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In this paper we study compact closed categories within the context ofhomotopical algebra.  We construct two new model category structures by localizing two (Quillenequivalent) model categories of symmetric monoidal categories with theobjective of establishing the free compact closed category on one generator asa 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 inour model categories are compact closed categories.

Compact closed categories and $\Gamma$-categories (with an appendix by Andr\'e Joyal)