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14
Canonical and opcanonical lax algebras
 Theory Appl. Categ
, 2005
"... Abstract. The definition of a category of (T, V)algebras, where V is a unital commutative quantale and T is a Setmonad, requires the existence of a certain lax extensionof T. In this article, we present a general construction of such an extension. This leads tothe formation of two categories of ( ..."
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Cited by 7 (2 self)
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Abstract. The definition of a category of (T, V)algebras, where V is a unital commutative quantale and T is a Setmonad, requires the existence of a certain lax extensionof T. In this article, we present a general construction of such an extension. This leads tothe formation of two categories of (
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.
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
On the categorical meaning of Hausdorff and Gromov distances
 I. Topology Appl
"... Abstract. Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every Vcategory X, provides the powerset of X with a suitable Vcategory structure, is part of a monad on VCat whose Eile ..."
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Cited by 2 (2 self)
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Abstract. Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every Vcategory X, provides the powerset of X with a suitable Vcategory structure, is part of a monad on VCat whose EilenbergMoore algebras are ordercomplete. The Gromov construction may be pursued for any endofunctor K of VCat. In order to define the Gromov “distance ” between Vcategories X and Y we use Vmodules between X and Y, rather than Vcategory structures on the disjoint union of X and Y. Hence, we first provide a general extension theorem which, for any K, yields a lax
EXTENSIONS IN THE THEORY OF LAX ALGEBRAS Dedicated
"... Abstract. Recent investigations of lax algebrasin generalization of Barr's relationalalgebrasmake an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monadthere may be many such lax extensions, an ..."
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Abstract. Recent investigations of lax algebrasin generalization of Barr's relationalalgebrasmake an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monadthere may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on these lax extensions, andpresent a symptomatic situation in which distinct monads yield isomorphic categories of lax algebras.
EXTENSIONS IN THE THEORY OF LAX ALGEBRAS Dedicated to Walter Tholen on the occasion of his 60th birthday
"... Abstract. Recent investigations of lax algebras—in generalization of Barr’s relational algebras—make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monad there may be many such lax extensions, and ..."
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Abstract. Recent investigations of lax algebras—in generalization of Barr’s relational algebras—make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monad there may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on these lax extensions, and present a symptomatic situation in which distinct monads yield isomorphic categories of lax algebras. 1.