Universality of Coproducts in Categories of Lax Algebras

Mahmoudi, Mojgan; Schubert, Christoph; Tholen, Walter

doi:10.1007/s10485-006-9019-6

Cited by 9 publications

(14 citation statements)

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“…In Section 1 we present the Yoneda embedding for V-categories as a subproduct of the fact that a V-matrix ψ : X−→ Y between V-categories (X, a) and (Y, b) is a V-bimodule if and only if, as a map ψ : X op ⊗ Y → V, is a V-functor (Theorem 1.5); then the monoidal-closed structure of V-Cat gives us the Yoneda Functor X → V X op . In the (Ì, V)-setting this construction becomes more elaborated (see In Section 5 we present the announced topological examples, with the exception of quasiuniform spaces, which are presented in the Appendix, due to the fact that their presentation as lax algebras does not fit in the (Ì, V)-setting (as shown in [20]). …”

Section: Introductionmentioning

confidence: 99%

“…This is the subject of Section 3. In addition we also prove that, under some conditions, the (Ì, V)-category V is Lawvere-complete.In Section In Section 5 we present the announced topological examples, with the exception of quasiuniform spaces, which are presented in the Appendix, due to the fact that their presentation as lax algebras does not fit in the (Ì, V)-setting (as shown in [20]). …”

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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: Introductionmentioning

confidence: 99%

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confidence: 99%

Lawvere Completeness in Topology

Clementino

Hofmann

2008

Appl Categor Struct

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“…Then [CHR18,Theorem 5.4] says the following: (f) (T, V)-Cat is infinitely extensive. This is proved in [MST06] under the condition that T is a taut functor [Man02], what comes for free from the assumption that T preserves weak pullbacks.…”

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

confidence: 99%

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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“…The previous lemma implies at once that the class of coalgebras induced by an abstract logic is closed under homomorphic images and subcoalgebras. To show closedness under the formation of sums, we note that the sum of a family (X i , ⊢ i ) i∈I of abstract logics can be calculated as the disjoint union X = i∈I X i , equipped with the consequence relation ⊢ defined by A ⊢ x whenever (A ∩ X i ) ⊢ i x, where x ∈ X i (see also [8]). By definition, every inclusion map k i : (X i , ⊢ i ) ֒→ (X, ⊢) is open.…”

Section: A Coalgebraic View On Logicmentioning

confidence: 99%