## Algebraic HomotopyThis book gives a general outlook on homotopy theory; fundamental concepts, such as homotopy groups and spectral sequences, are developed from a few axioms and are thus available in a broad variety of contexts. Many examples and applications in topology and algebra are discussed, including an introduction to rational homotopy theory in terms of both differential Lie algebras and De Rham algebras. The author describes powerful tools for homotopy classification problems, particularly for the classification of homotopy types and for the computation of the group homotopy equivalences. Applications and examples of such computations are given, including when the fundamental group is non-trivial. Moreover, the deep connection between the homotopy classification problems and the cohomology theory of small categories is demonstrated. The prerequisites of the book are few: elementary topology and algebra. Consequently, this account will be valuable for non-specialists and experts alike. It is an important supplement to the standard presentations of algebraic topology, homotopy theory, category theory and homological algebra. |

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### Contents

I Axioms for homotopy theory and examples of eofibration categories | 1 |

II Homotopy theory in a cofibration category | 83 |

III The homotopy spectral sequences in a cofibration category | 179 |

IV Extensions coverings and cohomology groups of a category | 229 |

V Maps between mapping cones | 258 |

VI Homotopy theory of CWcompIexes | 306 |

VII Homotopy theory of complexes in a cofibration category | 371 |

VIII Homotopy theory of Postnikov towers and the Sullivande Rham equivalence of rational homotopy categories ... | 404 |

IX Homotopy theory of reduced complexes | 436 |

455 | |

461 | |

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### Common terms and phrases

abelian groups action assume attaching map axiom based map based object basepoint bijection category Top chain algebras chain complex chain map classiﬁcation Clearly cochain cocycle coefﬁcients coﬁbrant objects coﬁbration category cohomology commutative diagram consider Corollary CW-complexes CW-spaces cylinder deﬁned Deﬁnition denotes derive differential dual element equivalence of categories example extension of categories ﬁber ﬁbrant model ﬁbration category Fil(C ﬁnite ﬁrst following diagram full subcategory given homology homomorphism homotopy category homotopy classes homotopy equivalence homotopy groups homotopy theory homotopy types inclusion initial object isomorphism isotropy groups kernel Lemma Let f linear extension map f mapping cone model category Moreover morphism natural system nilpotent obtain pair map path connected principal coﬁbration Proof properties Proposition push out diagram quotient R-module reduced complex relative respectively result satisﬁes shows simply connected spectral sequence surjective theorem topology tower of categories trivial twisted map weak equivalence yields