We investigate categorical models of the unitfree multiplicative and multiplicativeadditive fragments of linear logic by representing derivations as particular structures known as dinatural transformations. Suitable categories are considered to satisfy a property known as full completeness if all such entities are the interpretation of a correct derivation. It is demonstrated that certain HylandSchalk double glueings [HS03] are capable of transforming large numbers of degenerate models into more accurate ones. Compact closed categories with finite biproducts possess enough structure that their morphisms can be described as forms of linear arrays. We introduce the notion of an extended tensor (or 'extensor') over arbitrary semirings, and show that they uniquely describe arrows between objects generated freely from the tensor unit in such categories. It is made evident that the concept may be extended yet further to provide meaningful decompositions of more general arrows. We demonstrate how the calculus of extensors makes it possible to examine the combinatorics of certain double glueing constructions. From this we show that the HylandTan version [Tan97], when applied to compact closed categories satisfying a far weaker version of full completeness, produces genuine fully complete models of unitfree multiplicative linear logic. Research towards the development of a full completeness result for the multiplicativeadditive fragment is detailed. The proofs work for categories of finite arrays over certain semirings under both the HylandTan and Schalk [Sch04] constructions. We offer a possible route to finishing this proof. An interpretation of these results with respect to linear logic proof theory is provided, and possible further research paths and generalisations are discussed.
Date of Award  1 Aug 2013 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Andrea Schalk (Supervisor) 

 Combinatorics
 Permutations
 Multilinear Algebra
 Category Theory
 Linear Logic
 Categorical Logic
Combinatorial Arguments for Linear Logic Full Completeness
Steele, H. (Author). 1 Aug 2013
Student thesis: Phd