Exponentiable Functors Between Quantaloid-Enriched Categories

Clementino, Maria Manuel; Hofmann, Dirk; Stubbe, Isar

doi:10.1007/s10485-007-9104-5

Cited by 7 publications

(12 citation statements)

References 6 publications

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“…More precisely, while the space Met Q (X, Y ) of non-expansive functions can be endowed with a metric (the sup-metric d sup (f, g) = sup{d(f (x), g(x)) | x ∈ X}), this construction does not yield a right-adjoint to the categorical product. For this reason Met Q is not a cartesian closed category (although Met Q still admits some interesting cartesian closed sub-categories, see [21], [22]). This abstract issue is not the only one has to face, though.…”

Section: A Program Metrics and Higher-order Languagesmentioning

confidence: 99%

On Generalized Metric Spaces for the Simply Typed Lambda-Calculus

Pistone

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Generalized metrics, arising from Lawvere's view of metric spaces as enriched categories, have been widely applied in denotational semantics as a way to measure to which extent two programs behave in a similar, although non equivalent, way. However, the application of generalized metrics to higher-order languages like the simply typed lambda calculus has so far proved unsatisfactory. In this paper we investigate a new approach to the construction of cartesian closed categories of generalized metric spaces. Our starting point is a quantitative semantics based on a generalization of usual logical relations. Within this setting, we show that several families of generalized metrics provide ways to extend the Euclidean metric to all higher-order types.

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Section: A Program Metrics and Higher-order Languagesmentioning

confidence: 99%

On Generalized Metric Spaces for the Simply Typed Lambda-Calculus

Pistone

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…The following characterization of exponentiable objects in V -Cat was proved in [5, Corollary 3.5] under the condition that k is the top element of V , and later, in [7,Section 5], without this extra condition.…”

Section: Exponentiable V -Categoriesmentioning

confidence: 99%

“…Finally, the following characterization of exponentiable morphisms in V -Cat can be found in [5] and [7].…”

Section: Exponentiable V -Functorsmentioning

confidence: 99%

“…This framework was suggested by various results on topological spaces via convergence: the characterization of effective descent continuous maps given in [30] and [20], and the characterization of exponentiable topological spaces of [28] and [25,26]. With this motivation, we presented a detailed study of effective descent morphisms of categories enriched in a quantale V in [4], and a characterization of exponentiable V -functors in [5,7]. The purpose of this article is to complete the results obtained in these papers in the realm of quantale-enriched categories, with particular focus on specific examples such as probabilistic metric spaces and categories enriched in the unit interval equipped with a continuous t-norm.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we recall the characterization of exponentiable V -categories obtained in [5,7] and refine this description for the specific choices of quantales introduced in the previous section. Section 2 can be seen as a "warm-up" for Section 3 where we consider the less familiar notion of exponentiable V -functor; similarly to the object-case, we provide customised descriptions for our main examples.…”

Section: Introductionmentioning

confidence: 99%

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The rise and fall of V-functors

Clementino

Hofmann

2017

Fuzzy Sets and Systems

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Kantorovich Functors and Characteristic Logics for Behavioural Distances

Goncharov

Hofmann

Nora

et al. 2023

Lecture Notes in Computer Science

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Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest in generic approaches to behavioural distances. In particular, coalgebraic methods capture variations in the system type (nondeterministic, probabilistic, game-based etc.), and the notion of quantale abstracts over the actual values distances take, thus covering, e.g., two-valued equivalences, (pseudo)metrics, and probabilistic (pseudo)metrics. Coalgebraic behavioural distances have been based either on liftings of $$\textsf{Set}$$ Set -functors to categories of metric spaces, or on lax extensions of $$\textsf{Set}$$ Set -functors to categories of quantitative relations. Every lax extension induces a functor lifting but not every lifting comes from a lax extension. It was shown recently that every lax extension is Kantorovich, i.e. induced by a suitable choice of monotone predicate liftings, implying via a quantitative coalgebraic Hennessy-Milner theorem that behavioural distances induced by lax extensions can be characterized by quantitative modal logics. Here, we essentially show the same in the more general setting of behavioural distances induced by functor liftings. In particular, we show that every functor lifting, and indeed every functor on (quantale-valued) metric spaces, that preserves isometries is Kantorovich, so that the induced behavioural distance (on systems of suitably restricted branching degree) can be characterized by a quantitative modal logic.

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Exponentiable Functors Between Quantaloid-Enriched Categories

Cited by 7 publications

References 6 publications

On Generalized Metric Spaces for the Simply Typed Lambda-Calculus

On Generalized Metric Spaces for the Simply Typed Lambda-Calculus

The rise and fall of V-functors

Kantorovich Functors and Characteristic Logics for Behavioural Distances

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