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32
Identities among relations for higherdimensional rewriting systems
"... Abstract – We generalize the notion of identities among relations, well known for presentations of groups, to presentations of ncategories by polygraphs. To each polygraph, we associate a track ncategory, generalizing the notion of crossed module for groups, in order to define the natural system o ..."
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Abstract – We generalize the notion of identities among relations, well known for presentations of groups, to presentations of ncategories by polygraphs. To each polygraph, we associate a track ncategory, generalizing the notion of crossed module for groups, in order to define the natural system of identities among relations. We relate the facts that this natural system is finitely generated and that the polygraph has finite derivation type. Support – This work has been partially supported by ANR Inval project (ANR05BLAN0267).
The Structure of FirstOrder Causality
"... Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in firstorder propositional logic. One of the main difficulties that has to be fac ..."
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Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in firstorder propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by firstorder quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages. Denotational semantics were introduced to provide useful abstract invariants of proofs and programs modulo cutelimination or reduction. In particular, game semantics, introduced in the nineties, have been very successful in capturing precisely the interactive behaviour of programs. In these semantics, every type is interpreted as a game (that is as a set of moves that can be played during the game) together with the rules of the game (formalized by a partial order on the moves of the game indicating the dependencies between them). Every move is to be played by one of the two players, called Proponent and Opponent, who should be thought respectively as the program and its environment. A program is characterized by the sequences of moves that it can exchange with its environment during an
A NonStandard Semantics for Kahn Networks in Continuous Time
"... In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginn ..."
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In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginning, process networks have been drawn as particular graphs, but this syntax is never formalized. We take the opportunity to clarify it by giving a precise definition of these graphs,
Diagram rewriting and operads
, 2009
"... We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for prac ..."
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We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for practical computations, but also for theoretical results. Moreover, rewriting is strongly related to homotopy theory. For instance, it can be used to compute homological invariants of algebraic structures, or to prove coherence results.
A folk model structure on omegacat
, 2009
"... The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generat ..."
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The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while free objects are cofibrant. We further exhibit model structures of this type on ncategories for arbitrary n ∈ N, as specialisations of the ωcategorical one along right adjoints. In particular, known cases for n = 1 and n = 2 nicely fit into the scheme.
Computing Critical Pairs in Polygraphs
 In Workshop on Computer Algebra Methods and Commutativity of Algebraic Diagrams (CAMCAD
, 2009
"... Polygraphs generalize to 2categories the usual notion of equational theory, by describing them as quotients, modulo equations, of freely generated 2categories on a given set of generators. In order to work with morphisms modulo the equations, it is often convenient to orient the equations into a c ..."
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Polygraphs generalize to 2categories the usual notion of equational theory, by describing them as quotients, modulo equations, of freely generated 2categories on a given set of generators. In order to work with morphisms modulo the equations, it is often convenient to orient the equations into a confluent rewriting system. In the case of a terminating system, confluence can be checked by showing that critical pairs are joinable. However, the computation of the critical pairs is more complicated for polygraphs than for term rewriting systems: in particular, two left members of a rule don’t necessarily have a finite number of unifiers. We advocate here that a more general notion of rewriting system should be considered instead, and introduce an operad of compact contexts in a 2category, in which two rules have a finite number of unifiers. A concrete representation of contexts is proposed, as well as an unification algorithm for these.
Formal Relationships Between Geometrical and Classical Models for Concurrency
"... Abstract. A wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical fla ..."
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Abstract. A wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical flavor have been introduced, based on the notion of cubical set. These models are very rich and expressive since they can represent commutation between any number of events, thus generalizing the principle of true concurrency. While they seem to be very promising – because they make possible the use of techniques from algebraic topology in order to study concurrent computations – they have not yet been precisely related to the previous models, and the purpose of this paper is to fill this gap. In particular, we describe an adjunction between Petri nets and cubical sets which extends the previously known adjunction between Petri nets and asynchronous transition systems by Nielsen and Winskel. 1 1
Coherence in monoidal track categories
"... Abstract – We introduce homotopical methods based on rewriting on higherdimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and use rewriting ..."
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Abstract – We introduce homotopical methods based on rewriting on higherdimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and use rewriting methods on polygraphs to solve it. The setting is generalized to more general coherence problems, seen as 3dimensional word problems in a track category. We prove general results that, in the case of braided monoidal categories, yield the coherence theorem for braided monoidal categories.
Polygraphic resolutions and homology of monoids
, 2007
"... We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZMmodules. 1 ..."
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We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZMmodules. 1
A Combinatorial Definition of BaezDolan ωCategories
"... We give a new combinatorial definition of a sort of weak ! category originally devised by J. Baez and J. Dolan in finite dimensional cases. Our definition is a mixture of both inductive and coinductive definitions, and suitable for `computational category theory.' Keyword: weak ncategory, bica ..."
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We give a new combinatorial definition of a sort of weak ! category originally devised by J. Baez and J. Dolan in finite dimensional cases. Our definition is a mixture of both inductive and coinductive definitions, and suitable for `computational category theory.' Keyword: weak ncategory, bicategory, tricategory, formalized mathematics 1