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