From the interpretation of Linear Logic multiplicative disjunction as the epsilon product defined by Laurent Schwartz, we construct several models of Differential Linear Logic based on the usual mathematical notions of smooth maps. This improves on previous results by Blute, Ehrhard and Tasson based on convenient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-completeness, and an associated notion which is stable under duality called k-reflexivity, allowing for a star-autonomous category of k-reflexive spaces in which the dual of the tensor product is the reflexive version of the epsilon-product. We adapt Meise's definition of smooth maps into a first model of Differential Linear Logic, made of k-reflexive spaces. We also build two new models of Linear Logic with conveniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions). Varying slightly the notion of smoothness, one also recovers models of DiLL on the same star-autonomous categories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the epsilon-product (without reflexivization).
Keywords: Topological vector spaces, $\ast$-autonomous and dialogue categories, differential linear logic
2010 MSC: 03B47, 18C50, 18D15, 46A20, 46M05, 46E50, 68Q55
Theory and Applications of Categories, Vol. 34, 2019, No. 45, pp 1440-1525.