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Metric, Topology and Multicategory  A Common Approach
 J. Pure Appl. Algebra
, 2001
"... For a symmetric monoidalclosed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T , V)algebra and show that various old and new structures are instances of such algebras. Lawvere's presentation of a metric space as a Vcategory is ..."
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Cited by 18 (7 self)
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For a symmetric monoidalclosed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T , V)algebra and show that various old and new structures are instances of such algebras. Lawvere's presentation of a metric space as a Vcategory is included in our setting, via the BettiCarboniStreetWalters interpretation of a Vcategory as a monad in the bicategory of Vmatrices, and so are Barr's presentation of topological spaces as lax algebras, Lowen's approach spaces, and Lambek's multicategories, which enjoy renewed interest in the study of ncategories. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.
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.
Tholen: A functional approach to general topology
 Categorical Foundations (Cambridge
, 2004
"... ..."
Local homeomorphisms via ultrafilter convergence
 Proc. Am. Math. Soc
"... Abstract. Using the ultrafilterconvergence description of topological spaces, we generalize JanelidzeSobral characterization of local homeomorphisms between finite topological spaces, showing that local homeomorphisms are the pullbackstable discrete fibrations. ..."
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Cited by 2 (2 self)
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Abstract. Using the ultrafilterconvergence description of topological spaces, we generalize JanelidzeSobral characterization of local homeomorphisms between finite topological spaces, showing that local homeomorphisms are the pullbackstable discrete fibrations.
Exponentiability Of Perfect Maps: Four Approaches
, 2002
"... Two proofs of the exponentiability of perfect maps are presented and compared to two other recent approaches. One of the proofs is an elementary approach including a direct construction of the exponentials. The other, implicit in the literature, uses internal locales in the topos of setvalued sh ..."
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Cited by 1 (1 self)
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Two proofs of the exponentiability of perfect maps are presented and compared to two other recent approaches. One of the proofs is an elementary approach including a direct construction of the exponentials. The other, implicit in the literature, uses internal locales in the topos of setvalued sheaves on a topological space.
Contents
, 904
"... particular some aspects related to “internal ” (co)limits and to Cauchy completeness are considered. In the second part, we maintain that also some aspects of topology can be effectively synthesized in a (weak) balanced factorization category T, whose objects should be considered as possibly “infini ..."
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particular some aspects related to “internal ” (co)limits and to Cauchy completeness are considered. In the second part, we maintain that also some aspects of topology can be effectively synthesized in a (weak) balanced factorization category T, whose objects should be considered as possibly “infinitesimal ” and suitably “regular ” topological spaces. While in C the classes M and M ′ play the role of discrete fibrations and opfibrations, in T they play the role of local homeomorphisms and perfect maps, so that M/1 and M ′ /1 are the subcategories of discrete and compact spaces respectively. One so gets a direct abstract link between the subjects, with mutual benefits. For example, the slice projection X/x → X and the coslice projection x\X → X, obtained as the second factors of x: 1 → X according to (E, M) and (E ′ , M ′ ) in C, correspond in T to the “infinitesimal ” neighborhood of x ∈ X and to the closure of x. Furthermore, the openclosed complementation (generalized to reciprocal stability) becomes the key tool to internally treat, in a coherent way, some categorical concepts (such as (co)limits
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.
Under consideration for publication in Math. Struct. in Comp. Science Exponentiable morphisms of domains
, 2008
"... converse is not true. We find then the extra conditions needed on f exponentiable in Pos to be exponentiable in ωCpo, showing the existence of partial products of the twopoint ordered set S = {0 < 1} (Theorem 1.8). Using this characterization and the embedding via the Scott topology of ωCpo in ..."
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converse is not true. We find then the extra conditions needed on f exponentiable in Pos to be exponentiable in ωCpo, showing the existence of partial products of the twopoint ordered set S = {0 < 1} (Theorem 1.8). Using this characterization and the embedding via the Scott topology of ωCpo in the category Top of topological spaces, we can compare exponentiability in each setting, obtaining that a morphism in ωCpo, exponentiable both in Top and in Pos, is exponentiable also in ωCpo. Furthermore we show that the exponentiability in Top and in Pos are independent from each other.
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.