Abstract: Categorical topology applies the results and methods of the Category Theory to topological structures and, conversely, generalizes some results and methods from topological structures to more general categories. A category C is defined as a class of objects such as X,Y (topological spaces, groups, and lattices) together with sets C (X,Y ) of morphisms between objects X, Y (continuous maps, homomorphisms, and monotone maps) satisfying some natural conditions (composition of morphisms exists; it is associative and has units, that is, identity morphisms on objects exist) ñ. Thus, a category Set of sets and maps, Top of topological spaces with continuous maps, Unif of uniform spaces with uniformly continuous maps, and TopGr of topological groups with continuous homomorphisms are obtained. For metric spaces as objects, morphisms may be chosen—such as continuous maps or uniformly continuous maps or Lipschitz maps or contractions. A subclass Κ of objects of C together with subsets of morphisms that form a category under the same composition and units is called a subcategory of C (in case the sets of morphisms of Κ are the same as in C, Κ is called a full subcategory of C).
Publication Year: 2003
Publication Date: 2003-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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