Index Page numbers in bold point to the definitions of the indexed terms. adjunctions, 122-123 compatibility with (co)limit functors, 57, 151 counits of, 122, 122-123, 135-138 deformable, 14-15, 37, 51, 51-54, 119- 122, 133 derived adjunctions of, 8, 14, 36-37, 46, 53, 44-54, 121, 139, 138-143, see also derived adjunctions partial, 44, 53, 133, 133-138 Quillen, 40, see also Quillen adjunctions units of, 122, 122-123, 135-138 adjunctions of systems, 157 and dinatural transformations, 167 and Kan extensions along systems, 169 categorical uniqueness, 157 counits of, 157 deformable, 157 derived adjunctions of, 158 locally deformable, 157 sufficient conditions for existence, 157 units of, 157 all or none proposition for compositions, 131 alternate description of homotopy categories, 100 approximations, 13, 35-37, 42-44, 51, 84, 120-122, 127, 126-130 and total derived functors, 128 compositions of, 44, 120-121, 130-133 homotopical uniqueness of, 42, 51, 127 of deformable functors, 51, 127 of homotopical functors, 128 of Kan extensions and homotopical Kan extensions, 163 of natural transformations, 129, 129-130 of Quillen functors, 42 sufficient conditions for composability, 52 sufficient conditions for existence, 51, 127 approximations of systems, 154 homotopical uniqueness of, 154 sufficient conditions for existence, 155 arrow categories, 101, 101-112 T-diagrams of, 102 axiom factorization, 4, 19, 26 lifting, 26 limit, 3, 19, 25 retract, 19-20, 26 two out of six, 26 two out of three, 19-20 canonical natural isomorphisms, 45 canonical natural transformations, 139, 139- 142 canonically isomorphic objects, 10, 38, 82, 92, 112, see also categorical uniqueness canonically weakly equivalent objects, 10, 39, 83, 92, 114, see also homotopical uniqueness CAT , 95, 97-99 cat, 95 CAT W , 97, 97-99 cat-systems, see also systems left, 62, 62-65, 73 right, 62, 62-65, 73 categorical uniqueness, 38, 38-39, 92-93, 112, 112-114 of adjoints of systems, 157 of colimit systems, 160 of initial objects, 38, 92, 113 of limit systems, 160 of terminal objects, 38, 92, 113 categorically contractible categories, 10, 38, 82, 92, 113 full subcategories, 38, 82, 92, 112 unique objects, 10, 38, 82, 92, 112, see also categorical uniqueness categories, 4, 23, 22-23, 78-79, 89-90, 95, 95 arrow, 101, 101-112 categorically contractible, 10, 38, 82, 92, 113 closed model, 3-4, 19, 27-28 cocomplete, 25, 58, 58, 159, see also co- completeness complete, 25, 58, 58, 159, see also com- pleteness 171
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