> WxV and
epsilon to be (n !) : VxW -> I. We need to check the two triangular identities for adjunction.
The first: (Vx eta)(epsilon xV) = (Vx!)(Vxd)(nxV)(!xV)
                           = (Vx!)n delta (!xV)
                          = 1 1 =1 as requried.
The second triangular equation   (eta x W)(W x epsilon) = (!xW)(dxW)(Wxn)(Wx!)
                         =(!xW)(nabla d)(Wx!) =1 1 =1.

Remark Just as there is graphical language for Frobenius algebras, there is a useful graphical language here. The difference is that the connectors between components have orientation.