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
