## An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory (1997)

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Venue: | In Proceedings of CSL 96 |

Citations: | 17 - 6 self |

### BibTeX

@INPROCEEDINGS{Fiore97anextension,

author = {Marcelo Fiore and Gordon D. Plotkin},

title = {An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory},

booktitle = {In Proceedings of CSL 96},

year = {1997},

pages = {129--149},

publisher = {Springer LNCS}

}

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### Abstract

. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of non-elementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !-inductive fixed-point object (Definition 1...

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Citation Context ... T (B) ffi! T (A \Theta B) is an isomorphism with inverse T (A\ThetaB) hT1 ;T 2 i ! T (A)\ThetaT (B)ffiffi!T (A)\Omega T (B). Thus, the adjunction C \Gamma! ? /\Gamma C T is monoidal (in the sense of =-=[7]-=-). Linear exponentials , denoted with ffi) , are defined as the closed structure associated to tensor products (i.e. by the adjoint situation \Omega X a X ffi) : C T ! C T ). If C T has reflexive coeq... |

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Citation Context ... dominance ? : 1 ,! \Sigma , writing \Pi ? for the right adjoint to ? , we have, for every B 2 j C j, that (\Pi ?B) (?)s= B; hence obtaining a pullback diagram B ! 1 LB jB # " \Pi ?B ! \Sigma # &=-=quot; ? : (1) Moreover,-=- the ?-subobject jB : B ,! LB in (1) is a classifier of ?-partial maps with target B, in that every ?-partial map [A /- U ! B] appears in a pullback U ! B A # " ! LB # " jB for a unique char... |

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Citation Context ...hes, for every C 2 j S j, a bijective correspondence ( \Theta C) \Gamma1 ( ) : \Sigma Sub(F \Theta C)s= S(F \Theta C; \Sigma )s= S(I \Theta C; \Sigma )s= \Sigma Sub(I \Theta C) f 7! f ffi ( \Theta C) =-=(3)-=- where \Sigma Sub(A) denotes the collection of ?-subobjects of A. We characterise well-completeness in terms of orthogonality and study closure properties of well-complete objects. Theorem 2.7. 1. (In... |

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Citation Context ... description of the lifting monad. 1.2 Tensor products and linear exponentials We summarise standard material on tensor products and linear exponentials in categories of Eilenberg-Moore algebras (see =-=[20, 21, 6, 18]-=-) and study them in categories of lift-algebras over presheaf toposes. Let T = (T; j; ; st) be a commutative monad on a cartesian category C. We write st 0 for the composite T (A) \Theta Bs= B \Theta ... |

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Citation Context ... impose a non-elementary axiom (Axiom 3) stating that the initial lift-algebra in the ambient topos is inductive, in that it arises as a colimit of the standard chain. 2.1 Orthogonality Orthogonality =-=[12]-=- plays a crucial role in understanding well-complete objects (see Theorem 2.7 (1)), in this subsection we recall the notion together with various useful observations about it. Definition 2.1 (Orthogon... |

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Citation Context ... description of the lifting monad. 1.2 Tensor products and linear exponentials We summarise standard material on tensor products and linear exponentials in categories of Eilenberg-Moore algebras (see =-=[20, 21, 6, 18]-=-) and study them in categories of lift-algebras over presheaf toposes. Let T = (T; j; ; st) be a commutative monad on a cartesian category C. We write st 0 for the composite T (A) \Theta Bs= B \Theta ... |

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Citation Context ...of small cpos and partial continuous functions [27]); it has partial products, Kleisli (or partial) exponentials and finite coproducts (all yielding Cpo-functors), and it is Cpo-algebraically compact =-=[8]-=-. This example is representative of the class of models that we consider in this paper in that the lifting monad L on Cpo allows us to recover the other data. Indeed, we have that Cppo ?s= Cpo L (the ... |

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Citation Context ... characterised by the equation (1 ! ! succ ! !) = (1 ! !). 2 Synthetic domain theory We develop a non-elementary version of SDT. On the one hand, we follow closely the approach of Longley and Simpson =-=[22, 23]-=- in that we adopt their completeness axiom (Axiom 1) and identify domains with well-complete objects (Definition 2.6 (2)); on the other hand, we depart notably from their approach in that we impose a ... |

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Citation Context ...r a unique characteristic map A ! \Sigma . A ?-subobject classifier is called a dominance if ? : C =\Sigma ! C has a right adjoint, and ?-subobjects are closed under composition. (For details consult =-=[30, 17]-=-). ut Remark. In the context of SDT, the notion of dominance is only defined by the second requirement, as the first requirement generally holds in the universes of sets considered (e.g. locally carte... |

45 |
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Citation Context ...categories) and of recursive types. Finally, in Section 1.5 we define the class of ADT models that we will be interested in. 1.1 Lifting We present an axiomatisation of lifting in terms of dominances =-=[30]-=- which is by now traditional; its connection with partiality is discussed briefly. (For details consult [8]). Definition 1.1. (Dominance [30]) Let C be a category with a terminal object 1. An object \... |

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Citation Context ...the cone hoei :s: ! ! is colimiting and (1 ! ! succ ! !) = (1 ! !) where succ def = (! ae j! ! L!s= !). ut The essential difference between our fixed-point object and the one previously considered in =-=[5]-=- is its inductive nature, in that we require it to arise as the colimit of the standard chain generated by iterating the lifting functor. Definition 1.12. 1. (Monadic base) A (lifting) monadic (enrich... |

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Citation Context ...stic map A ! LB. The following is well-known (see e.g. [17, 25, 8]): Proposition 1.2. In a cartesian category, dominances induce commutative monads.sut Remark. For the notion of commutative monad see =-=[20, 24]-=-. The underlying endofunctor of the monad induced by a dominance ? : 1 ,! \Sigma is given by the composite C \Pi ? ! C =\Sigma ! C where C =\Sigma ! C is the domain functor. Convention. We refer to th... |

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Citation Context ...xed-point object is the ordinal ! + 1 equipped with its limit point !. A typical model, with respect to the above enrichment base, is pCpo (the category of small cpos and partial continuous functions =-=[27]-=-); it has partial products, Kleisli (or partial) exponentials and finite coproducts (all yielding Cpo-functors), and it is Cpo-algebraically compact [8]. This example is representative of the class of... |

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Citation Context ...DT models based on Eilenberg-Moore algebras to SDT models we only have partial results (among which is Corollary 3.3). -- On the synthetic side, the trend has been to investigate realisability models =-=[30, 26, 13, 23]-=- (though see [30, 36, 14]). Here our main means for finding models of ADT within models of SDT (via well-complete objects [23]) is a non-elementary axiom (Axiom 3) that is not satisfied in that settin... |

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Citation Context ...DT models based on Eilenberg-Moore algebras to SDT models we only have partial results (among which is Corollary 3.3). -- On the synthetic side, the trend has been to investigate realisability models =-=[30, 26, 13, 23]-=- (though see [30, 36, 14]). Here our main means for finding models of ADT within models of SDT (via well-complete objects [23]) is a non-elementary axiom (Axiom 3) that is not satisfied in that settin... |

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Citation Context ... characterised by the equation (1 ! ! succ ! !) = (1 ! !). 2 Synthetic domain theory We develop a non-elementary version of SDT. On the one hand, we follow closely the approach of Longley and Simpson =-=[22, 23]-=- in that we adopt their completeness axiom (Axiom 1) and identify domains with well-complete objects (Definition 2.6 (2)); on the other hand, we depart notably from their approach in that we impose a ... |

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Citation Context ... description of the lifting monad. 1.2 Tensor products and linear exponentials We summarise standard material on tensor products and linear exponentials in categories of Eilenberg-Moore algebras (see =-=[20, 21, 6, 18]-=-) and study them in categories of lift-algebras over presheaf toposes. Let T = (T; j; ; st) be a commutative monad on a cartesian category C. We write st 0 for the composite T (A) \Theta Bs= B \Theta ... |

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Citation Context ...on ! + 1, and 2. the monoid of stable endomorphisms on (! + 1) \Theta 2 (where 2 denotes the Sierpinski space) are GSDT models. Remark. Refinements of these models have been introduced and studied in =-=[14]-=-. ut In GSDT models, an extrinsic characterisation of the well-complete objects is available. Theorem 2.15 (Extrinsic characterisation of WC). For a GSDT model S, WC(S) is the largest full sub-cartesi... |

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Citation Context ... (5) Follows from a generalisation of [8, Theorems 7.1.5 and 7.3.11 (6)] in the context of [28]. ut Remark. There is an alternative synthetic viewpoint of domains (`a la [17, 36]) via replete objects =-=[16]-=-. In general a conservative-extension result in which domains are taken to be the replete objects (rather than the well-complete ones) is impossible, because embeddings that preserve dominances reflec... |

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Citation Context ...hetaC ! F \Theta C ! \Sigma ): (3) Using that \Sigma is well-complete and the closure under (Stability) of J one can adapt the argument of Proposition 1.6 to well-completeness (see [9, Lemma 11.16]). =-=(4) A well-co-=-mplete object A is also complete, by Proposition 2.2 (1) and 2.3 (2) because A ! 1 LA # " ! \Sigma # " in S. ut Corollary 2.8. Let W be a subcategory of S containing the terminal object and ... |

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Citation Context ...NNO N and equipped with a dominance ? : 1 ,! \Sigma . We have that \Sigma is a subobject of the subobject classifier\Omega ; in fact, 1 ! 1 \Sigma ? # " ? !\Omega # " true : The following is=-= folklore [31, 19, 34]-=-: Proposition 2.5. The lifting functor L has an initial algebra ' : LIs= I and a final coalgebra ' : Fs= LF with a unique global element 1 ! F such that (1 ! F ae j F ! LFs= F) = (1 ! F). Moreover, th... |

7 | Enrichment and representation theorems for categories of domains and continuous functions - Fiore - 1996 |

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Citation Context ... algebraic compactness is to provide a universal method of solving recursive type equations [15]; in this context algebraic compactness is guaranteed by the usual completeness conditions on the model =-=[28]-=-. The canonical example of an enrichment base is obtained by taking: -- C = Cpo, the category of small cpos ---!-complete partial orders--- and continuous functions; ? Research supported by Typed Lamb... |

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Citation Context ...unction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !-inductive fixed-point object (Definition 1.11 =-=(2)-=-). Roughly speaking, an !-inductive fixed-point object is an initial algebra (for the endofunctor underlying the monad induced by the adjunction) arising as the colimit of a standard !-chain, equipped... |

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Citation Context ... replete object and there are models of ADT in which not every object is replete (e.g. in the category Cpo as recently discovered by Makkai and Rosolini [33], and in the category VPoset introduced in =-=[10]-=-). 4 Comments on our approach This paper is an initial investigation of the relation between axiomatic and synthetic domain theory. For this, we had to restrict both the axiomatic and the synthetic (`... |

2 |
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Citation Context ...erve dominances reflect the property of being a replete object and there are models of ADT in which not every object is replete (e.g. in the category Cpo as recently discovered by Makkai and Rosolini =-=[33]-=-, and in the category VPoset introduced in [10]). 4 Comments on our approach This paper is an initial investigation of the relation between axiomatic and synthetic domain theory. For this, we had to r... |

1 |
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Citation Context ...constructors (as, for example, products, higher types, and sums) and is C-algebraically compact. The role of algebraic compactness is to provide a universal method of solving recursive type equations =-=[15]-=-; in this context algebraic compactness is guaranteed by the usual completeness conditions on the model [28]. The canonical example of an enrichment base is obtained by taking: -- C = Cpo, the categor... |

1 |
Mulry: Monads and algebras in the semantics of partial data types
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Citation Context ...r of ?-partial maps with target B, in that every ?-partial map [A /- U ! B] appears in a pullback U ! B A # " ! LB # " jB for a unique characteristic map A ! LB. The following is well-known =-=(see e.g. [17, 25, 8]-=-): Proposition 1.2. In a cartesian category, dominances induce commutative monads.sut Remark. For the notion of commutative monad see [20, 24]. The underlying endofunctor of the monad induced by a dom... |

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1 | Notes on Synthetic Domain Theory. Available from http://hypatia.dcs.qmw.ac.uk - Rosolini - 1995 |

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Private communication
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Citation Context ...cause 0 is initial in A. (2) The first part follows from Theorem 1.4 using that the ?-subobjects of a sheaf in ~ A are exactly the ?-subobjects in b A. The second part follows from: Proposition 1.6. (=-=[32]-=-) Let S be a topos equipped with a dominance ? : 1 ,! \Sigma . For every topology j in S, if \Sigma is a j-sheaf then the lifting in S preserves the property of being a j-sheaf. ut ut In the situation... |