## Homotopical algebra |

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### Contents

The loop and suspension functors | 2-1 |

Fibration and cofibration sequences | 2-3 |

Equivalence of homotopy theories | 4-1 |

Copyright | |

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### Common terms and phrases

abelian category adjoint functors axiom carries weak equivalences category of simplicial Ch.I closed model category closed simplicial model closed under finite cofibrant objects cofibration sequences commutative complex composition constant simplicial constructed correspondence cylinder object define Definition deformation retract deformation retract map denote diagram dotted arrow dual effective epimorphism f to g factored f fibrant objects fibration resp fibration sequence finite limits hence Hom(X,Y homology Homotopical Algebra homotopy equivalence homotopy from f homotopy theory induced isomorphism Lemma Let h:AxI lifting LLP with respect map f morphism path object plicial projective object projective resolution Proposition proved right homotopy right simplicial satisfies Seiten sheaf simplicial abelian group simplicial category simplicial functor simplicial groups simplicial model category simplicial objects simplicial R module simplicial ring simplicial set spectral sequence strong deformation retract structure surjective suspension functors trivial cofibration trivial fibration unique map weak equivalence