Summary: ABSTRACT PHYSICAL TRACES
SAMSON ABRAMSKY AND BOB COECKE
ABSTRACT. We revise our `Physical Traces' paper [Abramsky and Coecke CTCS`02]
in the light of the results in [Abramsky and Coecke LiCS`04]. The key fact is that the
notion of a strongly compact closed category allows abstract notions of adjoint, bipartite
projector and inner product to be defined, and their key properties to be proved. In
this paper we improve on the definition of strong compact closure as compared to the
one presented in [Abramsky and Coecke LiCS`04]. This modification enables an elegant
characterization of strong compact closure in terms of adjoints and a Yanking axiom,
and a better treatment of bipartite projectors.
In [Abramsky and Coecke CTCS`02] we showed that vector space projectors
V# W #
which have a onedimensional subspace of
V# W as fixedpoints, su#ce to implement
any linear map, and also the categorical trace [Joyal, Street and Verity 1996] of the cate
,# ) of finitedimensional vector spaces and linear maps over a base field K.