Rick Blute

(joint work with Prakash Panangaden and Sergey Slavnov.)

We propose a definition of categorical model of the deep inference system BV. BV was designed

as a logical framework for the consideration of Retoré's extension

of multiplicative linear logic to include a self-dual noncommutative connective.

Our definition is based on Cockett and Seely's notion of a linear functor.

A BV-category is a *-autonomous category
with an additional tensor product,

which when viewed as a bivariant functor, is linear with a degeneracy
condition. We show that this simple definition implies the isomorphisms required of the theory.

We show that coherence spaces, with Retoré's noncommutative tensor, form a model.

We then consider Girard's category of probabilistic coherence spaces, and show that it

contains an additional monoidal structure making it a BV-category.

We discuss possible applications of such categories to the discrete quantum causal dynamics

of Blute-Ivanov-Panangaden. The latter is ongoing work with Guglielmi, Panangaden,

and Strassburger.