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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 (
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 8 (4 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.
Universal properties of Span
 in The Carboni Festschrift, Theory and Applications of Categories 13 (2005
"... Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphi ..."
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Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphisms.
THE SPAN CONSTRUCTION
"... Abstract. We present two generalizations of the Span construction. The first generalization ..."
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Abstract. We present two generalizations of the Span construction. The first generalization
The enriched Vietoris monad on representable spaces
, 2012
"... Abstract. Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the downset monad becomes the filter monad, cocomplete ordered set translate ..."
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Abstract. Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the downset monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “upset monad ” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of KockZöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “upset monad”. We emphasize that these generic categorical notions and results can be indeed connected to more “classical ” topology: for topological spaces, the “upset monad ” becomes the lower Vietoris monad, and the statement “X is totally cocomplete if and only if Xop is totally complete” specialises to O. Wyler’s characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces.
On étale algebraic homomorphisms
"... a map is exponentiable if, and only if, it is étale (see [1]). Briefly, to prove that étale continuous maps between compact Hausdorff spaces are exponentiable, Cagliari and Mantovani build exponentials in the quasitopos of pseudotopological spaces and show that the exponentials belong to CompHaus ..."
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a map is exponentiable if, and only if, it is étale (see [1]). Briefly, to prove that étale continuous maps between compact Hausdorff spaces are exponentiable, Cagliari and Mantovani build exponentials in the quasitopos of pseudotopological spaces and show that the exponentials belong to CompHaus whenever the map is étale. Using techniques developed in the study of lax algebras (see [4, 2, 3]), this can be generalized to any category CT of EilenbergMoore algebras for a monad T on Set satisfying the BeckChevalley condition (BC). Indeed, every such category can be embedded in a quasitopos and it can be shown that exponentials of étale algebraic homomorphisms belong to CT. In this talk I will outline this proof and discuss the role of (BC) in this context.
ON EXTENSIONS OF LAX MONADS
"... Abstract. In this paper we construct extensions of Setmonads { and, more generally, lax Relmonads { into lax monads of the bicategory Mat(V) of generalized Vmatrices, whenever V is a wellbehaved lattice equipped with a tensor product. We add some guiding examples. ..."
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Abstract. In this paper we construct extensions of Setmonads { and, more generally, lax Relmonads { into lax monads of the bicategory Mat(V) of generalized Vmatrices, whenever V is a wellbehaved lattice equipped with a tensor product. We add some guiding examples.
ON EXTENSIONS OF LAX MONADS Dedicated to Aurelio Carboni on the occasion of his sixtieth birthday
"... Abstract. In this paper we construct extensions of Setmonads and, more generally, of lax Relmonads into lax monads of the bicategory Mat(V) of generalized Vmatrices, whenever V is a wellbehaved lattice equipped with a tensor product. Weadd some guiding examples. ..."
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Abstract. In this paper we construct extensions of Setmonads and, more generally, of lax Relmonads into lax monads of the bicategory Mat(V) of generalized Vmatrices, whenever V is a wellbehaved lattice equipped with a tensor product. Weadd some guiding examples.