Applied Categorical Structures – Official Blog

by Lowen, Bob

DOI: 10.1007/s10485-007-9101-8
Online Date: 6/29/2007
Print publication date: 6/1/2007
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by Bosi, Gianni; Herden, Gerhard

In this paper the continuous utility representation problem will be discussed in arbitrary concrete categories. In particular, generalizations of the utility representation theorems of Eilenberg, Debreu and Estévez and Hervés will be presented that also hold if the codomain of a utility function is an arbitrary totally ordered set and not just the real line. In addition, we shall prove and apply a general result on the characterization of structures that have the property that every continuous total preorder has a continuous utility representation. Finally, generalizations of the utility representation theorems of Debreu and Eilenberg will be discussed that are valid if we consider arbitrary binary relations and allow a utility function to have values in an arbitrary totally ordered set.

DOI: 10.1007/s10485-007-9097-0
Online Date: 6/28/2007
Print publication date: 10/1/2008
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by Facchini, Alberto; Fernández-Alonso, Rogelio

We study the notions of coproduct and subdirect product of preadditive categories and prove a Birkhoff type theorem showing that every skeletally small preadditive category is a subdirect product of subdirectly irreducible, skeletally small, preadditive categories. Moreover, we show that every direct-sum decomposition of the monoid $V({\cal A})$ of the isomorphism classes of objects of $\cal A$ is weakly induced by a coproduct decomposition of the preadditive category $\cal A$.

DOI: 10.1007/s10485-007-9093-4
Online Date: 6/23/2007
Print publication date: 4/1/2008
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by Baboolal, Dharmanand; Pillay, Paranjothi

Given any metric space, we construct its uniformly locally pathwise connected coreflection in the category of all metric spaces and uniformly continuous maps.

DOI: 10.1007/s10485-007-9096-1
Online Date: 6/23/2007
Print publication date: 8/1/2008
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by Janelidze, Zurab

A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini’s characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation – we show that a finitely complete pointed category is protomodular if and only if every binary internal relation R→A
² in it has this closedness property.

DOI: 10.1007/s10485-007-9072-9
Online Date: 6/23/2007
Print publication date: 6/1/2007
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by Seal, Gavin J.

By exploiting the description of topological spaces by either neighborhood systems or filter convergence, we obtain a neighborhood-like presentation of categories of lax algebras. The simplicity of this presentation pinpoints the importance of the Kleisli extension, which is introduced as a particular lax extension of the associated monad functor.

DOI: 10.1007/s10485-007-9080-9
Online Date: 6/5/2007
Print publication date: 2/1/2009
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by Abuhlail, Jawad Y.

In this paper we introduce and investigate top (bi)comodules of corings, that can be considered as dual to top (bi)modules of rings. The fully coprime spectra of such (bi)comodules attains a Zariski topology, defined in a way dual to that of defining the Zariski topology on the prime spectra of (commutative) rings. We restrict our attention in this paper to duo (bi)comodules (satisfying suitable conditions) and study the interplay between the coalgebraic properties of such (bi)comodules and the introduced Zariski topology. In particular, we apply our results to introduce a Zariski topology on the fully coprime spectrum of a given non-zero coring considered canonically as duo object in its category of bicomodules.

DOI: 10.1007/s10485-007-9088-1
Online Date: 6/1/2007
Print publication date: 4/1/2008
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by Raussen, Martin

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.

DOI: 10.1007/s10485-007-9085-4
Online Date: 6/1/2007
Print publication date: 8/1/2007
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