Linear logic and monoidal categories

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The goal of my project will be to continue to explore the theory of monoidal categories, especially using the techniques of linear logic. Examples of monoidal categories have arisen in a number of settings, from the theory of concurrent computation in computer science to topological and algebraic quantum field theory in physics. The monoidal structure is crucial in understanding these applications. In joint work with R. Seely and R. Cockett, we have invented the new notion of differential category. These are essentially models of linear logic with an additional differentiation operator, which allows one to differentiate morphisms. The obvious next goal is to consider categories of manifolds. In what category can the set of smooth functions between manifolds be considered a manifold? What is the logical structure of this category? There is a notion of manifold based on the convenient vector spaces of Frolicher and Kriegl; this will likely be quite relevant. In a related development, there is also a notion of holomorphic function between convenient vector spaces. How is the logical and categorical structure affected by this change in viewpoint? Algebraic Quantum Field Theory is a mathematically rigorous framework for modelling the interaction of quantum mechanics and relativity. I wish to combine AQFT with the recent abstract quantum mechanics of Abramsky and Coecke. There, quantum mechanics is reformulated away from the theory of C*-algebras of observables and expressed in abstract, categorical terms. This new notion of AQFT will assign a category to each open region in Minkowski space. The assignment will have properties suggested by both AQFT, and the Abramsky-Coecke axiomatics. Our work has already led to the new notion of a von Neumann category. We are currently developing this theory further, establishing crossed products for von Neumann categories, and extending the Doplicher-Roberts theorem to this setting.