Summary: Abstract Scalars, Loops, and Free Traced and
Strongly Compact Closed Categories
Oxford University Computing Laboratory
Wolfson Building, Parks Road, Oxford OX1 3QD, U.K.
Abstract. We study structures which have arisen in recent work by the
present author and Bob Coecke on a categorical axiomatics for Quantum
Mechanics; in particular, the notion of strongly compact closed category.
We explain how these structures support a notion of scalar which allows
quantitative aspects of physical theory to be expressed, and how the
notion of strong compact closure emerges as a significant refinement of
the more classical notion of compact closed category.
We then proceed to an extended discussion of free constructions for a
sequence of progressively more complex kinds of structured category,
culminating in the strongly compact closed case. The simple geometric
and combinatorial ideas underlying these constructions are emphasized.
We also discuss variations where a prescribed monoid of scalars can be
`glued in' to the free construction.