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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