Finitary Functors: From Set to Preord and Poset

Balan, Adriana; Kurz, Alexander

doi:10.1007/978-3-642-22944-2_7

Cited by 21 publications

(32 citation statements)

References 9 publications

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“…2 We show next that coalgebras for Pre(F) × Pre ⊆ (F) ξ 1 ,ξ 2 correspond to weak bisimulations. We have the next commuting diagram…”

Section: Lemma 1010 For Any Weak Pullback Preserving Set-functor T mentioning

confidence: 84%

“…This lifting restricts to a functor Rel(F) X : Rel X → Rel F X , which in this case is just a monotone function between posets. The monotone function ξ * : Rel F X → Rel X is the inverse image of the coalgebra ξ , mapping a relation R ⊆ (F X) 2 …”

Section: Fωmentioning

confidence: 99%

“…Extensions of Set-functors to preorders or posets have been studied via relators as in [25,53] and using presentations of functors and (enriched) Kan extensions [2,3]. We are interested in extending not only functors, but also natural transformations to an ordered setting.…”

Section: Lifting Functors From Sets To Preordersmentioning

confidence: 99%

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A general account of coinduction up-to

et al. 2016

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Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modeled as coalgebras. The fact that bisimulations up to context can be safely used in any language specified by GSOS rules can also be seen as an instance of our framework, using the well-known observation by Turi and Plotkin that such languages form bialgebras. In the second part of the paper, we provide a new categorical treatment of weak bisimilarity on labeled transition systems and we prove the soundness of up-to context for weak bisimulations of systems specified by cool rule formats, as defined by Bloom to ensure congruence of weak bisimilarity. The weak transition systems obtained from such cool rules give rise to lax bialgebras, rather than to bialgebras. Hence, to reach our goal, we extend the categorical framework developed in the first part to an ordered setting. B Daniela Petrişan

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“…2 We show next that coalgebras for Pre(F) × Pre ⊆ (F) ξ 1 ,ξ 2 correspond to weak bisimulations. We have the next commuting diagram…”

Section: Lemma 1010 For Any Weak Pullback Preserving Set-functor T mentioning

confidence: 84%

Section: Fωmentioning

confidence: 99%

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A general account of coinduction up-to

et al. 2016

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“…Proof. This was proved in [8] under the additional assumption that T is finitary. Here, we present an argument valid for all Set-functors.…”

Section: Remark 42mentioning

confidence: 90%

Positive fragments of coalgebraic logics

Balan¹,

Kurz²,

Velebil³

2015

Logical Methods in Computer Science

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Abstract. Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond.In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors.To facilitate this analysis we prove a number of category theoretic results on functors on the categories Set of sets and Pos of posets:Every functor Set → Pos has a Pos-enriched left Kan extension Pos → Pos. Functors arising in this way are said to have a presentation in discrete arities. In the case that Set → Pos is actually Set-valued, we call the corresponding left Kan extension Pos → Pos its posetification.A set functor preserves weak pullbacks if and only if its posetification preserves exact squares. A Pos-functor with a presentation in discrete arities preserves surjections.The inclusion Set → Pos is dense. A functor Pos → Pos has a presentation in discrete arities if and only if it preserves coinserters of 'truncated nerves of posets'. A functor Pos → Pos is a posetification if and only if it preserves coinserters of truncated nerves of posets and discrete posets.A locally monotone endofunctor of an ordered variety has a presentation by monotone operations and equations if and only if it preserves Pos-enriched sifted colimits. ACM CCS: [Theory of computation]:Logic.

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“…His lifting works for monopreserving functors, and when they preserve weak pullbacks, his lifting coincides with the one in Example 5. Balan and Kurz give liftings and extensions of finitary Setfunctors to endofunctors over Pre and Pos [2]. Their method uses the fact that every finitary Set-functor T is presented as Lan I (T • I), where I : Finord → Set is the inclusion functor.…”

Section: Conclusion and Related Workmentioning

confidence: 99%

Preorders on Monads and Coalgebraic Simulations

Katsumata

Sato

2013

Lecture Notes in Computer Science

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Abstract. We study the construction of preorders on Set-monads by the semantic -lifting. We show the universal property of this construction, and characterise the class of preorders on a monad as a limit of a Card op -chain. We apply these theoretical results to identifying preorders on some concrete monads, including the powerset monad, maybe monad, and their composite monad. We also relate the construction of preorders and coalgebraic formulation of simulations.

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Finitary Functors: From Set to Preord and Poset

Cited by 21 publications

References 9 publications

A general account of coinduction up-to

A general account of coinduction up-to

Positive fragments of coalgebraic logics

Preorders on Monads and Coalgebraic Simulations

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