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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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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
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 10 (5 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.
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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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 (
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 9 (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 Kleislibased approach to lax algebras
 Appl. Categ. Structures
, 2007
"... By exploiting the description of topological spaces by either neighborhood systems or filter convergence, we obtain a neighborhoodlike presentation of categories of lax algebras. A notable advantage of this approach is that it does not require the introduction of a lax extension of the associated m ..."
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By exploiting the description of topological spaces by either neighborhood systems or filter convergence, we obtain a neighborhoodlike presentation of categories of lax algebras. A notable advantage of this approach is that it does not require the introduction of a lax extension of the associated monad functor. As a byproduct, the different philosophies underlying the construction of fuzzy topological spaces on one hand, and approach spaces on the other, may be simply expressed in terms of lax algebras.
Injective spaces via adjunction
 J. Pure Appl. Algebra
, 2011
"... Abstract. Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relationx − → x between ultrafilters and points of a top ..."
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Cited by 5 (5 self)
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Abstract. Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relationx − → x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from (enriched) Category Theory for the investigation of (for instance) topological spaces. In this paper we study cocompleteness, adjoint functors and Kan extensions in the context of topological theories. We show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on Set. This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.
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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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
Approximation in quantaleenriched categories
 DIRK HOFMANN AND PAWE L WASZKIEWICZ
, 2010
"... ar ..."