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Notions of Lawvere theory
"... Categorical universal algebra can be developed either using Lawvere theories (singlesorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of ..."
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Categorical universal algebra can be developed either using Lawvere theories (singlesorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of
Towards “dynamic domains”: totally continuous cocomplete Qcategories
 Theoretical Computer Science
, 2007
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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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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.
RELATIVE INJECTIVITY AS COCOMPLETENESS FOR A CLASS OF DISTRIBUTORS
"... Dedicated to Walter Tholen on the occasion of his sixtieth birthday ..."
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Dedicated to Walter Tholen on the occasion of his sixtieth birthday
2007a), Components, Complements and the Reflection Formula, Theory and
"... ABSTRACT. Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations in categories over a base category X are considered. In particular, we illustrate the formulas (↓P)x = ten(x/X, P) ; (P↓)x = hom(X/x, P) ..."
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ABSTRACT. Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations in categories over a base category X are considered. In particular, we illustrate the formulas (↓P)x = ten(x/X, P) ; (P↓)x = hom(X/x, P)
DUALITY FOR CCD LATTICES
"... Abstract. The 2category of constructively completely distributive lattices is shown to be bidual to a 2category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2category of ordered sets. 1. ..."
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Abstract. The 2category of constructively completely distributive lattices is shown to be bidual to a 2category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2category of ordered sets. 1.
Completions of nonsymmetric metric spaces via enriched categories
 Georgian Math. J
"... Abstract. It is known from [13] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0,∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits ..."
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Abstract. It is known from [13] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0,∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0,∞] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the socalled dcpo completion.
Under consideration for publication in Math. Struct. in Comp. Science Quasivarieties and varieties of ordered algebras: Regularity and exactness
"... We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets. We also prove that finitary ..."
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We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets. We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, in the enriched sense). 1.