Probabilistic metric spaces as enriched categories

Hofmann, Dirk; Reis, Carla David

doi:10.1016/j.fss.2012.05.005

Cited by 45 publications

(50 citation statements)

References 19 publications

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“…which is equivalent to exponentiability of injective V-categories (see [HR13,Theorem 5.3]). Then [CHR18,Theorem 5.4] says the following: (f) (T, V)-Cat is infinitely extensive.…”

Section: The Case X=(t V)-catmentioning

confidence: 99%

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On generalized equilogical spaces

Ribeiro

2020

Topology and its Applications

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In this paper we carry the construction of equilogical spaces into an arbitrary category X topological over Set, introducing the category X-Equ of equilogical objects. Similar to what is done for the category Top of topological spaces and continuous functions, we study some features of the new category as (co)completeness and regular (co-)well-poweredness, as well as the fact that, under some conditions, it is a quasitopos. We achieve these various properties of the category X-Equ by representing it as a category of partial equilogical objects, as a reflective subcategory of the exact completion X ex , and as the regular completion X reg . We finish with examples in the particular cases, amongst others, of ordered, metric, and approach spaces, which can all be described using the (T, V)-Cat setting. IntroductionAs a solution to remedy the problem of non-existence of general exponentials in Top, Scott presents first in [Sco96], and later with his co-authors Bauer and Birkedal in [BBS04], the category Equ of equilogical spaces. Formed by equipping topological T 0 -spaces with arbitrary equivalence relations, Equ contains Top 0 (T 0 -spaces and continuous functions) as a full subcategory and it is cartesian closed. This fact is directly proven by showing an equivalence with the category PEqu of partial equilogical spaces, which is formed by equipping algebraic lattices with partial (not necessarily reflexive) equivalence relations. Also in [BBS04], equilogical spaces are presented as modest sets of assemblies over algebraic lattices, offering a model for dependent type theory. Contributing to the study of Equ, a more general categorical framework, explaining why such (sub)categories are (locally) cartesian closed, was presented in [BCRS98, CR00, Ros99]. It turned out that Equ is related to the free exact completion (Top 0 ) ex of Top 0 [Car95, CM82, CV98]. By the same token, suppressing the T 0 -separation condition on the topological spaces, the category Equ is a full reflective subcategory of the exact completion Top ex of Top. More, the reflector preserves products and special pullbacks, from where it is concluded in [BCRS98] that Equ is locally cartesian closed, since Top ex is so [BCRS98, Theorem 4.1]. It is shown in [Ros98] that Equ can be presented as the free regular completion of Top [Car95, CV98], and [Men00] provides conditions for such regular completions to be quasitoposes.

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“…which is equivalent to exponentiability of injective V-categories (see [HR13,Theorem 5.3]). Then [CHR18,Theorem 5.4] says the following: (f) (T, V)-Cat is infinitely extensive.…”

Section: The Case X=(t V)-catmentioning

confidence: 99%

“…• the identity monad I = (Id, 1, 1) on Set laxly extended to the identity lax monad on V-Rel; [HR13]). We assemble the table:…”

Section: The Case X=(t V)-catmentioning

confidence: 99%

On generalized equilogical spaces

Ribeiro

2020

Topology and its Applications

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“…An analogue of Proposition 3.6 for a commutative quantale V (u ⊗ v = v ⊗ u for every u, v ∈ V ) can be found in [11]. Proposition 3.6 appears in a slightly more restrictive form in [20,Proposition 1] and, in a more general form, in [26].…”

Section: From Quotients To Nucleimentioning

confidence: 99%

“…In [11], D. Hofmann and C. D. Reis represented the category ProbMet of probabilistic metric spaces of [23] as the category V -Cat for a unital quantale V . They also constructed a pair of functors ProbMet − → Met − → ProbMet, which serve as an instance of lax-algebraic quotients as is shown below.…”

Section: Probabilistic Metric Spacesmentioning

confidence: 99%

“…In a series of papers, M. M. Clementino, D. Hofmann, and W. Tholen [3,4,6,7,10,11] generalized this approach to an arbitrary monad T on Set and the category V -Rel of sets and V -relations, where V is an arbitrary unital quantale. In particular, they showed that many of the existing categories of topological structures (e.g., preordered sets, premetric spaces in the sense of F. W. Lawvere [17], approach spaces of R. Lowen [18], probabilistic metric spaces of B. Schweizer and A. Sklar [23]) can be represented as the categories (T, V )-Cat of lax algebras and lax homomorphisms with respect to a suitable monad T and a quantale V .…”

Section: Introductionmentioning

confidence: 99%

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On lax-algebraic (co)nuclei

Solovyov¹

2016

Math. Appl.

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Abstract. Quantic (co)nuclei provide a convenient technique for constructing quotients and subquantales of quantales. This paper shows its analogue for the laxalgebraic approach to topology of M. M. Clementino, D. Hofmann, and W. Tholen, based in a monad and a unital quantale. As a result, we get a machinery for constructing quotient categories and subcategories, which provides, in particular, several of the already defined ones by M. M. Clementino et al. We also get a representation theorem for the approach of M. M. Clementino et al.

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Categorical approach to modelling and to coupling of models

Gürlebeck

Hofmann

Legatiuk

2016

Math Methods in App Sciences

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Solution of any engineering problem starts with a modelling process, which typically involves a choice among different kinds of models. To create a realistic model, one has to think carefully about the modelling process. Particularly in the case of coupled problems when several models are coupled together to represent a given physical phenomenon. This paper presents an approach based on the category theory that allows to describe this modelling process on a more abstract level. Using the advantages of abstract level, one can describe the coupling process in a concise way and introduce certain criteria to check consistency of a coupled model. The main idea of the proposed approach is to introduce a structure in the modelling process, which allows to see how different models interact without a precise look into them.

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Probabilistic metric spaces as enriched categories

Abstract: In this paper we investigate Cauchy completeness and exponentiablity for quantale enriched categories, paying particular attention to probabilistic metric spaces.

Cited by 45 publications

References 19 publications

On generalized equilogical spaces

On generalized equilogical spaces

On lax-algebraic (co)nuclei

Categorical approach to modelling and to coupling of models

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