## A Theory of Recursive Domains with Applications to Concurrency (1997)

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Venue: | In Proc. of LICS ’98 |

Citations: | 24 - 14 self |

### BibTeX

@INPROCEEDINGS{Cattani97atheory,

author = {Gian Luca Cattani and Marcelo P. Fiore and Glynn Winskel},

title = {A Theory of Recursive Domains with Applications to Concurrency},

booktitle = {In Proc. of LICS ’98},

year = {1997},

pages = {214--225},

publisher = {IEEE Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2-categorical theory for recursively defined domains.

### Citations

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Citation Context ...for every A; B 2 j K j, equipped with identity functors 1 ! K(C;C) for every C 2 j K j, and composition functors K(B;C) \Theta K(A;B) ! K(A;C) for every A; B;C 2 j K j, subject to the usual laws (see =-=[15]-=-). As a convention, the action of the composition functors is denoted by juxtaposition. Also, for objects A; B 2 j K j, objects and morphisms of the hom-category K(A;B) are respectively called morphis... |

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Citation Context ... with the approach to Axiomatic Domain Theory adopted in [6, 24, 9, 10]. Conceptually, the categorical theory of domains that we put forward may be seen as the traditional theory of Smyth and Plotkin =-=[28]-=- where !-cpos (!-complete partial orders) are replaced with their categorical analogue (viz. small categories with colimits of !-chains). Technically, this 1 is not straightforward. For example, the c... |

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Citation Context ...We present a central result of the paper, namely a generalisation of the local characterisation of colimits of !-chains of embeddings in Cpo-categories [28] which yields the limit-colimit coincidence =-=[26]-=-. We generalise in two directions. First, we move from the notion of embedding-projection pair in a Cpo-category (viz. coreflection, in the categorical jargon) to consider adjunctions in an !Cat-categ... |

122 | Bisimulation from open maps
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Citation Context ...lating models [33]. The presentation of models for concurrency as categories makes possible a general definition of bisimulation based on open maps, once a distinguished subcategory of paths is given =-=[14]-=-. This definition of bisimulation suggested a much broader class of models for concurrency built directly from the categories of paths---presheaf models. The Yoneda embedding provides each presheaf ca... |

103 | Parametricity and local variables
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Citation Context ...trict initial object (typically denoted ?) to it. Recursive-type constructor: We use the results of Section 3, and take parameterisedsfree pseudo algebras (see [6]). 5 Relational structures Following =-=[20, 22]-=-, we consider relational structures in the spirit of categorical logic [16] (c.f. [11]). A relational structure R on an !Cat 0 -category K induces a !Cat 0 -category of relations fK j Rg with objects ... |

101 | Relational properties of domains
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Citation Context ...trict initial object (typically denoted ?) to it. Recursive-type constructor: We use the results of Section 3, and take parameterisedsfree pseudo algebras (see [6]). 5 Relational structures Following =-=[20, 22]-=-, we consider relational structures in the spirit of categorical logic [16] (c.f. [11]). A relational structure R on an !Cat 0 -category K induces a !Cat 0 -category of relations fK j Rg with objects ... |

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Citation Context ...unctors A n En // A n 'n \Gamma! A y A \Gamma! b A : The same construction yields pseudo colimits in Prof M . 3 Pseudo-algebraic compactness Algebraic compactness is a universal property due to Freyd =-=[7]-=- that provides canonical interpretations of recursive domains. In this section we show this property for so-called Kcats; these may be seen as a 2-categorical analogue 6 of !-cppos (!-complete pointed... |

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Citation Context ...gory of !-cpos and partial !-continuous functions, with hom-sets ordered pointwise), Prof , and Prof M . From Theorem 2.1, Corollary 2.2, and Lemma 3.1, we can deduce pseudoalgebraic compactness (see =-=[8, 6]-=-). Corollary 3.2 (Pseudo-algebraic compactness) Kcats are pseudo-algebraically compact with respect to pseudo !Cat-functors. Thus, every pseudo !Cat-functor T : K op \Theta K ! K on a Kcat K has a fre... |

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Citation Context ...= ! + , and Set I -bimodules where I is the category of finite cardinals and injections. The latter is a promising setting for working out a theory of presheaf models for higher-order process calculi =-=[25, 31]-=- with features of local name creation and name passing as in the -calculus. 20 Aiming at a general treatment of higher order concurrent process calculi, a clear next step is to extend the intensional ... |

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Citation Context ...e use the results of Section 3, and take parameterisedsfree pseudo algebras (see [6]). 5 Relational structures Following [20, 22], we consider relational structures in the spirit of categorical logic =-=[16]-=- (c.f. [11]). A relational structure R on an !Cat 0 -category K induces a !Cat 0 -category of relations fK j Rg with objects fC j Rg given 9 by an object C of K together with a relation R on it, maps ... |

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Citation Context ...= ! + , and Set I -bimodules where I is the category of finite cardinals and injections. The latter is a promising setting for working out a theory of presheaf models for higher-order process calculi =-=[25, 31]-=- with features of local name creation and name passing as in the -calculus. 20 Aiming at a general treatment of higher order concurrent process calculi, a clear next step is to extend the intensional ... |

45 | Presheaf models for concurrency
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Citation Context ...ategory Prof of colimit-preserving functors between presheaf categories (with natural transformations as 2-cells). The key facts here are: open maps and so bisimulation are preserved by such functors =-=[4]-=-; the 2-category is rich in constructions which can be summarised as those we expect from a model of classical linear logic [32, 3]; open maps are closed under a wide range of constructions [12, 13]. ... |

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Citation Context ...nctors [4]; the 2-category is rich in constructions which can be summarised as those we expect from a model of classical linear logic [32, 3]; open maps are closed under a wide range of constructions =-=[12, 13]-=-. We have the basics of a domain theory for concurrency with a compositional account of bisimulation, though at a cost; we have to move domain theory up a level to handle categories rather than just p... |

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Citation Context ...Delta; R) : Clearly, \Delta is a TR -bisimulation. Moreover, if TR (R; \Delta) = TR (\Delta; \Delta) (3) for all R 2 R(D), then free pseudo dialgebras satisfy the following coinduction property (c.f. =-=[21, 22, 5, 11]-=-): \Delta = W fR 2 R(D) j R is a TR -bisimulationg : Notice that the requirement (3) is vacuous when T # is essentially covariants; that is, when it factors through an endofunctor on fK j Rg via the p... |

37 | A coinduction principle for recursive data types based on bisimulation
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Citation Context ...erpretations of recursive domains. In this section we show this property for so-called Kcats; these may be seen as a 2-categorical analogue 6 of !-cppos (!-complete pointed partial orders). Following =-=[5]-=-, our approach is to obtain the result from the Local-Characterisation and Limit-Colimit Coincidence Theorems, together with the Basic Lemma [28]. Recall that the Basic Lemma provides conditions under... |

37 |
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Citation Context ...itional Cpo-enriched setting and make a start on applications to concurrency. The generalisation of domain theory from order-theoretic structures to category-theoretic ones has been considered before =-=[18, 19, 30, 1]-=-. In particular, Paul Taylor [30] investigated the limit-colimit coincidence for categories with filtered colimits. In some respects his work is a precursor to ours; however, we take a step further an... |

33 | A Presheaf Semantics of Value-Passing Processes
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Citation Context ...s here are: open maps and so bisimulation are preserved by such functors [4]; the 2-category is rich in constructions which can be summarised as those we expect from a model of classical linear logic =-=[32, 3]-=-; open maps are closed under a wide range of constructions [12, 13]. We have the basics of a domain theory for concurrency with a compositional account of bisimulation, though at a cost; we have to mo... |

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25 |
Fibrations in bicategories, Cahiers de topologie et géométrie différentielle 21
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Citation Context ...ect), the morphisms are functors that preserve these colimits, and the 2-cells are natural transformations. 2 Concerning exactness properties in 2-categories we will focus on bicategorical (co)limits =-=[29]-=-. We exemplify this notion with the most basic example. Bicategorical (or pseudo) initial object. An object 0 in a 2-category is said to be pseudo initial if, for every object C, there exists a morphi... |

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Citation Context ...s whose denotations are locally countable presheaves if we generalise the results here from !-colimits to ! 1 -colimits. This would follow the pioneering work on countable nondeterminism described in =-=[23]-=-. Of course an even greater degree of branching would require even larger cardinals. 18 7.2 Intensional relations We consider the following extension of the grammar in (4): t ::= # j P i2I t i j t ? j... |

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17 | An extension of models of axiomatic domain theory to models of synthetic domain theory
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Citation Context ...tered colimits. In some respects his work is a precursor to ours; however, we take a step further and develop an axiomatic theory in accordance with the approach to Axiomatic Domain Theory adopted in =-=[6, 24, 9, 10]-=-. Conceptually, the categorical theory of domains that we put forward may be seen as the traditional theory of Smyth and Plotkin [28] where !-cpos (!-complete partial orders) are replaced with their c... |

17 | Cuboidal sets in axiomatic domain theory
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Citation Context ...tered colimits. In some respects his work is a precursor to ours; however, we take a step further and develop an axiomatic theory in accordance with the approach to Axiomatic Domain Theory adopted in =-=[6, 24, 9, 10]-=-. Conceptually, the categorical theory of domains that we put forward may be seen as the traditional theory of Smyth and Plotkin [28] where !-cpos (!-complete partial orders) are replaced with their c... |

16 | Axiomatic Domain Theory
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Citation Context ...tered colimits. In some respects his work is a precursor to ours; however, we take a step further and develop an axiomatic theory in accordance with the approach to Axiomatic Domain Theory adopted in =-=[6, 24, 9, 10]-=-. Conceptually, the categorical theory of domains that we put forward may be seen as the traditional theory of Smyth and Plotkin [28] where !-cpos (!-complete partial orders) are replaced with their c... |

16 |
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Citation Context ... n : A n ,! A n for the embedding of the category A n into its Cauchy completion A n , and let E n : c A n ' \Gamma! c A n denote the induced equivalence of categories. It is known (see, for example, =-=[17, 2]-=-) that there exist functors H n : A n ! A n+1 such that F n can be seen as a left Kan extension as follows: F n = Lan yn (E n+1 H n E n ) : Let ' n : A n ! A be a colimit of the !-chain hH n : A n ! A... |

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Citation Context ...s here are: open maps and so bisimulation are preserved by such functors [4]; the 2-category is rich in constructions which can be summarised as those we expect from a model of classical linear logic =-=[32, 3]-=-; open maps are closed under a wide range of constructions [12, 13]. We have the basics of a domain theory for concurrency with a compositional account of bisimulation, though at a cost; we have to mo... |

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Citation Context ...itional Cpo-enriched setting and make a start on applications to concurrency. The generalisation of domain theory from order-theoretic structures to category-theoretic ones has been considered before =-=[18, 19, 30, 1]-=-. In particular, Paul Taylor [30] investigated the limit-colimit coincidence for categories with filtered colimits. In some respects his work is a precursor to ours; however, we take a step further an... |

8 |
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Citation Context ...esults of Section 3, and take parameterisedsfree pseudo algebras (see [6]). 5 Relational structures Following [20, 22], we consider relational structures in the spirit of categorical logic [16] (c.f. =-=[11]-=-). A relational structure R on an !Cat 0 -category K induces a !Cat 0 -category of relations fK j Rg with objects fC j Rg given 9 by an object C of K together with a relation R on it, maps are require... |

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Citation Context ...alise in two directions. First, we move from the notion of embedding-projection pair in a Cpo-category (viz. coreflection, in the categorical jargon) to consider adjunctions in an !Cat-category (c.f. =-=[30, 27]-=-). Next, for the reasons exposed above, we consider bicategorical and pseudo-colimits rather than strict ones. We start by recalling some definitions and fixing notation. Adjunctions. Let K be a 2-cat... |

3 |
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Citation Context ...itional Cpo-enriched setting and make a start on applications to concurrency. The generalisation of domain theory from order-theoretic structures to category-theoretic ones has been considered before =-=[18, 19, 30, 1]-=-. In particular, Paul Taylor [30] investigated the limit-colimit coincidence for categories with filtered colimits. In some respects his work is a precursor to ours; however, we take a step further an... |

2 |
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