Theory and Applications of Categories, Vol. 34, No. 45, 2019, pp. 1440–1525. MODELS OF LINEAR LOGIC BASED ON THE SCHWARTZ "-PRODUCT. YOANN DABROWSKI AND MARIE KERJEAN Abstract. 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 no- tion 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 con- veniently 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 cate- gories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the epsilon-product (without reflexivization). Contents 1 Introduction 1441 I. Models of MALL 1447 2 Preliminaries 1447 3 The original setting for the Schwartz "-product and smooth maps. 1457 4 Models of MALL coming from classes of smooth maps 1471 5 Schwartz locally convex spaces, Mackey-completeness and the ρ-dual. 1485 II. Models of LL and DiLL 1493 6 Smooth maps and new models of LL 1493 7 Models of DiLL 1502 8 Conclusion 1517 9 Appendix 1518 The authors are grateful to Richard Blute for his numerous suggestions and grammar corrections. Received by the editors 2018-04-19 and, in final form, 2019-12-16. Transmitted by Richard Blute. Published on 2019-12-18. 2010 Mathematics Subject Classification: 03B47, 18C50, 18D15, 46A20, 46M05, 46E50, 68Q55 . Key words and phrases: Topological vector spaces, ∗-autonomous and dialogue categories, differential linear logic. c Yoann Dabrowski and Marie Kerjean, 2019. Permission to copy for private use granted. 1440 MODELS OF LINEAR LOGIC BASED ON THE SCHWARTZ "-PRODUCT. 1441 1. Introduction Smooth models of classical Linear Logic. Since the discovery of linear logic by Girard [Gir87], thirty years ago, many attempts have been made to obtain denotational models of linear logic in the context of categories of vector spaces with linear proofs interpreted as linear maps [Blu96, Ehr02, Gir04, Ehr05, BET]. Models of linear logic are often inspired by coherent spaces, or by the relational model of linear logic. Coherent Banach spaces [Gir99], coherent probabilistic or coherent quantum spaces [Gir04] are Girard’s attempts to extend the first model, as finiteness spaces [Ehr05] or Kothe¨ spaces [Ehr02] were designed by Ehrhard as a vectorial version of the relational model. Following the construction of Differential linear logic [ER06], one would want moreover to find natural models of it where non-linear proofs are interpreted by some classes of smooth maps. This requires the use of more general objects of functional anal- ysis which were not directly constructed from coherent spaces. We see this as a strong point, as it paves the way towards new computational interpretations of functional analytic constructions, and a denotational interpretation of continuous or infinite data objects. A consequent categorical analysis of the theory of differentiation was tackled by Blute, Cockett and Seely [BCS06, BCS09]. They gave several structures in which a differentiation operator is well-behaved. Their definition then restricts to models of Intuitionistic Differential Linear Logic. Our paper takes another point of view as we look for models of classical DiLL, in which spaces equal some double dual. We want to emphasize the classical computational nature of Differential Linear Logic. Three difficulties appear in this semantical study of linear logic. The equivalence between a formula and its double negation in linear logic asks for the considered vector spaces to be iso- morphic to their double duals. This is constraining in infinite dimension. This infinite dimen- sionality is strongly needed to interpret exponential connectives. Then one needs to find a good category with smooth functions as morphisms, which should give a cartesian closed category. This is not at all trivial, and was solved by using a quantative setting, i.e. power series as the in- terpretation for non-linear proofs, in most of the previous works [Gir99, Gir04, Ehr05, Ehr02]. Finally, imposing a reflexivity condition to respect the first requirement usually implies issues of stability by natural tensor products of this condition, needed to model multiplicative connec- tives. This corresponds to the hard task of finding ∗-autonomous categories [Ba79]. As pointed out in [Ehr16], the only model of differential linear logic using smooth maps [BET] fails to satisfy the ∗-autonomous property for classical linear logic. Our paper solves all these issues simultaneously and produces several denotational models of classical linear logic with some classes of smooth maps as morphism in the Kleisli cate- gory of the monad. We will show that the constraint of finding a ∗-autonomous category in a compatible way with a cartesian closed category of smooth maps is even relevant to find better mathematical notions of smooth maps in locally convex spaces. Let us explain this mathemati- cal motivation first. A framework for differential calculus. It seems that, historically, the development of differential calculus beyond normed spaces suffered from the lack of interplay between analytic considerations and categorical, synthetic or logical ones. As a consequence, analysts often gave 1442 YOANN DABROWSKI AND MARIE KERJEAN up looking for good properties stable under duality and focused on one side of the topological or bornological viewpoint. An analytic summary of the early theory can be found in Keller’s book [Kel]. It already gives a unified and simplified approach based on continuity conditions of derivatives in various senses. But it is well-known that in order to look for strong categorical properties such as carte- sian closedness, the category of continuous maps is not a good starting point, the category of maps continuous on compact sets would be better. This appears strongly in all the developments made to recover continuity of evaluation on the topological product (instead of considering the product of a cartesian closed category), which is unavoidable for full continuity of composition of derivatives in the chain rule. This leads to considering convergence notions beyond topo- logical spaces on spaces of linear maps, but then, no abstract duality theory of those vector convergence spaces or abstract tensor product theory is developed. In the end, everything goes well only on restricted classes of spaces that lack almost any categorical stability properties, and nobody understands half of the notions introduced. The situation became slightly better when [Me] considered k-space conditions and obtained what analysts call kernel representation theorems (Seely isomorphisms for linear logicians), but still the only classes stable by products were Frechet´ spaces and (DFM)-spaces, which are by their very nature not stable under duality. The general lesson here is that, if one wants to stay within better studied and commonly used locally convex spaces, one had better not stick to functions continuous on products, and the corresponding projective topological tensor product, but always take tensor products that come from a ∗-autonomous category, since one also needs duality, or at least a closed category, to control the spaces of linear maps in which the derivatives take values. ∗-autonomous categories are the better behaved categories having all those data. Ideally, following developments inspired by game semantics [MT], we will be able to get more flexibility and allow larger dialogue categories containing such ∗-autonomous categories as their category of continuation. We will get slightly better categorical properties on those larger categories. A better categorical framework was later found and summarized in [FK,KM] the so-called convenient smoothness. A posteriori, as seen in [Ko], the notion is closely related to synthetic differential geometry as diffeological spaces are. It chooses a very liberal notion of smoothness, that does not imply continuity except on very special compact sets, the images of finite dimen- sional compact sets by smooth maps. It gives a nice cartesian closed category and this enabled [BET] to obtain a model of intuitionistic differential linear logic. As we will see, this may give the wrong idea that this very liberal notion of smoothness is the only way of getting cartesian closedness and it also takes the viewpoint of focusing on bornological properties. This is the main reason why, in our view, they don’t obtain ∗-autonomous categories since bornological locally convex spaces have complete duals which gives an asymmetric requirement on duals since they only need a much weaker Mackey-completeness on their spaces to work with their notion of smooth maps. We will obtain in this paper several models of linear logic using con- veniently smooth maps, and we will explain logically this Mackey-completeness