##
**Algebraic homotopy.**
*(English)*
Zbl 0688.55001

Cambridge Studies in Advanced Mathematics, 15. Cambridge etc.: Cambridge University Press. xix, 466 p. $ 89.50; £60.00 (1989).

Ever since the successful axiomatization of homology theory by Eilenberg and Steenrod, there have been repeated attempts to construct an analogous axiomatic foundation for homotopy theory. These attempts have mainly been motivated by the development of classical homotopy theory and by the appearance in the last few decades of a wide variety of homotopy theories, defined in different topological, combinatorial, and algebraic contexts.

The best known among the various axiomatizations of homotopy theory are those constructed by D. M. Kan [Proc. Natl. Acad. Sci. USA 42, 255- 258 (1956; Zbl 0071.167)], D. G. Quillen [Homotopical algebra (1967; Zbl 0168.209)], A. Heller [Bull. Am. Math. Soc. 74, 28-63 (1968; Zbl 0177.256)], and K. S. Brown [Trans. Am. Math. Soc. 186, 419-458 (1973; Zbl 0245.55007)]. A different approach, which concentrates on the homotopy category rather than on a preassigned concrete precursor, has recently been used by A. Heller [Mem. Am. Math. Soc. 71, No.383 (1988; Zbl 0643.55015)]. The most influential of these axiomatizations has proved to be that of Quillen. It is based on the concept of a (closed) model category; this is a category closed under finite limits and colimits and having three distinguished families of morphisms called cofibrations, fibrations, and weak equivalences, satisfying a set of axioms. An analysis of the various approaches shows that the problem of the axiomatic foundation of homotopy theory is a diffiult task; each of the proposed systems of axioms has its own advantages and failures.

In the present monograph, the author presents his own solution to the problem. The defining axioms are chosen according to the following two basic criteria: (a) they are sufficiently strong to allow the derivation of the fundamental constructions of homotopy theory; (b) they are as weak as possible, as to make the constructions of homotopy theory available in as many contexts as possible. The approach proves to be highly successful: it not only establishes a general, unifying framework for the various topological and algebraic homotopy theories, but also leads to the development of powerful new tools for homotopy classification problems, in particular for the classification of homotopy types and for the computation of groups of homotopy equivalences.

The basic concepts are those of a cofibration category and of a fibration category, dual to each other. A cofibration category is a category with two distinguished classes of morphisms, called cofibrations and weak equivalences and satisfying four axioms \(C_ 1,...,C_ 4\); a fibration category is obtained by replacing the cofibrations by fibrations and the four axioms by their duals \(F_ 1,...,F_ 4\). These axioms are introduced in Chapter I and are then compared to the various axiomatic systems which exist in the literature. Two more restrictive concepts are defined as well. The first is called an I-category (a category with a natural cylinder object) and admits the structure of a cofibration category; its dual is called a P-category (a category with a natural path object) and has the structure of a fibration category. These concepts are used to derive the important result that the topological category \({\mathcal T}op\) satisfies the axioms of both a cofibration and a fibration category. Many other examples of cofibration categories are then described in detail; these include various frequently encountered topological categories, as well as some important algebraic categories: chain complexes, chain algebras, commutative chain algebras, and chain Lie algebras. The derivation of the basic facts of homotopy theory in the general context of a cofibration category is performed in Chapter II. First, the appropriate versions of the central concepts are defined: homotopy groups, the action of the fundamental group, homotopy groups of function spaces, and homotopy groups of pairs. These concepts serve as the main ingredients of the fundamental exact sequences which are then constructed: (A) the cofiber sequence of a cofibration; (B) the exact sequence for homotopy groups of function spaces; and (C) the exact sequence for relative homotopy groups of pairs. These sequences are described in detail and their naturality with respect to functors between cofibration categories is proved. As consequences, many classical and new results in topology are derived. As an example, the cofiber sequence implies a long exact extension of the classical Puppe sequence. Other results concern the Whitehead products and the classification of maps in some special classes. A brief description is also provided for the homotopy theory in a fibration category. To demonstrate the usefulness of the general theory in different special contexts, an explicit description is given of the homotopy groups and homotopy groups of function spaces in the category of chain algebras.

The spectral sequences corresponding to the three fundamental exact sequences are constructed in Chapter III. These turn out to be far reaching generalizations of important constructions in topology: the Atiyah-Hirzebruch and the Bousfield-Kan spectral sequences in case (A), the Federer spectral sequence in case (B), and the homotopy exact couple in case (C). In the last case, a “certain exact sequence of J. H. C. Whitehead” in a cofibration category is derived. Many properties of these spectral sequences and of the morphisms induced by the functors between cofibration categories are proved; some of these results appear to be new even in the category \({\mathcal T}op\). On the other hand, appropriate concepts of complexes, chain complexes, and twisted chain complexes in a cofibration category, and their duals in a fibration category, are introduced; they generalize the CW-complexes and the Postnikov towers of topology. The associated cohomology and twisted cohomology (which generalize the classical cohomology with constant and local coefficients respectively) lead to the construction of the \(E_ 2\)-terms of the spectral sequences in the cases (A) and (B) respectively.

In Chapter IV the author introduces the concept of a detecting functor as a functor which satisfies both the sufficiency and the realizability conditions of J. H. C. Whitehead. The consideration of group actions on categories leads to the definition of linear extensions of categories (by natural systems of abelian groups) and of linear coverings of categories. The two concepts are combined in the construction of exact sequences of functors. Then the cohomology groups of a small category with coefficients in a natural system are introduced; these generalize the Hochschild-Mitchell cohomology. The main result here shows that the cohomology groups in degree 2 and 1 classify linear extensions and linear coverings respectively.

The motivation for the theory of natural group actions on categories is found in Chapter V, where it is applied to the study of maps between mapping cones. The discussion concentrates on a sequence \({\mathfrak PRIN}\subset {\mathfrak TWIST}\subset {\mathfrak PAIR}\) of subcategories of the homotopy category of a cofibration category; \({\mathfrak PRIN}\) contains the principal and \({\mathfrak TWIST}\) the twisted maps between mapping cones. The consideration of certain linear group actions on these categories leads to the description of \({\mathfrak PRIN}\) and \({\mathfrak TWIST}\) as linear extensions of model categories \({\mathfrak \Pr in}/\simeq\) and \({\mathfrak Twist}/\simeq\) respectively. On the other hand, quadratic group actions are constructed and used to characterize the equivalence relation \(\simeq\) on the categories \({\mathfrak \Pr in}\) and \({\mathfrak Twist}\). A brief account is also given of the dual theory of maps between fiber spaces in a fibration category. The application of the general theory in various topological contexts leads to the derivation of some important consequences. These include the solutions of some of J. H. C. Whitehead’s problems and proofs of the general suspension theorem under a space and of the general loop theorem over a space.

Chapter VI is devoted to the combinatorial homotopy theory of CW- complexes. The author first introduces the category of “homotopy systems”; then he defines the “CW-tower of categories” as a tower of homotopy n-systems together with “obstruction” maps out of each of these, giving a picture resembling a Postnikov tower. The CW-tower approximates the homotopy category (under D) of relative CW-complexes (X,D). It turns out to be a new and powerful tool which allows not only to give conceptually new and easy proofs of most of the results in J. H. C. Whitehead’s classical paper “Combinatorial homotopy II” [Bull. Am. Math. Soc. 55, 453-496 (1949; Zbl 0040.388)] and of the final theorem in his “Simple homotopy types” [Am. J. Math. 72, 1-57 (1950; Zbl 0040.389)], but also to derive extensions of these results to the relative case. Moreover, relative versions of C. T. C. Wall’s finiteness obstructions for CW-complexes are described. Other applications of the CW-tower concern the classification of homotopy types, the group of homotopy equivalences, and the realizability of chain maps; in particular, the classification of homotopy types of 4-dimensional CW- complexes is achieved.

The construction of the CW-tower is generalized in Chapter VII in the general context of an abstract cofibration category. Special classes of complexes with properties similar to those of the CW-complexes in topology are introduced. For each such class, a tower of categories is constructed, which approximates the corresponding homotopy category. Many new results on the homotopy classification problems are derived; these have important consequences in the various topological and algebraic contexts. In particular, it is shown that a cofibration in the category of chain algebras has in a canonical way the structure of a complex in this category.

The towers of categories also provide a powerful tool for the development of rational homotopy theory; this is shown in Chapter VIII. The consideration of topological fibrations and fiber preserving maps over a fixed base space enables the construction of a tower of categories which approximates the corresponding homotopy category; this tower is deduced from the Postnikov decompositions of the fibrations. A de Rham theorem for cohomology groups with local coefficients is proved. Then, the Sullivan-de Rham functor is introduced; it yields a structure preserving map between the “rational” towers of categories in the cofibration categories of commutative cochain algebras (under fixed algebras) and the “integral” towers of categories for Postnikov decompositions in the fibration categories of topological spaces (over fixed spaces). The study of this map leads in the end to elementary proofs, and to generalizations, of some of Sullivan’s fundamental results.

In the final Chapter IX, some examples of towers of categories are described, for which the underlying chain complexes are not twisted. It appears that these towers can also be used to deduce some of the fundamental results of homotopy theory, among them Quillen’s equivalence of rational homotopy categories and Whitehead’s classification of simply connected 4-dimensional CW-complexes. A few new results are obtained as well.

A substantial part of the material contained in this book has never appeared in published form before; it includes a wealth of results which reach up to the very forefront of current research. Yet the book is still accessible to many people, including graduate students, since the prerequisites are restricted to elementary topology and algebra. Altogether, the author presents us with an impressive achievement, which might reasonably be expected to become a standard reference for both teaching and research in the years to come.

The best known among the various axiomatizations of homotopy theory are those constructed by D. M. Kan [Proc. Natl. Acad. Sci. USA 42, 255- 258 (1956; Zbl 0071.167)], D. G. Quillen [Homotopical algebra (1967; Zbl 0168.209)], A. Heller [Bull. Am. Math. Soc. 74, 28-63 (1968; Zbl 0177.256)], and K. S. Brown [Trans. Am. Math. Soc. 186, 419-458 (1973; Zbl 0245.55007)]. A different approach, which concentrates on the homotopy category rather than on a preassigned concrete precursor, has recently been used by A. Heller [Mem. Am. Math. Soc. 71, No.383 (1988; Zbl 0643.55015)]. The most influential of these axiomatizations has proved to be that of Quillen. It is based on the concept of a (closed) model category; this is a category closed under finite limits and colimits and having three distinguished families of morphisms called cofibrations, fibrations, and weak equivalences, satisfying a set of axioms. An analysis of the various approaches shows that the problem of the axiomatic foundation of homotopy theory is a diffiult task; each of the proposed systems of axioms has its own advantages and failures.

In the present monograph, the author presents his own solution to the problem. The defining axioms are chosen according to the following two basic criteria: (a) they are sufficiently strong to allow the derivation of the fundamental constructions of homotopy theory; (b) they are as weak as possible, as to make the constructions of homotopy theory available in as many contexts as possible. The approach proves to be highly successful: it not only establishes a general, unifying framework for the various topological and algebraic homotopy theories, but also leads to the development of powerful new tools for homotopy classification problems, in particular for the classification of homotopy types and for the computation of groups of homotopy equivalences.

The basic concepts are those of a cofibration category and of a fibration category, dual to each other. A cofibration category is a category with two distinguished classes of morphisms, called cofibrations and weak equivalences and satisfying four axioms \(C_ 1,...,C_ 4\); a fibration category is obtained by replacing the cofibrations by fibrations and the four axioms by their duals \(F_ 1,...,F_ 4\). These axioms are introduced in Chapter I and are then compared to the various axiomatic systems which exist in the literature. Two more restrictive concepts are defined as well. The first is called an I-category (a category with a natural cylinder object) and admits the structure of a cofibration category; its dual is called a P-category (a category with a natural path object) and has the structure of a fibration category. These concepts are used to derive the important result that the topological category \({\mathcal T}op\) satisfies the axioms of both a cofibration and a fibration category. Many other examples of cofibration categories are then described in detail; these include various frequently encountered topological categories, as well as some important algebraic categories: chain complexes, chain algebras, commutative chain algebras, and chain Lie algebras. The derivation of the basic facts of homotopy theory in the general context of a cofibration category is performed in Chapter II. First, the appropriate versions of the central concepts are defined: homotopy groups, the action of the fundamental group, homotopy groups of function spaces, and homotopy groups of pairs. These concepts serve as the main ingredients of the fundamental exact sequences which are then constructed: (A) the cofiber sequence of a cofibration; (B) the exact sequence for homotopy groups of function spaces; and (C) the exact sequence for relative homotopy groups of pairs. These sequences are described in detail and their naturality with respect to functors between cofibration categories is proved. As consequences, many classical and new results in topology are derived. As an example, the cofiber sequence implies a long exact extension of the classical Puppe sequence. Other results concern the Whitehead products and the classification of maps in some special classes. A brief description is also provided for the homotopy theory in a fibration category. To demonstrate the usefulness of the general theory in different special contexts, an explicit description is given of the homotopy groups and homotopy groups of function spaces in the category of chain algebras.

The spectral sequences corresponding to the three fundamental exact sequences are constructed in Chapter III. These turn out to be far reaching generalizations of important constructions in topology: the Atiyah-Hirzebruch and the Bousfield-Kan spectral sequences in case (A), the Federer spectral sequence in case (B), and the homotopy exact couple in case (C). In the last case, a “certain exact sequence of J. H. C. Whitehead” in a cofibration category is derived. Many properties of these spectral sequences and of the morphisms induced by the functors between cofibration categories are proved; some of these results appear to be new even in the category \({\mathcal T}op\). On the other hand, appropriate concepts of complexes, chain complexes, and twisted chain complexes in a cofibration category, and their duals in a fibration category, are introduced; they generalize the CW-complexes and the Postnikov towers of topology. The associated cohomology and twisted cohomology (which generalize the classical cohomology with constant and local coefficients respectively) lead to the construction of the \(E_ 2\)-terms of the spectral sequences in the cases (A) and (B) respectively.

In Chapter IV the author introduces the concept of a detecting functor as a functor which satisfies both the sufficiency and the realizability conditions of J. H. C. Whitehead. The consideration of group actions on categories leads to the definition of linear extensions of categories (by natural systems of abelian groups) and of linear coverings of categories. The two concepts are combined in the construction of exact sequences of functors. Then the cohomology groups of a small category with coefficients in a natural system are introduced; these generalize the Hochschild-Mitchell cohomology. The main result here shows that the cohomology groups in degree 2 and 1 classify linear extensions and linear coverings respectively.

The motivation for the theory of natural group actions on categories is found in Chapter V, where it is applied to the study of maps between mapping cones. The discussion concentrates on a sequence \({\mathfrak PRIN}\subset {\mathfrak TWIST}\subset {\mathfrak PAIR}\) of subcategories of the homotopy category of a cofibration category; \({\mathfrak PRIN}\) contains the principal and \({\mathfrak TWIST}\) the twisted maps between mapping cones. The consideration of certain linear group actions on these categories leads to the description of \({\mathfrak PRIN}\) and \({\mathfrak TWIST}\) as linear extensions of model categories \({\mathfrak \Pr in}/\simeq\) and \({\mathfrak Twist}/\simeq\) respectively. On the other hand, quadratic group actions are constructed and used to characterize the equivalence relation \(\simeq\) on the categories \({\mathfrak \Pr in}\) and \({\mathfrak Twist}\). A brief account is also given of the dual theory of maps between fiber spaces in a fibration category. The application of the general theory in various topological contexts leads to the derivation of some important consequences. These include the solutions of some of J. H. C. Whitehead’s problems and proofs of the general suspension theorem under a space and of the general loop theorem over a space.

Chapter VI is devoted to the combinatorial homotopy theory of CW- complexes. The author first introduces the category of “homotopy systems”; then he defines the “CW-tower of categories” as a tower of homotopy n-systems together with “obstruction” maps out of each of these, giving a picture resembling a Postnikov tower. The CW-tower approximates the homotopy category (under D) of relative CW-complexes (X,D). It turns out to be a new and powerful tool which allows not only to give conceptually new and easy proofs of most of the results in J. H. C. Whitehead’s classical paper “Combinatorial homotopy II” [Bull. Am. Math. Soc. 55, 453-496 (1949; Zbl 0040.388)] and of the final theorem in his “Simple homotopy types” [Am. J. Math. 72, 1-57 (1950; Zbl 0040.389)], but also to derive extensions of these results to the relative case. Moreover, relative versions of C. T. C. Wall’s finiteness obstructions for CW-complexes are described. Other applications of the CW-tower concern the classification of homotopy types, the group of homotopy equivalences, and the realizability of chain maps; in particular, the classification of homotopy types of 4-dimensional CW- complexes is achieved.

The construction of the CW-tower is generalized in Chapter VII in the general context of an abstract cofibration category. Special classes of complexes with properties similar to those of the CW-complexes in topology are introduced. For each such class, a tower of categories is constructed, which approximates the corresponding homotopy category. Many new results on the homotopy classification problems are derived; these have important consequences in the various topological and algebraic contexts. In particular, it is shown that a cofibration in the category of chain algebras has in a canonical way the structure of a complex in this category.

The towers of categories also provide a powerful tool for the development of rational homotopy theory; this is shown in Chapter VIII. The consideration of topological fibrations and fiber preserving maps over a fixed base space enables the construction of a tower of categories which approximates the corresponding homotopy category; this tower is deduced from the Postnikov decompositions of the fibrations. A de Rham theorem for cohomology groups with local coefficients is proved. Then, the Sullivan-de Rham functor is introduced; it yields a structure preserving map between the “rational” towers of categories in the cofibration categories of commutative cochain algebras (under fixed algebras) and the “integral” towers of categories for Postnikov decompositions in the fibration categories of topological spaces (over fixed spaces). The study of this map leads in the end to elementary proofs, and to generalizations, of some of Sullivan’s fundamental results.

In the final Chapter IX, some examples of towers of categories are described, for which the underlying chain complexes are not twisted. It appears that these towers can also be used to deduce some of the fundamental results of homotopy theory, among them Quillen’s equivalence of rational homotopy categories and Whitehead’s classification of simply connected 4-dimensional CW-complexes. A few new results are obtained as well.

A substantial part of the material contained in this book has never appeared in published form before; it includes a wealth of results which reach up to the very forefront of current research. Yet the book is still accessible to many people, including graduate students, since the prerequisites are restricted to elementary topology and algebra. Altogether, the author presents us with an impressive achievement, which might reasonably be expected to become a standard reference for both teaching and research in the years to come.

Reviewer: J.Weinstein

### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

55P15 | Classification of homotopy type |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |