  In this paper we study compact closed categories within the context of
homotopical algebra.
  We construct two new model category structures by localizing two (Quillen
equivalent) model categories of symmetric monoidal categories with the
objective of establishing the free compact closed category on one generator as
a fibrant replacement of the free symmetric monoidal category on one generator,
in our localized model categories. We go on to show that the fibrant objects in
our model categories are compact closed categories.
