The multiplicative fragment of Linear Logic is the formal system in this family with the best understood proof theory, and the categorical models which best capture this theory are the fully complete ones. We demonstrate how the Hyland-Tan double glueing construction produces such categories, either with or without units, when applied to any of a large family of degenerate models. This process explains as special cases a number of such models from the literature.
In order to achieve this result, we develop a tensor calculus for compact closed categories with finite biproducts. We show how the combinatorial properties required for a fully complete model are obtained by this glueing construction adding to the structure already available from the original category.

Comment: 72 pages. An extended abstract of this work appeared in the proceedings of LICS 2012

• 03F52 - Proof-theoretic aspects of linear logic and other substructural logics
• 03G30 - Categorical logic, topoi
• 18D15 - Closed categories (closed monoidal and Cartesian closed categories, etc.)