Dagger categories and idempotents
Peter Selinger
Dagger compact closed categories, which describe the main
structure of the category of finite dimensional Hilbert spaces, have a
long tradition in category theory. In the 1980's, mathematical
physicists called them " *-categories" (a name derived from
C*-algebras); in the 1990's, John Baez called them "monoidal
categories with duals" (or for the less faint of heart: k-tuply
monoidal n-categories with duals); and most recently, Abramsky and
Coecke gave an interesting application to quantum protocols under the
name "strongly compact closed categories". In a recent paper, I showed
that the passage from "pure" to "mixed" quantum computation can be
described as a construction on dagger compact closed categories called
the CPM construction. A minor inconvenience of this construction is
that it does not automatically yield biproducts (i.e., classical
types); one has to add them freely after the fact. In this talk, I
will show that classical types can be equivalently obtained by
splitting self-adjoint idempotents. I will discuss general properties
of idempotents in dagger categories, and discuss their relationship to
classical types.