We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed -calculus and dierential calculus can be combined; we give a few examples of computations. 1

modality”) and a differential combinator, satisfying a number of coherence conditions. In

Abstract. In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigating differential structure in the context of linear logic. Consequently, within this setting, I introduce a notion of creation operator (as considered by physicists for bosonic Fock space in the context of quantum field theory), provide an equivalent description of creation operators in terms of creation maps, and show that they induce a differential operator satisfying all the basic laws of differentiation (the product and chain rules, the commutation relations, etc.). 1

This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and ∗-autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic. No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus. 0

We study the Taylor expansion of lambda-terms in a non-deterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambda-calculus we introduced recently. This operation is not possible in the general untyped case where reduction can produce unbounded coefficients. We endow resource terms with a finiteness structure (in the sense of our earlier work on finiteness spaces) and show that the Taylor expansions of terms typeable in Girard’s system F are finitary by a reducibility method.

We introduce the notion of differential λ-category as an extension of Blute-Cockett-Seely’s differential Cartesian categories. We prove that differential λ-categories can be used to model the simply typed versions of: (i) the differential λ-calculus, a λ-calculus extended with a syntactic derivative operator; (ii) the resource calculus, a non-lazy axiomatisation of Boudol’s λ-calculus with multiplicities. Finally, we provide two

Abstract. We define an extension of λ-calculus with linear combinations, endowing the set of terms with a structure of R-module, where R is a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector space. We then extend β-reduction on those algebraic λ-terms as follows: at + u reduces to at ′ + u as soon as term t reduces to t ′ and a is a non-zero scalar. We prove that reduction is confluent. Under the assumption that the set R of scalars is positive (i.e. a sum of scalars is zero iff all of them are zero), we show that this algebraic λcalculus is a conservative extension of ordinary λ-calculus. On the other hand, we show that if R admits negative elements, then every term reduces to every other term. We investigate the causes of that collapse, and discuss some possible fixes. Preliminary definitions and notations. Recall that a rig (also known as “semiring with zero and unit”) is the same as a ring, without the condition that every element admits an opposite for addition. Let R be a rig. We write R • for R \ {0}. We denote by letters a, b, c the elements of R, and say that R is positive if, for all a, b ∈ R, a + b = 0 implies a = 0 and b = 0. An example of positive rig is N, the set of natural numbers, with usual addition and multiplication. If i, j ∈ N, we write [i; j] for the set {k ∈ N; i ≤ k ≤ j}. Also, we write application of λ-terms à la Krivine: (s)t denotes the application of term s to term t. 1

of linear logic

We define an extension of Herbelin’s ¯ λµ-calculus, introducing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new semantical constructions, extending models of Ehrhard-Regnier’s differential interaction nets, along the lines of Laurent’s polarization of linear logic. We demonstrate this correspondence by providing this calculus with a denotational semantics inside a lambda-model in the category of sets and relations. 1

a comonad (a “coalgebra modality”) and a differential combinator. The morphisms of a differential category should be thought of as the linear maps; the differentiable or smooth maps would then be morphisms of the coKleisli category. The purpose of the present paper is to directly axiomatize differentiable maps and thus to move the emphasis from the linear notion to structures resembling the coKleisli category. The result is a setting with a more evident and intuitive relationship to the familiar notion of calculus on smooth maps. Indeed a primary example is the category whose objects are Euclidean spaces and whose morphisms are smooth maps.