vi CONTENTS 26. Homotopical categories 78 27. The hom-sets of the homotopy categories 80 28. Homotopical uniqueness 82 29. Deformable functors 83 30. Homotopy colimit and limit functors and homotopical ones 85 Chapter VI. Homotopical Categories and Homotopical Functors 89 31. Introduction 89 32. Universes and categories 93 33. Homotopical categories 96 34. A colimit description of the hom-sets of the homotopy category 101 35. A Grothendieck construction 103 36. 3-arrow calculi 107 37. Homotopical uniqueness 112 38. Homotopically initial and terminal objects 115 Chapter VII. Deformable Functors and Their Approximations 119 39. Introduction 119 40. Deformable functors 123 41. Approximations 126 42. Compositions 130 43. Induced partial adjunctions 133 44. Derived adjunctions 138 45. The Quillen condition 143 Chapter VIII. Homotopy Colimit and Limit Functors and Homotopical Ones 147 46. Introduction 147 47. Homotopy colimit and limit functors 148 48. Left and right systems 152 49. Homotopical cocompleteness and completeness (special case) 159 50. Homotopical colimit and limit functors 161 51. Homotopical cocompleteness and completeness (general case) 166 Index 171 Bibliography 181
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