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Cubical Sets And Their Site
 Theory Appl. Categ
, 2003
"... Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicia ..."
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Cited by 15 (3 self)
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Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid ; in fact, K is the classifying category of a monoidal algebraic theory of such monoids. Analogous results are given for the restricted cubical site I of ordinary cubical sets (just faces and degeneracies) and for the intermediate site J (including connections). We also consider briefly the reversible analogue, !K.
First Order Linear Logic in Symmetric Monoidal Closed Categories
, 1991
"... There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encodin ..."
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Cited by 11 (0 self)
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There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.
Explicit Substitution Internal Languages for Autonomous and *Autonomous Categories
 In Proc. Category Theory and Computer Science (CTCS'99), Electron
, 1999
"... We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. We show tha ..."
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Cited by 7 (2 self)
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We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. We show that the eight equality and three commutation congruence axioms of the autonomous type theory characterise autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. 1 Introduction In this paper we introduce a family of type theories which can be regarded as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the standard simplytyped calculus with surjective pairing is...
Type Theories for Autonomous and *Autonomous Categories: I. Type Theories and Rewrite Systems  II. Internal Languages and Coherence Theorems
, 1998
"... We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that simplytyped calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian clos ..."
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Cited by 5 (4 self)
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We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that simplytyped calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian closed categories. The rules are presented in the style of Gentzen's Sequent Calculus. A key feature is the systematic treatment of naturality conditions by explicitly representing the categorical composition, or cut in the type theory, by explicit substitution, and the introduction of new letconstructs, one for each of the three type constructors ?;\Omega and (, and a Parigotstyle ¯abstraction to give expression to the involutive negation. The commutation congruences of these theories are precisely those imposed by the naturality conditions. In particular the type theory for autonomous categories may be regarded as a term assignment system for the multiplicative (\Omega ; (;?;?)fragmen...
Physics, Topology, Logic and Computation: A Rosetta Stone
, 2009
"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."
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Cited by 5 (1 self)
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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps
A language for multiplicativeadditive linear logic
 In Proc. Category Theory in Computer Science 2004, ENTCS 122 (2005
"... A term calculus for the proofs in multiplicativeadditive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive categories with additives. It is also shown that proof equivalence ..."
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Cited by 5 (2 self)
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A term calculus for the proofs in multiplicativeadditive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive categories with additives. It is also shown that proof equivalence is decidable by showing that the cut elimination rewrites supply a confluent rewriting system modulo equations. 0
HigherOrder Categorical Grammars
 Proceedings of Categorial Grammars 04
"... into two principal paradigms: modeltheoretic syntax (MTS), which ..."
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Cited by 4 (1 self)
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into two principal paradigms: modeltheoretic syntax (MTS), which
Covert Movement in Logical Grammar
"... From the mid1970s until the emergence of Chomsky’s Minimalist Program (MP, Chomsky 1995) in the 1990s, the mainstream of research on naturallanguage syntax in much of the world embraced a theoretical architecture for syntactic derivations that came to be known as the Tmodel. According to this mod ..."
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Cited by 2 (1 self)
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From the mid1970s until the emergence of Chomsky’s Minimalist Program (MP, Chomsky 1995) in the 1990s, the mainstream of research on naturallanguage syntax in much of the world embraced a theoretical architecture for syntactic derivations that came to be known as the Tmodel. According to this model,
Internal Languages for Autonomous and *Autonomous Categories
"... We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. The rules for the typing judgeme ..."
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Cited by 1 (0 self)
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We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. The rules for the typing judgements are presented in the style of Gentzen's Sequent Calculus. A notable feature is the systematic treatment of naturality conditions by expressing the categorical composition, or cut in the type theory, by explicit substitution. We use letconstructs, one for each of the three type constructors ?;\Omega and (, to witness the leftintroduction rules, and a Parigotstyle ¯abstraction to express the involutive negation ?. We show that the eight equality and three commutation congruence axioms of the autonomous type theory characterise autonomous categories exactly. More precisely we prove that there is a canonical interpretation of the (autonomous) type theories in autonomous categorie...