This book presents the general theory of categorical closure operators toยญ gether with a number of examples, mostly drawn from topology and algeยญ bra, which illustrate the general concepts in several concrete situations. It is aimed mainly at researchers and graduate students in the area of cateยญ gorical topology, and to those interested in categorical methods applied to the most common concrete categories. Categorical Closure Operators is self-contained and can be considered as a graduate level textbook for topics courses in algebra, topology or category theory. The reader is expected to have some basic knowledge of algebra, topology and category theory, however, all categorical concepts that are recurrent are included in Chapter 2. Moreover, Chapter 1 contains all the needed results about Galois connections, and Chapter 3 presents the theยญ ory of factorization structures for sinks. These factorizations not only are essential for the theory developed in this book, but details about them canยญ not be found anywhere else, since all the results about these factorizations are usually treated as the duals of the theory of factorization structures for sources. Here, those hard-to-find details are provided. Throughout the book I have kept the number of assumptions to a minยญ imum, even though this implies that different chapters may use different hypotheses. Normally, the hypotheses in use are specified at the beginning of each chapter and they also apply to the exercise set of that chapter
CONTENT
I GENERAL THEORY -- 1 Galois Connections -- 2 Some Categorical Concepts -- 3 Factorization Structures For Sinks -- 4 Closure Operators: Definition and Examples -- 5 Idempotency, Weak Heredity and Factorization Structures -- 6 Additivity, Heredity, Suprema and Infima of Closure Operators -- 7 Additional Descriptions of ? and ? and Subobject Orthogonality -- 8 A Diagram of Galois Connections of Closure Operators -- 9 Regular Closure Operators -- 10 Hereditary Regular Closure Operators -- 11 APPLICATIONS -- 11 Epimorphisms -- 12 Separation -- 13 Compactness -- 14 Connectedness -- 15 Connectedness in Categories with a Terminal Object -- 16 A Link between two Connectedness Notions -- 17 Different Constructions Related -- References -- List of Symbols
SUBJECT
Mathematics
Category theory (Mathematics)
Homological algebra
Partial differential equations
Applied mathematics
Engineering mathematics
Statistics
Mathematics
Category Theory
Homological Algebra
Partial Differential Equations
Applications of Mathematics
Statistics for Business/Economics/Mathematical Finance/Insurance
