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An elementary overview of mathematical structures. Algebra, topology and categories. (English) Zbl 1468.18001

Hackensack, NJ: World Scientific (ISBN 978-981-12-2031-9/hbk; 978-981-12-2033-3/ebook). x, 382 p. (2021).
The book provides a convenient introduction into the basic structures of algebra, topology, and category theory, with a view to emphasize an important scientific shift (taking its origin in an article from the middle of the last century [S. Eilenberg and S. MacLane, Trans. Am. Math. Soc. 58, 231–294 (1945; Zbl 0061.09204)]) from the study of particular entities on their own (e.g., an abelian group or a topological space) to the study of those entities as a whole (united in a category) and their possible interconnections (as a functor or, more generally, a natural transformation).
The author takes a distinctly pedagogical stance suggesting the book for the undergraduates as well as self-study. He divides it into seven chapters, where the first four recall the most important concepts from universal algebra and general topology, and the next two introduce a number of basic categorical tools (e.g., category, functor, and natural transformation) and show how the classical algebraic or topological notions find a convenient category-theoretic analogue. For a better understanding of the presented material, the most crucial book chapters are equipped with a number of exercises for self-study and advancement, with the last book chapter being devoted to their solutions and hints (leaving a plenty of space for the imagination and the creativity of the interested reader).
The book essentially consists of three parts. Every part takes two chapters, with the first chapter being a preliminary one, and the second chapter building a more advanced theory. The concepts are not presented from the simplest to the most complex one, but rather from a better known (called a “rich structure” in the introduction) to a more general one. For example, the algebraic part (Chapters 1 and 2) starts from the field of reals and real vector spaces, then continues with rings and abelian groups, and then touches semigroups, monoids, and groups. Additionally, the author considers ordered sets and (complete) lattices. Afterwards, goes a more advanced theory of rings, including modules and algebras over them. An important place take the so-called universal constructions in algebra, e.g., products and sums, free algebraic structures, subalgebras and quotient algebras. These universal constructions make a plethora of examples of the more general categorical ones. The topological part (Chapters 3 and 4) starts with Euclidean spaces and the notion of continuity as defined in the classical calculus. It then introduces abstract topological spaces (as sets equipped with subsets of the respective powersets satisfying certain conditions) and also a general notion of continuity of a map (the preimage of of an open set is open). The author then considers universal topological constructions, e.g., topological products and sums, subspaces and quotient spaces. Apart from them, the reader will also find separation axioms (e.g., Hausdorff spaces) as well as relationships between topology and order (e.g., specialization preorder and Alexandrov spaces). The more advanced part deals with connectedness and compactness of topological spaces, and also touches algebraic topology and topological algebra. The category-theoretic part (Chapters 5 and 6) starts with the concepts of category, functor, and natural transformation, and then considers free objects, subobjects and quotient objects as well as representable functors. The more advanced part considers limits and colimits, adjoint functors, and different applications of the theory of the latter (e.g., Galois connections between ordered sets). As a bonus, the author gives a brief introduction (a couple of pages) into the theory of abelian categories and monads. An upgraded and far more advanced version of the category-theoretic chapters can be found in yet another book of the same author, namely, in [M. Grandis, Category theory and applications. A textbook for beginners. Hackensack, NJ: World Scientific (2018; Zbl 1393.18001)].
The book is extremely well written and is really enjoyable to read. The author carefully follows his line of thread, without sudden jumps into an uncharted territory. The new concepts never appear out the blue, but are motivated by examples and introduced in a gradual and timely manner. The book thus will be of primary interest to the undergraduates (to broaden their horizons), but could also be worth looking into by the experienced researchers (as a reference book for the most essential concepts of algebra, topology, and category theory).

MSC:

18-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory
06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
08-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general algebraic systems
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
46H05 General theory of topological algebras
54-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology
55M99 Classical topics in algebraic topology
55U99 Applied homological algebra and category theory in algebraic topology
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