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• Contents
  I. Simplicial Sets 1
  I.l Triangulated Spaces 1
  I.3 Simplicial Topological Spaces and the Eilenberg-Zilber Theorem 17
  I.4 Homology and Cohomology 23
  II. Main Notions of the Category Theory 57
  II.1 The Language of Categories and Functors 57
  II.2 Categories and Structures, Equivalence of Categories 69
  II.3 Structures and Categories. Representable Functors 78
  II.4 Category Approach to the Construction of Geometrical Objects 93
  II.5 Additive and Abelian Categories 109
  II.6 FunctorsIn Abelian Categories 122
  III. Derived Categories and Derived Functors 139
  III.1 Complexes as Generalized Objects 139
  III.2 Derived Categories and Localization 144
  III.3 Triangles as Generalized Exact Triples 153
  III.4 Derived Category as the Localization of Homotopic Category 159
  III.5 The Structure of the Derived Category 164
  III.6 Derived Functors 185
  III.7 Derived Functor of the Composition. Spectral Sequence 200
  III.8 Sheaf Cohomology 218
  IV. Triangulated Categories 239
  IV.1 Triangulated Categories 239
  IV.2 Derived Categories Are Triangulated 251
  IV.3 An Example: The Triangulated Category of A-Modules 267
  IV.4 Cores 278
  V.Introduction to Homotopic Algebra 291
  V.l Closed Model Categories 291
  V.2 Homotopic Characterization of Weak Equivalences 299
  V.3 DG-Algebras as a Closed Model Category 333
  V.4 Minimal Algebras 342
  V.5 Equivalence of Homotopy Categories 352