Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 30 Nov 2022 11:21:25 GMT2022-11-30T11:21:25Z50421Separated and Connected Mapshttp://hdl.handle.net/10316/7758Title: Separated and Connected Maps
Authors: Clementino, Maria Manuel; Tholen, Walter
Abstract: Using on the one hand closure operators in the sense of Dikranjan and Giuli and on the other hand left- and right-constant subcategories in the sense of Herrlich, Preuß, Arhangel'skii and Wiegandt, we apply two categorical concepts of connectedness and separation/disconnectedness to comma categories in order to introduce these notions for morphisms of a category and to study their factorization behaviour. While at the object level in categories with enough points the first approach exceeds the second considerably, as far as generality is concerned, the two approaches become quite distinct at the morphism level. In fact, left- and right-constant subcategories lead to a straight generalization of Collins' concordant and dissonant maps in the category $$\mathcal{T}op$$ of topological spaces. By contrast, closure operators are neither able to describe these types of maps in $$\mathcal{T}op$$, nor the more classical monotone and light maps of Eilenberg and Whyburn, although they give all sorts of interesting and closely related types of maps. As a by-product we obtain a negative solution to the ten-year-old problem whether the Giuli–Hušek Diagonal Theorem holds true in every decent category, and exhibit a counter-example in the category of topological spaces over the 1-sphere.
Thu, 01 Jan 1998 00:00:00 GMThttp://hdl.handle.net/10316/77581998-01-01T00:00:00ZOn bornological semi-abelian algebrashttp://hdl.handle.net/10316/101221Title: On bornological semi-abelian algebras
Authors: Borceux, Francis; Clementino, Maria Manuel
Abstract: If T is a semi-abelian algebraic theory, we prove that the category
BornT of bornological T-algebras is homological with semi-direct products.
We give a formal criterion for the representability of actions in BornT
and, for a bornological T-algebra X, we investigate the relation between the
representability of actions on X as a T-algebra and as a bornological T-
algebra. We investigate further the algebraic coherence and the algebraic
local cartesian closedness of BornT and prove in particular that both properties
hold in the case of bornological groups.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10316/1012212021-01-01T00:00:00ZOn some special classes of continuous mapshttp://hdl.handle.net/10316/11336Title: On some special classes of continuous maps
Authors: Clementino, Maria Manuel; Hofmann, Dirk
Abstract: We present a survey on recent study of special continuous maps, like
biquotient, triquotient, proper, perfect, open and étale maps and a selection of
open problems in this area.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/113362006-01-01T00:00:00ZOne Setting for All: Metric, Topology, Uniformity, Approach Structurehttp://hdl.handle.net/10316/7757Title: One Setting for All: Metric, Topology, Uniformity, Approach Structure
Authors: Clementino, Maria; Hofmann, Dirk; Tholen, Walter
Abstract: Abstract For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T,V)-algebras and of (T,V)-proalgebras turn out to be topological over Set.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/10316/77572004-01-01T00:00:00ZProlocalisations of homological categorieshttp://hdl.handle.net/10316/11316Title: Prolocalisations of homological categories
Authors: Borceux, F.; Clementino, M. M.; Gran, M.; Sousa, L.
Abstract: A prolocalisation of an homological (resp. semi-abelian) category is a
regular full re
ective subcategory, whose re
ection preserves short exact sequences.
We study the closure operator and the torsion theory associated with such a situation.
We pay special attention to the bered, the epire
ective and the monore
ective
cases. We give examples in algebra, topos theory, functional analysis.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/113162006-01-01T00:00:00ZSeparação e compacidade em categorias. Coimbra, ed. aut., 1992, p. 161.http://hdl.handle.net/10316/1567Title: Separação e compacidade em categorias. Coimbra, ed. aut., 1992, p. 161.
Authors: Clementino, Maria Manuel Pinto Lopes Ribeiro
Abstract: Desde a publicação de um artigo pioneiro de D. Dikranjan E. Giuli, onde os autores introduzem a noção categórica de operador de fecho, o estudo de operadores de fecho tem sido uma área importante de investigação em Topologia Categórica.
Utilizando a noção de operador de fecho, nesta dissertação desenvolvemos uma teoria abstracta de separação e compacidade numa categoria arbitrária.
Com esse objectivo, a nossa primeira contribuição consiste na formulação de um conceito relativo de ponto e numa análise cuidada das suas propriedades. A escolha axiomática de subobjectos, pontos e de um operador de fecho fornece-nos um 'ambiente topológico' onde surgem de uma forma natural generalizações dos axiomas de separação, a que dedicamos uma parte significativa do nosso trabalho.
O estudo de separação conduz-nos ainda a uma unificação de conceitos. De entre os resultados obtidos salientamos os que relacionam desconexões e teorias de torsão com operadores de fecho regulares fracamente hereditários.
Por fim estudamos uma noção de compacidade e estabelecemos diversas relações interessantes com outras abordagens de compacidade.
Wed, 01 Jan 1992 00:00:00 GMThttp://hdl.handle.net/10316/15671992-01-01T00:00:00ZCovering Morphisms in Categories of Relational Algebrashttp://hdl.handle.net/10316/43758Title: Covering Morphisms in Categories of Relational Algebras
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Montoli, Andrea
Abstract: In this paper we use Janelidze’s approach to the classical theory of topological coverings via categorical Galois theory to study coverings in categories of relational algebras. Moreover, we present characterizations of effective descent morphisms in the categories of M-ordered sets and of multi-ordered sets.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10316/437582013-01-01T00:00:00ZThe Other Closure and Complete Sublocaleshttp://hdl.handle.net/10316/90476Title: The Other Closure and Complete Sublocales
Authors: Clementino, Maria Manuel; Picado, Jorge; Pultr, Aleš
Abstract: Sublocales of a locale (frame, generalized space) can be equivalently represented by frame congruences. In this paper we discuss, a.o., the sublocales corresponding to complete congruences, that is, to frame congruences which are closed under arbitrary meets, and present a “geometric” condition for a sublocale to be complete. To this end we make use of a certain closure operator on the coframe of sublocales that allows not only to formulate the condition but also to analyze certain weak separation properties akin to subfitness or T_1. Trivially, every open sublocale is complete. We specify a very wide class of frames, containing all the subfit ones, where there are no others. In consequence, e.g., in this class of frames, complete homomorphisms are automatically Heyting.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10316/904762018-01-01T00:00:00ZRepresentable (T,V)-categorieshttp://hdl.handle.net/10316/43644Title: Representable (T,V)-categories
Authors: Chikhladze, Dimitri; Clementino, Maria Manuel; Hofmann, Dirk
Abstract: Working in the framework of (T,V)-categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad T, we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for (T,V)-categories.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10316/436442015-01-01T00:00:00ZThe rise and fall of V -functorshttp://hdl.handle.net/10316/43642Title: The rise and fall of V -functors
Authors: Clementino, Maria Manuel; Hofmann, Dirk
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.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10316/436422017-01-01T00:00:00ZRelative injectivity as cocompleteness for a class of distributorshttp://hdl.handle.net/10316/42771Title: Relative injectivity as cocompleteness for a class of distributors
Authors: Clementino, Maria Manuel; Hofmann, Dirk
Abstract: Notions and techniques of enriched category theory can be used to study
topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the category of injective spaces and left adjoints over Set. In this paper we generalise these results, studying cocompleteness with respect to a given class of distributors. We show in particular that the description of several semantic domains presented in [M. Escardó and B. Flagg, Semantic domains, injective spaces and monads, Electronic Notes in Theoretical Computer Science 20 (1999)] can be translated into the V-enriched setting.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/427712008-01-01T00:00:00ZProper maps for lax algebras and the Kuratowski-Mrówka Theoremhttp://hdl.handle.net/10316/42770Title: Proper maps for lax algebras and the Kuratowski-Mrówka Theorem
Authors: Clementino, Maria Manuel; Tholen, Walter
Abstract: The characterization of stably closed maps of topological spaces as the
closed maps with compact fibres and the role of the Kuratowski-Mrowka' Theorem in this characterization are being explored in the general context of lax (T; V )-algebras, for a quantale V and a Set-monad T with a lax extension to V -relations. The general results are being applied in standard (topological and metric) and non-standard (labeled graphs) contexts.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10316/427702013-01-01T00:00:00ZSemidirect products of (topological) semi-abelian algebrashttp://hdl.handle.net/10316/43641Title: Semidirect products of (topological) semi-abelian algebras
Authors: Clementino, Maria Manuel; Montoli, Andrea; Sousa, Lurdes
Abstract: We give a description of the semidirect products in any semi-abelian variety. Moreover, we internalize this description to the models of semi-abelian theories in any finitely complete category. As an application, we characterize the topology of the semidirect products in the topological models of any semi-abelian theory.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10316/436412015-01-01T00:00:00ZOn regular and homological closure operatorshttp://hdl.handle.net/10316/43635Title: On regular and homological closure operators
Authors: Clementino, Maria Manuel; Gutierres, Gonçalo
Abstract: Observing that weak heredity of regular closure operators in Top and of homological closure operators in homological categories identifies torsion theories, we study these closure operators in parallel, showing that regular closure operators play the same role in topology as homological closure operators do algebraically.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10316/436352010-01-01T00:00:00ZOn exponentiability of étale algebraic homomorphismshttp://hdl.handle.net/10316/43640Title: On exponentiability of étale algebraic homomorphisms
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Janelidze, George
Abstract: In this paper we show that the theorem, by Cagliari and Mantovani, stating that in
the category of compact Hausdorff spaces every étale map is exponentiable, can be formulated
in a general category Alg(T) of Eilenberg-Moore T-algebras, for a monad T, and proved in
case T satisfies the so-called Beck-Chevalley condition. For that, Alg(T) is embedded in the
(topological) category RelAlg(T) of relational T-algebras, where a suitable notion of étale
morphism can be studied, it is shown that morphisms between T-algebras are exponentiable in
RelAlg(T), and, moreover, these exponentials belong to Alg(T) whenever the morphisms are
étale.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10316/436402013-01-01T00:00:00ZFibrewise injectivity in order and topologyhttp://hdl.handle.net/10316/43643Title: Fibrewise injectivity in order and topology
Authors: Cagliari, Francesca; Clementino, Maria Manuel; Mantovani, Sandra
Abstract: This paper studies injectivity for continuous maps between T0-spaces. The new characterizations obtained establish a parallelism between characterizations of injective monotone maps between ordered sets and of injective continuous maps between T0-spaces.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10316/436432016-01-01T00:00:00ZExponentiability in categories of lax algebrashttp://hdl.handle.net/10316/42772Title: Exponentiability in categories of lax algebras
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Tholen, Walter
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/10316/427722003-01-01T00:00:00ZThe monads of classical algebra are seldom weakly cartesianhttp://hdl.handle.net/10316/44482Title: The monads of classical algebra are seldom weakly cartesian
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Janelidze, George
Abstract: This paper begins a systematic study of weakly cartesian properties of monads that determine familiar varieties of universal algebras. While these properties clearly fail to hold for groups, rings, and many other related classical algebraic structures, their analysis becomes non-trivial in the case of semimodules over semirings, to which our main results are devoted. In particular necessary and sufficient conditions on a semiring S, under which the free semimodule monad has: (a) its underlying functor weakly cartesian, (b) its unit a weakly cartesian natural transformation, (c) its multiplication a weakly cartesian natural transformation, are obtained.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10316/444822013-01-01T00:00:00ZOn Exponentiable Morphisms in Classical Algebrahttp://hdl.handle.net/10316/41050Title: On Exponentiable Morphisms in Classical Algebra
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Janelidze, George
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10316/410502016-01-01T00:00:00ZLax orthogonal factorisation systemshttp://hdl.handle.net/10316/43633Title: Lax orthogonal factorisation systems
Authors: Clementino, Maria Manuel; Lopez Franco, Ignacio
Abstract: This paper introduces lax orthogonal algebraic weak factorisation systems on
2-categories and describes a method of constructing them. This method rests in
the notion of simple 2-monad, that is a generalisation of the simple
reflections studied by Cassidy, Hébert and Kelly. Each simple 2-monad on a
finitely complete 2-category gives rise to a lax orthogonal algebraic weak
factorisation system, and an example of a simple 2-monad is given by completion
under a class of colimits. The notions of KZ lifting operation, lax natural
lifting operation and lax orthogonality between morphisms are studied.
Sat, 22 Oct 2016 00:00:00 GMThttp://hdl.handle.net/10316/436332016-10-22T00:00:00ZFrom lax monad extensions to Topological theorieshttp://hdl.handle.net/10316/43927Title: From lax monad extensions to Topological theories
Authors: Clementino, Maria Manuel; Tholen, Walter
Abstract: We investigate those lax extensions of a Set-monad T = (T,m, e) to the category V-Rel of sets and V-valued relations for a quantale V = (V,⊗, k) that are fully determined by ξ maps : TV ⟶ V. We pay
special attention to those maps ξ that make V a T-algebra and, in fact, (V,⊗, k) a monoid in the category Set^T with its cartesian structure. Any such map ξ forms the main ingredient to Hofmann’s notion of topological theory.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/439272014-01-01T00:00:00ZOn the categorical behaviour of preordered groupshttp://hdl.handle.net/10316/89416Title: On the categorical behaviour of preordered groups
Authors: Clementino, Maria Manuel; Martins-Ferreira, Nelson; Montoli, Andrea
Abstract: We study the categorical properties of preordered groups. We first give a description of limits and colimits in this category, and study some classical exactness properties. Then we point out a strong analogy between the algebraic behaviour of preordered groups and monoids, and we apply two different recent approaches to relative categorical algebra to obtain some homological properties of preordered groups.
Tue, 01 Oct 2019 00:00:00 GMThttp://hdl.handle.net/10316/894162019-10-01T00:00:00ZOn extensions of lax monadshttp://hdl.handle.net/10316/27771Title: On extensions of lax monads
Authors: Clementino, Maria Manuel; Hofmann, Dirk
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/10316/277712004-01-01T00:00:00ZOn the representability of actions for topological algebrashttp://hdl.handle.net/10316/43925Title: On the representability of actions for topological algebras
Authors: Borceux, Francis; Clementino, Maria Manuel; Montoli, Andrea
Abstract: The actions of a group B on a group X correspond bijectively to the group homomorphisms B ⟶ Aut(X), proving that the functor “actions on X” is representable by the group of automorphisms of X. Making the detour through pseudotopological spaces, we generalize this result to the topological case, for quasi-locally compact groups and some other algebraic structures. We investigate next the case of arbitrary topological algebras for a semi-abelian theory and prove that the representability of topological actions reduces to the preservation of coproducts by the functor Act(−,X).
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/439252014-01-01T00:00:00ZAnother note on effective descent morphisms of topological spaces and relational algebrashttp://hdl.handle.net/10316/89423Title: Another note on effective descent morphisms of topological spaces and relational algebras
Authors: Clementino, Maria Manuel; Janelidze, George
Abstract: We make three independent observations on characterizing effective descent morphisms in the category of topological spaces. The first of them proposes a new modification of known characterizations of effective descent morphisms of general spaces, while the other two are devoted to locally finite and Hausdorff spaces, respectively. The Hausdorff case is considered, as far as we could, at the more general level of relational algebras in the sense of M. Barr.
Sun, 01 Mar 2020 00:00:00 GMThttp://hdl.handle.net/10316/894232020-03-01T00:00:00ZTopological groups have representable actionshttp://hdl.handle.net/10316/89424Title: Topological groups have representable actions
Authors: Cagliari, Francesca; Clementino, Maria Manuel
Abstract: This paper shows that the group of auto-homeomorphisms of a topological group can be endowed with a topology so that the resulting topological group plays, for topological groups, the role of the group of automorphisms of a group: it represents the internal actions on the given topological group.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10316/894242019-01-01T00:00:00ZCartesian closed exact completions in topologyhttp://hdl.handle.net/10316/89417Title: Cartesian closed exact completions in topology
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Ribeiro, Willian
Abstract: Using generalized enriched categories, in this paper we show that Rosický's proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over Set, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiability of (T,V)-categories and show that, under suitable conditions, every injective (T,V)-category is exponentiable in (T,V)-Cat.
Sat, 01 Feb 2020 00:00:00 GMThttp://hdl.handle.net/10316/894172020-02-01T00:00:00ZLax orthogonal factorisations in monad-quantale-enriched categorieshttp://hdl.handle.net/10316/89419Title: Lax orthogonal factorisations in monad-quantale-enriched categories
Authors: Clementino, Maria Manuel; López-Franco, Ignacio
Abstract: We show that, for a quantale V and a Set-monad T laxly extended to V-Rel, the presheaf monad on the category of (T,V)-categories is simple, giving rise to a lax orthogonal factorisation system (lofs) whose corresponding weak factorisation system has embeddings as left part. In addition, we present presheaf submonads and study the LOFSs they deﬁne. This provides a method of constructing weak factorisation systems on some well-known examples of topological categories over Set.
Fri, 01 Sep 2017 00:00:00 GMThttp://hdl.handle.net/10316/894192017-09-01T00:00:00ZSome remarks on protolocalizations and protoadditive reﬂectionshttp://hdl.handle.net/10316/89422Title: Some remarks on protolocalizations and protoadditive reﬂections
Authors: Clementino, Maria Manuel; Gran, Marino; Janelidze, George
Abstract: We investigate additional properties of protolocalizations, introduced and studied by Borceux, Clementino, Gran, and Sousa, and of protoadditive reﬂections, introduced and studied by Everaert and Gran. Among other things, we show that there are no nontrivial (protolocalizations and) protoadditive reﬂections of the category of groups, and establish a connection between protolocalizations and Kurosh–Amitsur radicals of groups with multiple operators whose semisimple classes form subvarieties.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10316/894222018-01-01T00:00:00ZLocal homeomorphisms via ultrafilter convergencehttp://hdl.handle.net/10316/11436Title: Local homeomorphisms via ultrafilter convergence
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Janelidze, George
Abstract: Using the ultrafilter-convergence description of topological spaces,
we generalize Janelidze-Sobral characterization of local homeomorphisms
between finite topological spaces, showing that local homeomorphisms are
the pullback-stable discrete fibrations.
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/10316/114362003-01-01T00:00:00ZExponentiation in V-categorieshttp://hdl.handle.net/10316/4614Title: Exponentiation in V-categories
Authors: Clementino, Maria Manuel; Hofmann, Dirk
Abstract: For a Heyting algebra V which, as a category, is monoidal closed, we obtain characterizations of exponentiable objects and morphisms in the category of V-categories and apply them to some well-known examples. In the case these characterizations of exponentiable morphisms and objects in the categories (P)Met of (pre)metric spaces and non-expansive maps show in particular that exponentiable metric spaces are exactly the almost convex metric spaces, while exponentiable complete metric spaces are the complete totally convex ones.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/46142006-01-01T00:00:00ZTorsion theories and radicals in normal categorieshttp://hdl.handle.net/10316/4611Title: Torsion theories and radicals in normal categories
Authors: Clementino, M. M.; Dikranjan, D.; Tholen, W.
Abstract: We introduce a relativized notion of (semi)normalcy for categories that come equipped with a proper stable factorization system, and we use radicals (in the sense of module theory) and normal closure operators in order to study torsion theories in such categories. Our results generalize and complement recent studies in the realm of semi-abelian and, in part, homological categories. In particular, we characterize both, torsion-free and torsion classes, in terms of their closure under extensions. We pay particular attention to the homological and, for our purposes more importantly, normal categories of topological algebra, such as the category of topological groups. But our applications go far beyond the realm of these types of categories, as they include, for example, the normal, but non-homological category of pointed topological spaces, which is in fact a rich supplier for radicals of topological groups.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/46112006-01-01T00:00:00ZProtolocalisations of homological categorieshttp://hdl.handle.net/10316/4577Title: Protolocalisations of homological categories
Authors: Borceux, Francis; Clementino, Maria Manuel; Gran, Marino; Sousa, Lurdes
Abstract: A protolocalisation of a homological (resp. semi-abelian) category is a regular full reflective subcategory, whose reflection preserves short exact sequences. We study the closure operator and the torsion theory associated with such a situation. We pay special attention to the fibered, the regular epireflective and the monoreflective cases. We give examples in algebra, topos theory and functional analysis.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/45772008-01-01T00:00:00ZTopological protomodular algebrashttp://hdl.handle.net/10316/4613Title: Topological protomodular algebras
Authors: Borceux, F.; Clementino, Maria Manuel
Abstract: Topological groups have very striking properties, which have already been generalized to weaker "group like" structures, like various kinds of loops. This paper intends to show evidence that this generalization holds for a much wider class of theories, known as the protomodular theories, and which admit both an elegant categorical characterization and an easy description in universal algebra terms. Thus we propose a synthetic approach which allows to prove in a unique framework that the most striking properties of topological groups hold as well for loops or even semi-loops, rings with or without unit, associative algebras with or without unit, Lie algebras, Jordan algebras, Boolean algebras, Heyting algebras, Boolean rings, Heyting semi-lattices, and so on.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/46132006-01-01T00:00:00ZTopological semi-abelian algebrashttp://hdl.handle.net/10316/4633Title: Topological semi-abelian algebras
Authors: Borceux, F.; Clementino, Maria Manuel
Abstract: Given an algebraic theory whose category of models is semi-abelian, we study the category of topological models of and generalize to it most classical results on topological groups. In particular, is homological, which includes Barr regularity and forces the Mal'cev property. Every open subalgebra is closed and every quotient map is open. We devote special attention to the Hausdorff, compact, locally compact, connected, totally disconnected and profinite -algebras.
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/10316/46332005-01-01T00:00:00ZOn regular and homological closure operatorshttp://hdl.handle.net/10316/13629Title: On regular and homological closure operators
Authors: Clementino, Maria Manuel; Gutierres, Gonçalo
Abstract: Observing that weak heredity of regular closure operators in Top and
of homological closure operators in homological categories identifies torsion theories,
we study these closure operators in parallel, showing that regular closure operators
play the same role in topology as homological closure operators do algebraically.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/136292009-01-01T00:00:00ZExponentiable functors between quantaloid-enriched categorieshttp://hdl.handle.net/10316/11324Title: Exponentiable functors between quantaloid-enriched categories
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Stubbe, Isar
Abstract: Exponentiable functors between quantaloid-enriched categories are characterized
in elementary terms. The proof goes as follows: the elementary conditions
on a given functor translate into existence statements for certain adjoints that obey
some lax commutativity; this, in turn, is precisely what is needed to prove the existence
of partial products with that functor; so that the functor’s exponentiability
follows from the works of Niefield [1980] and Dyckhoff and Tholen [1987].
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10316/113242006-01-01T00:00:00ZOn Connectedness via Closure Operatorshttp://hdl.handle.net/10316/7747Title: On Connectedness via Closure Operators
Authors: Clementino, Maria Manuel
Abstract: This paper describes a convenient modification of the approach presented in the paper “Closure operators and connectedness” by G. Castellini and D. Hajek, which is shown to give a suitable generalization of left- and right-constant subcategories, both at the object and the morphism levels. We show in particular that the framework we introduce here allows the simultaneous study of the classes of connected topological spaces, of concordant continuous maps and of monotone continuous maps.
Mon, 01 Jan 2001 00:00:00 GMThttp://hdl.handle.net/10316/77472001-01-01T00:00:00ZTopological Features of Lax Algebrashttp://hdl.handle.net/10316/7752Title: Topological Features of Lax Algebras
Authors: Clementino, Maria Manuel; Hofmann, Dirk
Abstract: Having as starting point Barr's description of topological spaces as lax algebras for the ultrafilter monad, in this paper we present further topological examples of lax algebras – such as quasi-metric spaces, approach spaces and quasi-uniform spaces – and show that, in a suitable setting, the categories of lax algebras have indeed a topological nature. Furthermore, we generalize to this setting known properties of special categories of lax algebras and, extending the construction of Manes, we describe the Cech–Stone compactification of lax algebras.
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/10316/77522003-01-01T00:00:00ZEffective Descent Morphisms in Categories of Lax Algebrashttp://hdl.handle.net/10316/7742Title: Effective Descent Morphisms in Categories of Lax Algebras
Authors: Clementino, Maria Manuel; Hofmann, Dirk
Abstract: In this paper we investigate effective descent morphisms in categories of reflexive and transitive lax algebras. We show in particular that open and proper maps are of effective descent, result that extends the corresponding results for the category of topological spaces and continuous maps.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/10316/77422004-01-01T00:00:00ZRelative injectivity as cocompleteness for a class of distributorshttp://hdl.handle.net/10316/11241Title: Relative injectivity as cocompleteness for a class of distributors
Authors: Clementino, Maria Manuel; Hofmann, Dirk
Abstract: Notions and techniques of enriched category theory can be used to
study topological structures, like metric spaces, topological spaces and approach
spaces, in the context of topological theories. Recently in [D. Hofmann, Injective
spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding
allowed to identify injectivity of spaces as cocompleteness and to show
monadicity of the category of injective spaces and left adjoints over Set. In this
paper we generalise these results, studying cocompleteness with respect to a given
class of distributors. We show in particular that the description of several semantic
domains presented in [M. Escard´o and B. Flagg, Semantic domains, injective spaces
and monads, Electronic Notes in Theoretical Computer Science 20 (1999)] can be
translated into the V-enriched setting.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/112412008-01-01T00:00:00ZOn the categorical meaning of Hausdorff and Gromov distances, Ihttp://hdl.handle.net/10316/11196Title: On the categorical meaning of Hausdorff and Gromov distances, I
Authors: Akhvlediani, Andrei; Clementino, Maria Manuel; Tholen, Walter
Abstract: Hausdor and Gromov distances are introduced and treated in the
context of categories enriched over a commutative unital quantale V. The Hausdor
functor which, for every V-category X, provides the powerset of X with a suitable
V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras
are order-complete. The Gromov construction may be pursued for any endofunctor
K of V-Cat. In order to de ne the Gromov \distance" between V-categories X and
Y we use V-modules between X and Y , rather than V-category structures on the
disjoint union of X and Y . Hence, we rst provide a general extension theorem
which, for any K, yields a lax extension ~K to the category V-Mod of V-categories,
with V-modules as morphisms.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/111962009-01-01T00:00:00Z