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One Setting for All: Metric, Topology, Uniformity, Approach Structure
"... For a complete lattice V which, as a category, is monoidal closed, and for a suitable Setmonad 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 laxalgebraic setting, u ..."
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Cited by 18 (10 self)
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For a complete lattice V which, as a category, is monoidal closed, and for a suitable Setmonad 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 laxalgebraic 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.
The Convergence Approach to Exponentiable Maps
 352 MARIA MANUEL CLEMENTINO, DIRK HOFMANN AND WALTER
, 2000
"... Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the ..."
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Cited by 9 (7 self)
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Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the classical result for compact Hausdorff spaces. Furthermore, in generalization of the WhiteheadMichael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.
Exponentiability in Categories of Lax Algebras
, 2003
"... For a complete cartesianclosed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable BeckChevalleytype condition, it is shown that the category of lax reflexive (T , V)algebras is a quasitopos. This result encompasses many known and new examp ..."
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Cited by 6 (2 self)
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For a complete cartesianclosed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable BeckChevalleytype condition, it is shown that the category of lax reflexive (T , V)algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. 1.
Effective descent morphisms in categories of lax algebras
 Appl. Categ. Structures
, 2002
"... 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 effective descent, result that extends the corresponding results for the category of topological spaces and continuous maps. ..."
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Cited by 6 (3 self)
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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 effective descent, result that extends the corresponding results for the category of topological spaces and continuous maps.
Lawvere completeness in Topology
, 2008
"... It is known since 1973 that Lawvere’s notion of (Cauchy)complete enriched category is meaningful for metric spaces: it captures exactly Cauchycomplete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)categories and show that it has an interestin ..."
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Cited by 5 (3 self)
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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 Cauchycomplete 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 quasiuniform 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 Lawverecomplete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of (Ì, V)categories.
A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES
"... Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “l ..."
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Cited by 4 (0 self)
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Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous
Ordered topological structures
 Topology Appl
, 2009
"... The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general ..."
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Cited by 4 (2 self)
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The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general level of lax algebras, so that our categories do not concern just ordered topological spaces, but also sets with two interacting orders, approach spaces with an additional metric, etc. Key words: modular topological space, closedordered topological space, openordered topological space, lax (T, V)algebra, (T, V)category
A Monadic Approach to PolyCategories
 Theory Appl. Categ
, 2002
"... Polycategories form a rather natural generalization of multicategories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Multicategories are known to have an elegant global characterization as monads in a suitable bicategory of special spans with free m ..."
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Cited by 2 (0 self)
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Polycategories form a rather natural generalization of multicategories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Multicategories are known to have an elegant global characterization as monads in a suitable bicategory of special spans with free monoid as domains. To describe polycategories in similar terms, we investigate distributive laws in the sense of Beck between cartesian monads as tools for constructing new bicategories of modi ed spans. Three very simple such laws produce a bicategory in which the monads are precisely the planar polycategories (where composition only is de ned if the corresponding circuit diagram is planar). General polycategories, which only satisfy a local planarity condition, require a slightly more complicated construction.
A ∈ ˙x ⇐ ⇒ x ∈ A,
, 2007
"... 2.1. Some monads on Set and their lax extensions. For sets X, Y the natural bijection translates to Rel(X, Y) ∼ � � Rel(Y, X), r ↦− → r ◦, Set(X, P Y) ∼ � � Set(Y, P X) = Set op (P X, Y), showing the selfadjointness of the contravariant powerset functor P ⊢ P op, with P: Set − → Set op, (f: X − ..."
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2.1. Some monads on Set and their lax extensions. For sets X, Y the natural bijection translates to Rel(X, Y) ∼ � � Rel(Y, X), r ↦− → r ◦, Set(X, P Y) ∼ � � Set(Y, P X) = Set op (P X, Y), showing the selfadjointness of the contravariant powerset functor P ⊢ P op, with P: Set − → Set op, (f: X − → Y) ↦− → (P f: B ↦− → f −1 [B]). The induced monad P 2 = (P op P, e, m) is given by eX: X − → P 2 X, x ↦− → ˙x (the principal filter on x),
LAWVERE COMPLETENESS AS A TOPOLOGICAL PROPERTY
"... Abstract. Lawvere’s notion of completeness for quantaleenriched categories has been extended to the theory of lax algebras under the name of Lcompleteness. In this paper we introduce the corresponding morphism concept and examine its properties. We explore some important relativized topological co ..."
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Abstract. Lawvere’s notion of completeness for quantaleenriched categories has been extended to the theory of lax algebras under the name of Lcompleteness. In this paper we introduce the corresponding morphism concept and examine its properties. We explore some important relativized topological concepts like separatedness, denseness, compactness and compactification with respect to Lcomplete morphisms. Moreover, we show that separated Lcomplete morphisms belong to a factorization system. 1.