The rise and fall of V-functors

Clementino, Maria Manuel; Hofmann, Dirk

doi:10.1016/j.fss.2016.09.005

Cited by 11 publications

(9 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The isomorphism [0, 1] ⊕ -Cat → Met 1 assigns to each [0, 1] ⊕ -category (X, a) the metric space (X, a), with a(x, x ′ ) = 1 − a(x, x ′ ) for every x, x ′ ∈ X, and keeps morphisms unchanged. (For more information on tensor products on ([0, 1], ≤) see for instance [12] and the references there.) (4) When V is the quantale ∆ of distribution functions, that is,…”

Section: -Categoriesmentioning

confidence: 99%

See 1 more Smart Citation

On the categorical behaviour of $V$-groups

Clementino¹,

Montoli²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We consider compatible group structures on a V -category, where V is a quantale, and we study the topological and algebraic properties of such groups. Examples of such structures are preordered groups, metric and ultrametric groups, probabilistic (ultra)metric groups. In particular, we show that, when V is a frame, symmetric V -groups satisfy very strong categorical-algebraic properties, typical of the category of groups. In particular, symmetric V -groups form a protomodular category.

show abstract

Section: -Categoriesmentioning

confidence: 99%

“…then ∆-Cat is the category ProbMet of probabilistic metric spaces and non-expansive maps, as studied in [16] (see also [12]).…”

Section: -Categoriesmentioning

confidence: 99%

On the categorical behaviour of $V$-groups

Clementino¹,

Montoli²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Probabilistic metric spaces, as defined by [27,30], are a generalization of metric spaces in which the distance is defined as a distance distribution rather than a non-negative real number, and it has been known in [4,5,12] that probabilistic metric spaces can be considered as categories enriched in the quantale of distance distributions.…”

Section: The Quantale Of Distance Distributions Wrt a Continuous T-normmentioning

confidence: 99%

Towards probabilistic partial metric spaces: Diagonals between distance distributions

He¹,

Lai²,

Shen³

2019

Fuzzy Sets and Systems

View full text Add to dashboard Cite

The quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories. Motivated by the categorical interpretation of partial metric spaces, we are led to investigate the quantaloid of diagonals between distance distributions, which is expected to establish the categorical foundation of probabilistic partial metric spaces. Observing that the quantale of distance distributions w.r.t. an arbitrary continuous t-norm is non-divisible, we precisely characterize diagonals between distance distributions, and prove that one-step functions are the only distance distributions on which the set of diagonals coincides with the generated down set.for all t ∈ [0, ∞], called a distance distribution, rather than a non-negative real number. As discovered by Chai [4] and investigated by Hofmann-Reis [12], (generalized) probabilistic metric spaces are categories enriched in the quantale ∆ * = (∆, ⊗ * , κ 0,1 ) of distance distributions w.r.t. a continuous t-norm * on the unit interval [0, 1] (see Proposition 3.6 for further explanations) which, in elementary words, are precisely sets X equipped with a map α :for all x, y, z ∈ X and r, s ∈ [0, ∞]. Partial metric spaces are metric spaces in which the distance from a point to itself may not be zero. Explicitly, (generalized) partial metric spaces are sets X equipped a with a map α :for all x, y, z ∈ X. As discovered by Höhle-Kubiak [17] and Pu-Zhang [29] and formalized later by Stubbe [35] in a more general setting, although partial metric spaces are not categories enriched in the quantale [0, ∞] + , one may construct a quantaloid of diagonals of the quantale [0, ∞] + , usually denoted by D[0, ∞] + , such that partial metric spaces are precisely categories enriched in the quantaloid D[0, ∞] + .It is now natural to ask whether it is possible to consider the probabilistic version of partial metric spaces or, equivalently, the partial version of probabilistic metric spaces. As far as we know, this topic has been considered recently by several authors under the name probabilistic partial metric spaces or fuzzy partial metric spaces; see, e.g. [1,31,37,38,39]. Following the categorical interpretation of partial metric spaces, this paper aims at a full-scale investigation of the quantaloid D∆ * of diagonals of the quantale ∆ * of distance distributions w.r.t. a continuous t-norm * on [0, 1], so that a categorical foundation of probabilistic partial metric spaces can be established, which was more or less neglected in the existing references.In general, let Q = (Q, &, 1) be a commutative and integral quantale (i.e., complete residuated lattice), in which p & q ≤ r ⇐⇒ p ≤ q → r for all p, q, r ∈ Q. With DQ denoting the quantaloid of diagonals of Q (see Proposition 2.2), a DQ-category (also called partial Q-category, see [14]) consists of a set X and a map α : X × X

show abstract

“…The notion of (T , V)-categories, introduced in [10], generalizes both enriched categories and various notions of spaces. By studying effective descent morphisms in categories of (T , V)-categories, Clementino and Hofmann were able to give further descent results and understanding in various contexts, including, for instance, the reinterpretation of the topological results mentioned above and many other interesting connections (see, for instance, [6][7][8][9]).…”

Section: Introductionmentioning

confidence: 99%

Descent for internal multicategory functors

Prezado

Nunes

2023

Appl Categor Struct

View full text Add to dashboard Cite

We give sufficient conditions for effective descent in categories of (generalized) internal multicategories. Two approaches to study effective descent morphisms are pursued. The first one relies on establishing the category of internal multicategories as an equalizer of categories of diagrams. The second approach extends the techniques developed by Ivan Le Creurer in his study of descent for internal essentially algebraic structures.

show abstract

The rise and fall of V-functors

Abstract: In this article we study function spaces (rise) and descent (fall) in quantale-enriched categories, paying particular attention to enrichment in the non-negative reals, the quantale of distribution functions and the unit interval equipped with a continuous t-norm.

Cited by 11 publications

References 24 publications

On the categorical behaviour of $V$-groups

On the categorical behaviour of $V$-groups

Towards probabilistic partial metric spaces: Diagonals between distance distributions

Descent for internal multicategory functors

Contact Info

Product

Resources

About