  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.
