We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored -categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored -categories, all morphisms are nuclear, and in the tensored -category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored -categories, in which integration plays the role of composition. In the first, mor...

We present a denotational semantics based on Banach spaces; it is inspired from the familiar coherent semantics of linear logic, the role of coherence being played by the norm: coherence is rendered by a supremum, whereas incoherence is rendered by a sum, and cliques are rendered by vectors of norm at most 1. The basic constructs of linear (and therefore intuitionistic) logic are implemented in this framework: positive connectives yield ℓ 1-like norms and negative connectives yield ℓ ∞-like norms. The problem of non-reflexivity of Banach spaces is handled by specifying the dual in ¡ advance, whereas the exponential connectives (i.e. intuitionistic implication) are handled by means of analytical functions on the open unit ball. The fact that this ball is open (and not closed) explains the absence of a simple solution to the question of a topological cartesian closed

The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the self-dual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambda-calculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. 1 Introduction Linear logic was introduced by J.-Y. Girard as a "logic behind logic." It separates logical reasoning into a core linear part in which formulas are merely moved around, and an auxiliary nonlinear part in which formulas may be deleted and copied. The core, multiplicative linear logic (MLL), is a substructural logic whose basic connect...

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 give a definition of categorical model for the multiplicative fragment of non-commutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cut-elimination. We then focus on several methods of building entropic categories. Our first models are constructed via the notion of a partial bimonoid acting on a cocomplete category. We also explore an entropic version of the Chu construction, and apply it in this setting. It has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic. We extend these ideas to the entropic setting by developing a new type of Hopf algebra, which we call entropic Hopf algebras. We show that the category of modules over an entropic Hopf algebra is an entropic category, (possibly after application of the Chu construction). Several examples are discussed, based first on the notion of a bigroup. Finally the Tannaka-Krein reconstruction theorem is extended to the entropic setting.

We match up types and processes by putting values in correspondence with events, coproduct with (noninteracting) parallel composition, and tensor product with orthocurrence. We then bring types and processes into closer correspondence by broadening and unifying the semantics of both using Chu spaces and their transformational logic. Beyond this point the connection appears to break down; we pose the question of whether the failures of the corrrespondence are intrinsic or cultural. 1 Introduction Types-as-processes modernizes data-as-programs. It is the Curry-Howard propositions-as-types correspondence with propositions replaced by processes. To the extent that types and processes are both part of the working programmer 's toolkit, even more than propositions, the types-as-processes correspondence is more central to the practice of programming than propositions-astypes. Moreover the connection works out very well mathematically, at least up to a point. The similarities and differences ...

Most tensored \Lambda-categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored \Lambda-categories, all morphisms are nuclear, and in the tensored \Lambda-category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored \Lambda-categories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations.

Proof Theory is the result of a tumultuous history, developed on the periphery of mainstream mathematics. Hence, its language is often idiosyncratic: sequent calculus, cutelimination, subformula property, etc. This survey is designed to guide the novice reader and the itinerant mathematician on a smooth and engaging path through the subject – focused on the symbolic mechanisms of cut-elimination, and their transcription as coherence diagrams in categories with structure. This spiritual journey at the meeting point of linguistic and algebra is demanding at times, but unusually rewarding: to date, no language (formal or informal) has been studied as thoroughly in mathematics as the language of proofs. We start the survey by a short introduction to Proof Theory (Chapter 1) followed by an informal explanation of the principles of Denotational Semantics (Chapter 2) analogous to a Representation Theory for proofs, generating invariants modulo cutelimination. After describing in full detail the cut-elimination procedure of linear logic (Chapter 3), we explain how to transcribe it into the language of categories with structure. We review two alternative constructions of a ∗-autonomous category, or monoidal category with duality (Chapter 4). After giving a 2-categorical account of lax and colax monoidal adjunctions (Chapter 5) and recalling the notions of monoids and monads (Chapter 6) we relate four different categorical axiomatizations of propositional linear logic appearing in the litterature (Chapter 7).

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