Fundamental duality of abstract categories and its applications


Demirci M.

FUZZY SETS AND SYSTEMS, vol.256, pp.73-94, 2014 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 256
  • Publication Date: 2014
  • Doi Number: 10.1016/j.fss.2013.08.015
  • Journal Name: FUZZY SETS AND SYSTEMS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.73-94
  • Keywords: Category theory, Duality, Adjoint situation, Stone duality, Categorical topology, Many-valued topology, Fuzzy topology, SEPARATION AXIOMS, REPRESENTATION, FOUNDATIONS, COMPACTNESS, OBJECTS
  • Akdeniz University Affiliated: Yes

Abstract

Given an abstract category C, C-M-L-spaces that are categorical generalization of fixed-basis fuzzy topological spaces in C and their category C-M-L-Top are introduced. It is pointed out, as one of the main contributions of this paper, that C is dually adjoint to C-M-L-Top. By defining L-spatiality in C and L-sobriety in C-M-L-Top, this adjunction induces a dual equivalence between the full subcategory of C of all L-spatial objects and the full subcategory of C-M-L-Top of all L-sober objects. The present adjunction and duality are fruitful categorical extensions of the classical Top-Loc adjunction and SobTop-SpatLoc duality to abstract categories with a great deal of applications. In particular, their applications to Q-categories, quasivarieties and augmented posets are given. (C) 2013 Elsevier B.V. All rights reserved.