Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 06 Aug 2020 07:36:02 GMT2020-08-06T07:36:02Z5061One Setting for All: Metric, Topology, Uniformity, Approach Structurehttp://hdl.handle.net/10316/7757Title: One Setting for All: Metric, Topology, Uniformity, Approach Structure
Authors: Clementino, Maria; Hofmann, Dirk; Tholen, Walter
Abstract: Abstract For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T,V)-algebras and of (T,V)-proalgebras turn out to be topological over Set.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/10316/77572004-01-01T00:00:00ZSeparated and Connected Mapshttp://hdl.handle.net/10316/7758Title: Separated and Connected Maps
Authors: Clementino, Maria Manuel; Tholen, Walter
Abstract: Using on the one hand closure operators in the sense of Dikranjan and Giuli and on the other hand left- and right-constant subcategories in the sense of Herrlich, Preuß, Arhangel'skii and Wiegandt, we apply two categorical concepts of connectedness and separation/disconnectedness to comma categories in order to introduce these notions for morphisms of a category and to study their factorization behaviour. While at the object level in categories with enough points the first approach exceeds the second considerably, as far as generality is concerned, the two approaches become quite distinct at the morphism level. In fact, left- and right-constant subcategories lead to a straight generalization of Collins' concordant and dissonant maps in the category $$\mathcal{T}op$$ of topological spaces. By contrast, closure operators are neither able to describe these types of maps in $$\mathcal{T}op$$, nor the more classical monotone and light maps of Eilenberg and Whyburn, although they give all sorts of interesting and closely related types of maps. As a by-product we obtain a negative solution to the ten-year-old problem whether the Giuli–Hušek Diagonal Theorem holds true in every decent category, and exhibit a counter-example in the category of topological spaces over the 1-sphere.
Thu, 01 Jan 1998 00:00:00 GMThttp://hdl.handle.net/10316/77581998-01-01T00:00:00ZProper maps for lax algebras and the Kuratowski-Mrówka Theoremhttp://hdl.handle.net/10316/42770Title: Proper maps for lax algebras and the Kuratowski-Mrówka Theorem
Authors: Clementino, Maria Manuel; Tholen, Walter
Abstract: The characterization of stably closed maps of topological spaces as the
closed maps with compact fibres and the role of the Kuratowski-Mrowka' Theorem in this characterization are being explored in the general context of lax (T; V )-algebras, for a quantale V and a Set-monad T with a lax extension to V -relations. The general results are being applied in standard (topological and metric) and non-standard (labeled graphs) contexts.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10316/427702013-01-01T00:00:00ZExponentiability in categories of lax algebrashttp://hdl.handle.net/10316/42772Title: Exponentiability in categories of lax algebras
Authors: Clementino, Maria Manuel; Hofmann, Dirk; Tholen, Walter
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/10316/427722003-01-01T00:00:00ZFrom lax monad extensions to Topological theorieshttp://hdl.handle.net/10316/43927Title: From lax monad extensions to Topological theories
Authors: Clementino, Maria Manuel; Tholen, Walter
Abstract: We investigate those lax extensions of a Set-monad T = (T,m, e) to the category V-Rel of sets and V-valued relations for a quantale V = (V,⊗, k) that are fully determined by ξ maps : TV ⟶ V. We pay
special attention to those maps ξ that make V a T-algebra and, in fact, (V,⊗, k) a monoid in the category Set^T with its cartesian structure. Any such map ξ forms the main ingredient to Hofmann’s notion of topological theory.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/439272014-01-01T00:00:00ZOn the categorical meaning of Hausdorff and Gromov distances, Ihttp://hdl.handle.net/10316/11196Title: On the categorical meaning of Hausdorff and Gromov distances, I
Authors: Akhvlediani, Andrei; Clementino, Maria Manuel; Tholen, Walter
Abstract: Hausdor and Gromov distances are introduced and treated in the
context of categories enriched over a commutative unital quantale V. The Hausdor
functor which, for every V-category X, provides the powerset of X with a suitable
V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras
are order-complete. The Gromov construction may be pursued for any endofunctor
K of V-Cat. In order to de ne the Gromov \distance" between V-categories X and
Y we use V-modules between X and Y , rather than V-category structures on the
disjoint union of X and Y . Hence, we rst provide a general extension theorem
which, for any K, yields a lax extension ~K to the category V-Mod of V-categories,
with V-modules as morphisms.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/111962009-01-01T00:00:00Z