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18
An Algebraic Presentation of Term Graphs, via GSMonoidal Categories
 Applied Categorical Structures
, 1999
"... . We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particula ..."
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Cited by 38 (25 self)
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. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are onetoone with the arrows of the free gsmonoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gsmonoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...
An Axiomatization of the Algebra of Petri Net Concatenable Processes
 Theoretical Computer Science
, 1996
"... . The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category P(N ). However, this is only a partial axiomatization, since it is based on a concrete, ad hoc chosen, category of symmetries Sym N . In this paper we give a completely abstr ..."
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Cited by 19 (8 self)
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. The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category P(N ). However, this is only a partial axiomatization, since it is based on a concrete, ad hoc chosen, category of symmetries Sym N . In this paper we give a completely abstract characterization of the category of concatenable processes of N , thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets. Introduction Concatenable processes of Petri nets have been introduced in [3] to account, as their name indicates, for the issue of process concatenation. Let us briefly reconsider the ideas which led to their definition. The development of theory Petri nets, focusing on the noninterleaving aspects of concurrency, brought to the foreground various notions of process, e.g. [14, 5, 2, 12, 3]. Generally speaking, Petri net processeswhose standard version is given by the GoltzReisig nonsequential processes [5]are structures needed to acc...
Theory and Applications of Crossed Complexes
, 1993
"... ... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for t ..."
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Cited by 15 (2 self)
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... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for the chain complex version. Given simplicial sets K 0 ; : : : ; K r , we discuss the rcube of homotopies induced on (K 0 \Theta : : : \Theta K r ) and show these form a coherent system. We introduce a definition of a double crossed complex, and of the associated total (or codiagonal) crossed complex. We introduce a definition of homotopy colimits of diagrams of crossed complexes. We show that the homotopy colimit of crossed complexes can be expressed as the
An axiomatization of the category of Petri net computations
 Math. Struct. in Comput. Sci
, 1998
"... Abstract. We introduce the notion of strongly concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a ..."
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Cited by 13 (5 self)
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Abstract. We introduce the notion of strongly concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a
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...
1 INTRODUCTION TO ANNCATEGORIES
, 2007
"... Abstract. In this paper, we present new concepts of Anncategories, Annfunctors, and a transmission of the structure of categories based on Annequivalences. We build Anncategory ..."
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Cited by 6 (4 self)
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Abstract. In this paper, we present new concepts of Anncategories, Annfunctors, and a transmission of the structure of categories based on Annequivalences. We build Anncategory
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...
Strong Concatenable Processes: An Approach to the Category of Petri Net Computations
 BRICSComputer Science Department, University of Aarhus
, 1994
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS ..."
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Cited by 4 (1 self)
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS
Axiomatizing Petri Net Concatenable Processes
, 1995
"... . The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category P[N ]. Yet, this is only a partial axiomatization, since P[N ] is built on a concrete, ad hoc chosen, category of symmetries. In this paper we give a fully equational desc ..."
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Cited by 3 (3 self)
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. The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category P[N ]. Yet, this is only a partial axiomatization, since P[N ] is built on a concrete, ad hoc chosen, category of symmetries. In this paper we give a fully equational description of the category of concatenable processes of N , thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets. Introduction C oncatenable processes of Petri nets have been introduced in [3] to account, as their name indicates, for the issue of process concatenation. Let us briefly reconsider the ideas which led to their definition. The development of theory Petri nets, focusing on the noninterleaving aspects of concurrency, brought to the foreground various notions of process, e.g. [14, 5, 2, 12, 3]. Generally speaking, Petri net processeswhose standard version is given by the GoltzReisig nonsequential processes [5]are structures needed to account for th...