Topological theories and closed objects

Hofmann, Dirk

doi:10.1016/j.aim.2007.04.013

Cited by 50 publications

(105 citation statements)

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“…In this section we wish to obtain the analogue result to Corollary 1.5 in the setting of (Ì, V)-categories. This in turn requires an understanding of the right adjoint to X ⊗ : (Ì, V)-Cat → (Ì, V)-Cat, a problem studied in [16]. From there we import the following result.…”

Section: A Yoneda Lemma For (ì V)-categoriesmentioning

confidence: 99%

“…Recall that we assume T f = T o f for each Set-map f : X → Y ; this condition requires and implies equality in the latter inequality (see [16]). …”

Section: As a (ì V)-categorymentioning

confidence: 99%

“…It is well-known that in general the functor X ⊗ : (Ì, V)-Cat → (Ì, V)-Cat has no right adjoint as, for example, Top is not Cartesian closed. The problem of characterising those (Ì, V)-categories X = (X, a) such that tensoring with X has a right adjoint is studied in [16].…”

Section: Corollary (V Hom ξ ) Is a (ì V)-categorymentioning

confidence: 99%

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Lawvere Completeness in Topology

Clementino

Hofmann

2008

Appl Categor Struct

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It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that V has a canonical (Ì, V)-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of (Ì, V)-categories. (2000): 18A05, 18D15, 18D20, 18B35, 18C15, 54E15, 54E50. Mathematics Subject ClassificationKey words: V-category, bimodule, monad, (Ì, V)-category, completeness. IntroductionLawvere in his 1973 paper Metric spaces, generalized logic, and closed categories formulates a notion of complete V-category and shows that for (generalised) metric spaces it means Cauchy completeness. This notion of completeness deserved the attention of the categorical community, and the notion of Cauchy-complete category, or Freyd-Karoubi complete category is well-known, mostly in the context of Ab-enriched categories. However, it never got the attention of the topological community. In this paper we interpret Lawvere's completeness in topological settings. We extend Lawvere's notion of complete V-category to the (topological) setting of (Ì, V)-categories (for a symmetric and unital quantale V), and show that it encompasses well-known notions in topological categories, meaning weakly sober space in the category of topological spaces and continuous maps, weakly sober approach space in the category of approach spaces and non-expansive maps, and Cauchy-completeness in the category of quasi-uniform spaces and uniformly continuous maps.We present also a first step towards a possible construction of completion. Indeed, in the setting of V-categories, it is well-known that the completion of a V-category may be built out of the Yoneda embedding X → V X op . In the (Ì, V)-setting, we could prove that V has a canonical (Ì, V)-categorical structure and that every (Ì, V)-category X has a canonical dual * The authors acknowledge partial financial assistance by Centro de Matemática da Universidade de Coimbra/FCT and Unidade de Investigação e Desenvolvimento Matemática e Aplicações da Universidade de Aveiro/FCT. 1 X op . Using this structure and the free Eilenberg-Moore algebra structure |X| on T X, we get two "Yoneda-like" morphismsFor the latter one we prove a Yoneda Lemma (see 4.2). Furthermore, we show that, under suitable conditions, V is a Lawvere-complete (Ì, V)-category, a first step towards a completion construction which will be the subject of a forthcoming paper.In order to make the presentation of this paper smoother, in Section 1 we recall the notions and properties of V-categories we will generalize throughout. First we introduce V-categories and V-bimodules, and de...

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Section: A Yoneda Lemma For (ì V)-categoriesmentioning

confidence: 99%

“…Recall that we assume T f = T o f for each Set-map f : X → Y ; this condition requires and implies equality in the latter inequality (see [16]). …”

Section: As a (ì V)-categorymentioning

confidence: 99%

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Lawvere Completeness in Topology

Clementino

Hofmann

2008

Appl Categor Struct

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“…As a result, we arrive at a convenient technique for producing quotients and subcategories of the categories (T, V )-Cat, thereby obtaining not only the above-mentioned five examples, but also new ones, which are related to the categories of H-labeled graphs and multi-ordered sets of, e.g., [7]. We also show an application of our (co)nuclei technique to (op-)canonical extensions of monads of G. Seal [24] and topological theories of D. Hofmann [10].…”

Section: Introductionmentioning

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%

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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Topological theories and closed objects

Cited by 50 publications

References 14 publications

Lawvere Completeness in Topology

Lawvere Completeness in Topology

On lax-algebraic (co)nuclei

Categorical approach to modelling and to coupling of models

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