Search results for "Concrete category"
showing 8 items of 8 documents
On the category Set(JCPos)
2006
Category Set(JCPos) of lattice-valued subsets of sets is introduced and studied. We prove that it is topological over SetxJCPos and show its ''natural'' coalgebraic subcategory.
Some remarks on the category SET(L), part III
2004
This paper considers the category SET(L) of L-subsets of sets with a fixed basis L and is a continuation of our previous investigation of this category. Here we study its general properties (e.g., we derive that the category is a topological construct) as well as some of its special objects and morphisms.
Topological systems and Artin glueing
2012
Abstract Using methods of categorical fuzzy topology, the paper shows a relation between topological systems of S. Vickers and Artin glueing of M. Artin. Inspired by the problem of interrelations between algebra and topology, we show the necessary and sufficient conditions for the category, obtained by Artin glueing along an adjoint functor, to be (co)algebraic and (co)monadic, incorporating the respective result of G. Wraith. As a result, we confirm the algebraic nature of the category of topological systems, showing that it is monadic.
Categories of lattice-valued sets as categories of arrows
2006
In this paper we introduce a category X(A) which is a generalization of the category of lattice-valued subsets of sets Set(JCPos) introduced by us earlier. We show the necessary and sufficient conditions for X(A) to be topological over XxA.
On limits and colimits of variety-based topological systems
2011
The paper provides variety-based extensions of the concepts of (lattice-valued) interchange system and space, introduced by Denniston, Melton and Rodabaugh, and shows that variety-based interchange systems incorporate topological systems of Vickers, state property systems of Aerts, Chu spaces (over the category of sets in the sense of Pratt) of P.-H. Chu and contexts (of formal concept analysis) of Wille. The paper also provides an explicit description of (co)limits in the category of variety-based topological systems and applies the obtained results to extend the claim of Denniston et al. that the category of topological systems of Vickers is small initially topological over the category o…
Category, Measure, Inductive Inference: A Triality Theorem and Its Applications
2002
The famous Sierpinski-Erdos Duality Theorem [Sie34b, Erd43] states, informally, that any theorem about effective measure 0 and/or first category sets is also true when all occurrences of "effective measure 0" are replaced by "first category" and vice versa. This powerful and nice result shows that "measure" and "category" are equally useful notions neither of which can be preferred to the other one when making formal the intuitive notion "almost all sets." Effective versions of measure and category are used in recursive function theory and related areas, and resource-bounded versions of the same notions are used in Theory of Computation. Again they are dual in the same sense.We show that in…
Hypergraph functor and attachment
2010
Using an arbitrary variety of algebras, the paper introduces a fuzzified version of the notion of attachment in a complete lattice of Guido, to provide a common framework for the concept of hypergraph functor considered by different authors in the literature. The new notion also gives rise to a category of variable-basis topological spaces which is a proper supercategory of the respective category of Rodabaugh.
On a generalization of Goguen's category Set(L)
2007
The paper considers a category which generalizes Goguen's category Set(L) of L-fuzzy sets with a fixed basis L. We show the necessary and sufficient conditions for the generalized category to be a quasitopos and consider additional inner structure supplied by the latter property.