Categorical structure of closure operators. With applications to topology, algebra and discrete mathematics.(English)Zbl 0853.18002

Mathematics and its Applications (Dordrecht). 346. Dordrecht: Kluwer Academic Publishers. xvii, 356 p. (1995).
This book provides the first comprehensive investigation in bookform of closure operators in suitable categories with a rich array of examples drawn mainly from topology and algebra. A closure operator in a category $${\mathcal C}$$ supplied with a suitable notion of subobject is defined as an extensional, monotone, continuous operator on subobjects. Alternatively a closure operator can be considered quite naturally as a pointed endofunctor of the category of all subobjects of $${\mathcal C}$$. Closure operators give rise to factorizations of morphisms (via the closure of the image). Nice closure operators (e.g., idempotent, weakly hereditary ones) provide nice factorization structures (e.g., dense-closed) for morphisms. Besides idempotency and (weak) hereditariness, additivity turns out to be an important property a closure operator may or may not have. These properties are thoroughly investigated and illuminated by many interesting examples. As a main application of the theory the authors demonstrate how in various categories $${\mathcal C}$$ (particularly of a topological nature) epimorphisms can be characterized via suitable closure operators and how the often hard question whether $${\mathcal C}$$ is cowellpowered can be solved in these cases. Other aspects of the use of closure operators, e.g., the generalizations of fundamental topological concepts like compactness and separatedness are only sketched and are likely to be pursued in some further publication. The book contains a bibliography with 179 entries, and each of the 9 chapters ends with a collection of challenging exercises and valuable historical notes.

MSC:

 18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms 18-02 Research exposition (monographs, survey articles) pertaining to category theory 18A32 Factorization systems, substructures, quotient structures, congruences, amalgams 18B30 Categories of topological spaces and continuous mappings (MSC2010) 54B30 Categorical methods in general topology