more dagger problems

[Juergen Koslowski <koslowj@iti.cs.tu-bs.de>]

Besides their funny name, dagger-compact categories present, or
rather expose, another terminological dilemma: what's an adjoint?

Although the notion of dagger-compact category (dcc) was originally
defined for symmetric monoidal categories, let's try to makes sense of
it without symmetry. In fact, this should work in a (linear)
bicategory or even a poly-bicategory.

- The dagger operation flips 2-cells vertically (in view of the
  picture calculus).  f^{\dagger} is called the
  ``adjoint'' of f, which matches the terminology of functional
  analysis and physics.  In case of a linear bicategory, the two 1-cell
  compositions \tensor for the domain 1-cells and \par for the
  codomain 1-cells get interchanged as well.

- The definition of a dcc also calls for ``duals'' A^* of 1-cells,
  which in graphical terms flips 1-cells horizontally. 

- Finally, there are``units'' (or ``Bell states'' in physics terms) 
  \eta_A: I_Y ==> A^*\tensor A, when A:X --> Y.

The axioms for a symmetric dcc turn the ``duals'' turn into
categorical adjoints with unit \eta_A and counit (\eta_A)^{\dagger}
(the ``adjoint'' of the unit).  This seems to require symmetry, but it
really does not.  The correct interpretation of A^* should be that of
a 2-sided (categorical) adjoint for A (linear adjoint in the case of
poly-bicategories), i.e., A^* -| A -| A^*.  Hence there are 2
categorical adjunctions and hence 2 units, besides \eta_A also
\eta_A^*: I_X ==> A\tensor A^*.  Without symmetry (\eta_A)^{\dagger}
cannot be the counit for the adjunction A^* -| A, but for the other
adjunction A -| A^*.  Unfortunately, the functional analysis
terminology would refer to the counit of the second adjunction as the
adjoint of the first adjunction's unit, which I find rather confusing.

This bicategorical view also clarifies that the star operation is an
involution on 1-cells, while dagger is an involution on 2-cells.
While the name ``{1,2}-involutive bicategory'' may be adequate,
``{1,2}-involutive monoidal category'' is quite a mouthful.

I seem to recall reading not too long ago that Kahn did _not_ pursue
an (often rumoured) analogy with functional analysis when introducing
(categorical) adjunctions, and here we see the actual mismatch.

-- Jürgen

-- 
Juergen Koslowski               If I don't see you no more on this world
ITI, TU Braunschweig               I'll meet you on the next one
koslowj@iti.cs.tu-bs.de               and don't be late!
http://www.iti.cs.tu-bs.de/~koslowj      Jimi Hendrix (Voodoo Child, SR)