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Categorical topology of compact Hausdorff spaces. (English) Zbl 1032.54004
Mathematical Studies Monograph Series. 5. Lviv: VNTL Publishers. 263 p. (1999).
Several well known constructions of compact Hausdorff spaces may be considered as functors from the category $${\mathbf C}{\mathbf o}{\mathbf m}{\mathbf p}$$ of compact Hausdorff spaces into itself. By identifying the similarities between these functors, E. V. Shchepin introduced in 1981 the concept of a normal functor $$F:{\mathbf C}{\mathbf o}{\mathbf m}{\mathbf p}\to {\mathbf C}{\mathbf o}{\mathbf m}{\mathbf p}$$ [Russ. Math. Surv. 36, No. 3, 1–71 (1981); translation from Usp. Mat. Nauk 36, No. 3(219), 3–62 (1981; Zbl 0463.54009)]. This notion led to a rich theory which was mainly developed by Russian and Ukrainian topologists. The monograph provides a comprehensive introduction to this theory. It consists of 5 chapters which are organized as follows.
Chapter 1 contains some prerequisites from general topology and category theory including Shchepin’s spectral theorem and some basic facts concerning the Eilenberg-Moore category of monads.
In Chapter 2 the theory of normal and related functors (i.e., weakly normal functors, almost normal functors) is presented. It contains many examples including hyperspaces, hyperspaces of convex compacta, superextensions in the sense of J. de Groot, probability measure functors, functors of order-preserving functionals, $$G$$-symmetric power functors, subfunctors of the free topological group functor, functors generated by the Hartman-Mycielsky construction and projective power functors. It also contains the basic facts concerning the generalization of the concept of a normal functor to endofunctors of the category $${\mathbf T}{\mathbf y}{\mathbf c}{\mathbf h}$$ of Tychonoff spaces due to A. C. Chigogidze.
Chapter 3 is dedicated to the investigation of normal monads, i.e. monads $$(F,\eta,\mu)$$ where the functor $$F$$ is normal (either in the sense of Shchepin or Chigogidze). It is shown that some of the normal functors of Chapter 2 can be viewed as the functorial part of normal monads. Some of the corresponding Eilenberg-Moore categories are characterized internally. Additionally, perfectly metrizable monads in the sense of V. V. Fedorchuk are studied.
Chapter 4 connects the theory of normal functors and normal monads with some parts of geometric topology (absolute neighborhood retracts, $$LC^n$$-spaces, $$G$$-$$ANR$$-spaces, $$Q$$-manifolds, shape theory, homotopy theory).
Finally, in Chapter 5 the behaviour of some normal functors when applied to compact spaces of the form $$X^\kappa$$, $$\kappa> \omega$$, is studied. The monograph is an interesting addition to the literature in that it presents the main results of the theory of normal functors and normal monads in a uniform way. However, proofreading could have been better. Due to (too) many little mistakes, it is not easy to read.

##### MSC:
 54B30 Categorical methods in general topology 54-02 Research exposition (monographs, survey articles) pertaining to general topology 18B30 Categories of topological spaces and continuous mappings (MSC2010) 18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D30 Compactness 54B20 Hyperspaces in general topology