## Coalgebras and modal logics for parameterised endofunctors (2000)

Citations: | 3 - 3 self |

### BibTeX

@TECHREPORT{Kurz00coalgebrasand,

author = {Alexander Kurz and Dirk Pattinson},

title = {Coalgebras and modal logics for parameterised endofunctors},

institution = {},

year = {2000}

}

### OpenURL

### Abstract

### Citations

475 |
Institutions: Abstract Model Theory for Specification and Programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ...sh this duality formally, we begin with translating the satisfaction condition, as known from institutions, to (co-) brational setting. 4.1 Institutions, Fibrationally Institutions were introduced in =-=[6]-=- in order to describe the e ect of signature morphisms wrt. logics for di erent signatures. In an institution I =(S, Mod, Sen, (|=Σ)Σ∈S), one associates a contravariant functor Mod(σ) :Mod(Σ ′ ) → Mod... |

227 |
Abstract and concrete categories
- Adámek, Herrlich, et al.
- 1990
(Show Context)
Citation Context ...ituations: (i) In case that C =Set(because ΩL preserves (split) equalisers, hence UL creates them); (ii) In case that bres have factorisation structures for sinks (E,M) with sinks in E being epi (see =-=[2]-=-, 15.7); (iii) In case that C is locally presentable and the ΩL are accessible (then the bres are locally presentable and hence complete, see [19]). In the remainder of this section we investigate whe... |

162 |
A nal coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ...It follows from (C,UL(ci)) being a colimit of ULG that (D,UM(di)) is a colimit of UMλ † G. Since UM creates colimits, ((D,δ),di) is a colimit of λ † G. Generalising the de nition of Aczel and Mendler =-=[1]-=- to arbitrary categories by taking a bisimulation between two coalgebras (C,γ) and (D,δ) in the same bre EL to be a monic span (πC : R → C,πD : R → D) in C which can be equipped with a transition stru... |

138 |
Handbook of Categorical Algebra
- Borceux
- 1994
(Show Context)
Citation Context ...∈ L for every A over L, the assignment A ↦→ λ † (A) extends to a functor λ † : EL → EM . A functor obtained in this way is called a co-reindexing. We will use the bred terminology freely and refer to =-=[4, 9]-=- regarding further reading on this subject. That the co bration associated to a parameterised signature is indeed a co bration follows from (λ, idC) being a cocartesian lifting of λ : L → M ∈ L for ev... |

112 |
Categorical Logic and Type Theory
- Jacobs
- 1999
(Show Context)
Citation Context ...∈ L for every A over L, the assignment A ↦→ λ † (A) extends to a functor λ † : EL → EM . A functor obtained in this way is called a co-reindexing. We will use the bred terminology freely and refer to =-=[4, 9]-=- regarding further reading on this subject. That the co bration associated to a parameterised signature is indeed a co bration follows from (λ, idC) being a cocartesian lifting of λ : L → M ∈ L for ev... |

110 | A hidden agenda
- Goguen, Malcolm
- 2000
(Show Context)
Citation Context ...oach). 15s16 5. Hidden and Multiplicative Signatures In speci cation formalisms using algebras and/or coalgebras one often restricts attention to special signature functors, namely, hidden signatures =-=[7]-=- in case of algebras and multiplicative functors in case of coalgebras. We show here that we can characterise, roughly speaking, hidden signatures as functors on Set n having a right adjoint (and thes... |

89 | The Logic of
- Baltag, Moss, et al.
- 1998
(Show Context)
Citation Context ...nd, the semantical approach covers any logic where formulas are bisimulation invariant predicates on carriers, independently of how the logic is given. An important example is Moss' coalgebraic logic =-=[16]-=- (which ts in the semantical but not in the precicate lifting approach). 15s16 5. Hidden and Multiplicative Signatures In speci cation formalisms using algebras and/or coalgebras one often restricts a... |

61 |
Specifying Coalgebras with Modal Logic
- Kurz
- 1998
(Show Context)
Citation Context ...dual of condition (2) wrt. two di erent conceptions of logic for coalgebras.s4.2 Predicate Liftings and (Modal) Logic Viewing coalgebras on the category of sets as transition systems, several authors =-=[10, 11, 15,22,21]-=- have developed modal logics, where formulas are interpreted as bisimulation invariant predicates on the state space. The logics we will be concerned with here are interpreted (and assumed to be given... |

54 |
Logics for Coalgebras and Applications to Computer Science
- Kurz
- 2000
(Show Context)
Citation Context ...les. (ii) Theorem and proof generalise to coalgebras over arbitrary categories admitting a factorisation system which allows to form unions of subcoalgebras. Such categories have been investigated in =-=[14]-=-. (iii) The proof of the theorem shows how to actually calculate cartesian liftings of monos as certain subcoalgebras. This allows to determine cartesian liftings in our examples. Reindexing wrt. outp... |

33 | The temporal logic of coalgebras via galois algebras
- Jacobs
- 2002
(Show Context)
Citation Context ...dual of condition (2) wrt. two di erent conceptions of logic for coalgebras.s4.2 Predicate Liftings and (Modal) Logic Viewing coalgebras on the category of sets as transition systems, several authors =-=[10, 11, 15,22,21]-=- have developed modal logics, where formulas are interpreted as bisimulation invariant predicates on the state space. The logics we will be concerned with here are interpreted (and assumed to be given... |

32 |
Abstract families and the adjoint functor theorem
- Paré, Schumacher
- 1978
(Show Context)
Citation Context ... idC) ◦ γ). L ↦→ CΩL, (L λ → L ′ ) ↦→ λ † In this way, every parameterised signature Ω can be seen to de ne a split co-indexed category I(Ω), a concept originally introduced by Paré and Schumacher in =-=[17]-=-. Instead of working with coindexed categories, it is technically more convenient (and aesthetically more pleasing) to describe the phenomenon of variation over a parameter category in terms of (co-) ... |

30 | Semantical principles in the modal logic of coalgebras
- Pattinson
- 2001
(Show Context)
Citation Context ...or according to its syntactical structure. Closer investigation reveals that predicate liftings also occur implicitly in the logics discussed in [15, 22, 21]. Also, the concept of natural relation in =-=[18]-=- can be seen as a special instance. This leads us to consider logics interpreted by means of a set S of (arbitrary) predicate liftings for the signature functor \Omega : Set ! Set. Note that every pre... |

24 | A co-variety-theorem for modal logic
- Kurz
- 1998
(Show Context)
Citation Context ...s x ∈ X describes a quotient of the free algebra over X) and generalises the interpretation of modal formulas wrt. Kripke frames to coalgebras for arbitrary signature functors. Example 4.8. Following =-=[13]-=-, we show how Kripke frames are generalised in a coalgebraic setting. Consider ΩX = PκX where κ is a cardinal and PkX = {Y ⊂ X : |Y | <κ}. 4 Ω-coalgebras are Kripke frames, the notion frame referring ... |

21 | On the structure of categories of coalgebras
- Johnstone, Power, et al.
- 2001
(Show Context)
Citation Context ...s for sinks (E; M) with sinks in E being epi (see [2], 15.7); (iii) In case that C is locally presentable and the\Omega L are accessible (then the bres are locally presentable and hence complete, see =-=[19]-=-). In the remainder of this section we investigate when one can dispense with the assumption that bres have equaliseres. The crucial observation is that, for the use of the adjoint lifting theorem, it... |

17 | A complete calculus for equational deduction in coalgebraic specification
- Corradini
- 1998
(Show Context)
Citation Context ...t hidden algebras are given by precisely those signatures which give rise to coalgebras via proposition 5.1. Conversely, the use of equational logic for coalgebras (as opposed to modal logic) in e.g. =-=[5, 20, 8]-=- implicitly relies on multiplicative signatures giving rise to categories of algebras via proposition 5.1. Acknowledgments The authors would like to thank Horst Reichel for the organisation of the wor... |

14 |
Adjoint machines, state-behaviour machines, and duality
- Arbib, Manes
- 1975
(Show Context)
Citation Context ...ase of signatures over Set, multiplcative and hidden signatures are even characterised by this property. The key to this result is the following lemma which generalises theorem 5.7 in Arbib and Manes =-=[3]-=- from Set to Set n . 6 Lemma 5.6. Let Σ be a functor on Set n . Then the following are equivalent. (i) Σ has a right adjoint. (ii) Σ preserves coproducts. (iii) There isa(n × n)-matrix M over Set such... |

11 |
Towards a duality result in coalgebraic modal logic
- Jacobs
(Show Context)
Citation Context ...dual of condition (2) wrt. two di erent conceptions of logic for coalgebras.s4.2 Predicate Liftings and (Modal) Logic Viewing coalgebras on the category of sets as transition systems, several authors =-=[10, 11, 15,22,21]-=- have developed modal logics, where formulas are interpreted as bisimulation invariant predicates on the state space. The logics we will be concerned with here are interpreted (and assumed to be given... |

1 |
Notes on coalgebras, co- brations and concurrency
- Kurz, Pattinson
- 2000
(Show Context)
Citation Context ...om reindexing being right adjoint, see (3). Let us comment on this theorem. First, the proof of this theorem does not exhibit how limits and cartesian liftings can actually be calculated. As shown in =-=[12]-=-, both can be obtained by factoring 2 F is bred (dualise for `U co bred') i pF = q and F preserves cartesian liftings. Note that this de nition makes sense even when p fails to be a bration. 3 Note th... |

1 |
Xi)-logic: On the algebraic extension of coalgbraic specications
- Hennicker, Kurz
- 1952
(Show Context)
Citation Context ...t hidden algebras are given by precisely those signatures which give rise to coalgebras via proposition 5.1. Conversely, the use of equational logic for coalgebras (as opposed to modal logic) in e.g. =-=[5, 20, 8]-=- implicitly relies on multiplicative signatures giving rise to categories of algebras via proposition 5.1. Acknowledgments The authors would like to thank Horst Reichel for the organisation of the wor... |

1 |
Notes on coalgebras, co-brations and concurrency
- Kurz, Pattinson
- 1992
(Show Context)
Citation Context ...om reindexing being right adjoint, see (3). Let us comment on this theorem. First, the proof of this theorem does not exhibit how limits and cartesian liftings can actually be calculated. As shown in =-=[12]-=-, both can be obtained by factoring appropriate sinks with cofree codomain thus giving a possibility to calculate limits and cartesian liftings in concrete examples. Second, (1) and (2) hold whenever ... |

1 |
A Birkhooe-like axiomatisability result for hidden algebra and coalgebra
- Rou
(Show Context)
Citation Context ...t hidden algebras are given by precisely those signatures which give rise to coalgebras via proposition 5.1. Conversely, the use of equational logic for coalgebras (as opposed to modal logic) in e.g. =-=[5, 20, 8]-=- implicitly relies on multiplicative signatures giving rise to categories of algebras via proposition 5.1. Acknowledgments The authors would like to thank Horst Reichel for the organisation of the wor... |