Re: dagger?

[selinger@mathstat.dal.ca (Peter Selinger)]

Lawrence Stout wrote:
>
> Could someone please post a definition of a dagger category and a
> reference for typical useful examples?

For reference: an involutive category (or dagger category) is a
category C equipped with a contravariant, involutive,
identity-on-objects functor "+" (my ASCII rendition of the TeX symbol
$\dagger$).

The main reason to consider dagger structure is that it allows the
following definitions (and other similar ones):

* a morphism f:A->B is _unitary_ if f is invertible and f^{-1} = f^+

* a morphism f:A->A is _hermitian_ if f = f^+.

* a morphism f:A->A is _hermitian positive_ if there exists some
  object B and g:A->B such that f = g^+ o g.

* a morphism f:A->B is called an _isometry_ if f^+ o f = id.
  ("Isometry" is to "unitary" like "mono" to "iso").

The main example is the category of finite-dimensional Hilbert
spaces. In it, for a map f : A -> B, the map f^+ : B -> A is given as
the adjoint of f (in the linear-algebra sense). Note that the
definition of the adjoint requires inner products, hence *Hilbert* and
not just vector spaces.

The category of finite-dimensional Hilbert spaces is additionally
compact closed, so that for a morphism f : A -> B, we also have
f^* : B^* -> A^*.  While the functor (-)^* is also contravariant and
involutive, it is not to be confused with the dagger structure. A^* is
the dual space, which is not naturally isomorphic to A. Also, relative
to chosen bases, the matrix of f^* is the transpose of that of f,
whereas the matrix of f^+ is the adjoint (complex conjugate transpose).

Dagger compact closed categories were axiomatized by Abramsky and
Coecke [LICS 2004], and also by Baez and Dolan [ArXiV:q-alg/9503002,
1995].  One requires the following compatibilities between the two
structures:

* (f tensor g)^+ = f^+ tensor g^+,
* the structural natural isomorphisms (associativity, symmetry, etc)
  are unitary,
* the maps I -> (A^* tensor A) and (A^* tensor A) -> I are each
  other's adjoints.

As Abramsky and Coecke have shown, many interesting constructions from
Hilbert spaces can be done in a dagger compact closed category.

Another example of a dagger compact closed category is the category of
sets and relations, but it is degenerate, in the sense that A^* = A
and f^* = f^+.

-- Peter