dagger, involutive and pivotal categories

dagger, involutive and pivotal categories

[Clemens.BERGER]

Dear all,

   this message continues the discussion about the "evil" aspect of the 
definition of a dagger category. In part I would like to give some 
support to Dusko Pavlovic's message and also stress the relationship 
with another concept (that of a pivotal, aka sovereign category).

  Call a functor (-)^*:E->E^{op} an *involution* of E if (-)^* is 
faithful and self-adjoint. The latter means that there is a binatural 
bijection between maps A->B^* and B->A^* for all objects A,B of E. The 
unit (and counit) of this adjunction is a natural transformation 
i_A:A->A^** which is supposed to be monic in E by faithfulness of (-)^*, 
and which fulfills the triangular identity id_{A^*}=(i_A)^*i_(A^*) by 
self-adjointness of (-)^*.

  Call a category E *involutive *if E comes equipped with an involution 
(-)^*:E-> E^{op}. Involutions may be transported along adjoint 
equivalences. Now, to any involutive category (E,(-)^*) is naturally 
associated a dagger category, which I shall call the *dagger core* of 
(E,(-)^*). Namely, the objects of the dagger core are pairs (A,\phi_A) 
consisting of an object A of E together with an isomorphism 
\phi_A:A->A^* which is invariant under the self-adjunction (i.e. 
phi_A=(phi_A)^*i_A). The morphisms f: (A,\phi_A)->(B,\phi_B) are simply 
the morphisms A->B in E. The dagger operation on the dagger core is 
given by f^{dagger}=phi_A^{-1}f^*\phi_B.

Equivalent involutive categories have dagger-equivalent dagger-cores. 
What I am trying to say is that involutive categories are "large" 
objects (in Bob Paré's terminology) giving rise (among others) to 
"small'' objects such as dagger categories (in which it isn't "evil" to 
speak about "equality" of objects).

In order to get the dagger-category of Hilbert spaces and continuous 
maps, one has to start with the involutive category of Banach spaces and 
continuous (i.e. bounded) maps. Over the reals, the involution is given 
by the continuous dual; over the complex numbers however, the involution 
is given by homming into C equipped with the complex-conjugate C-action.

Let me briefly discuss two possible strengthenings of the notion of 
involutive category (relevant to quantum algebra). Call a category E 
*closed involutive* if  E is at once closed monoidal and involutive in 
such a way that the involution (-)^*:E->E^{op} is given by homming into 
the unit of the monoidal structure. Define the *core *of a closed 
involutive category to be the full subcategory spanned by the dualizable 
objects A (i.e. those for which i_A:A->A^** is an isomorphism). Then the 
core is again closed involutive, and is actually a pivotal category. 
Indeed, pivotal categories are precisely closed involutive categories in 
which all objects are dualizable. I guess that most of the pivotal 
categories occuring in nature are cores of "large" closed involutive 
categories. This is at least the case when Tannaka-reconstruction applies.

Finally, call a category E *star-involutive* if E is at once closed 
monoidal and involutive in such a way that the involution is given by 
homming into a "dualising" object D. The only thing we require is that 
i_D:D->D^** be an isomorphism. Then the core of a star-involutive 
category (spanned by the dualizable objects) is again star-involutive.

In analogy with above, a star-involutive category in which all objects 
are dualizable, could be called *star-pivotal*; star-pivotal categories 
bear the same relationship to pivotal categories as Michael Barr's 
star-autonomous categories bear to autonomous categories.

All the best,
                    Clemens.
 


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