# Talk:Dagger compact category

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## examples of f-dagger

It may help to make the article more useful if somebody (more proficient at this than I) adds what is f^\dagger in each of the examples.

All the dagger does is reverse the direction of arrows. So, if ${\displaystyle f:A\longrightarrow B}$, then ${\displaystyle f^{\dagger }:B\longrightarrow A}$. I will have a look at the examples... --Radjenef (talk) 01:39, 28 April 2009 (UTC)
In Hilb, f^\dagger is given by the adjoint of f, in Rel it is given by the opposite relation, and in the category of finitely generated projective modules it is... what, the transpose of the matrix representation of a linear map? —Preceding unsigned comment added by 130.54.16.83 (talk) 07:36, 26 May 2009 (UTC)

## TODO List

Things that need clarification:

• The bit about defining a basis here makes use of relations from coalgebra to define it. Much the same as in Frobenius algebra, except that the dagger version of the axioms are not given there ('Frobenius law'). This needs to be firmed up; the dagger version needs to be added there.
• The Frobenius part has something to do with the movement of classical information (viz not quantum information) since frobenius stuff can be cloned/delete. This needs to be explained in detail.
• Quantum mechanics technically lives on complex projective hilbert space, not hilbert space in general. What would need to be added to make a category be projective? Is there a projective category?
• So, for example, normally, unitarity arises from the projective nature .. here, we might expect unitarity to be a 2-cell between morphisms ... right!?
• Hilbert space is defined for any field; including finite fields; again, QM works only for the field being the complex numbers. Can anything intelligent be said about this?
• What, exactly, is the deal with finite vs. infinite-dim hilbert space? Can we explicitly demonstrate why this doesn't work for the infinite case (yes, compactness. but spell it out in detail...)
• Stuff like the basis, eigenvectors, etc. should be worked out in detail for the category of relations, viz to have a clearer view of what it looks like in a different cat.

User:Linas (talk) 07:35, 27 November 2013 (UTC)