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)