Tensor products and *-autonomous categories

The main use of ∗-autonomous categories is in the semantic study of Linear Logic. For this reason, it is thus natural to look for a ∗-autonomous category of locally convex topological vector spaces (tvs). On one hand, Linear Logic inherits its semantics from Linear Algebra, and it is thus natural to build models of Linear Logic from vector spaces [3,5,6,4]. On the other hand, denotational semantics has sought continuous models of computation through Scott domains [9]. Moreover, the infinite nature of the exponential of Linear Logic calls for infinite dimensional spaces, for which topology becomes necessary. One of the first intuitions that comes to mind when thinking about objects in a ∗-autonomous category is the notion of reflexive vector space, i.e. a a tvs which equals its double dual. When A is a vector space, the transpose dA : A→ (A → ⊥) → ⊥ of the evaluation map evA : (A → ⊥) × A → ⊥ is exactly the canonical injection of a vector space in its bidual. Then, requiring dA to be an isomorphism amounts to requiring A to be reflexive. However, the category of reflexive topological vector spaces is not ∗-autonomous, as it is not closed. Barr [2] constructs two closed subcategories of the category of tvs by restricting to tvs endowed with their weak topology (wtvs) or with their Mackey topology (mtvs), which are both polar topologies. Indeed, if E is a tvs, one can define its dual E′ as the space of all continuous linear form on E. Enforcing E with its weak or with its Mackey topology doesn’t change E′. The weak topology is exactly the coarsest among the polar topologies, while the Mackey topology is the finest.