Index

df, see also fiber diagram

Fun(I,C), see also diagram category

Funb(1L,C),

see also bounded diagram

Ftm^ (L,C), see also bounded diagram,relatively

to map

/-bounded, see also bounded diagram,relatively

to map

/-cofibrant, see also cofibrant,relative

/-cofibration, see also cofibration,relative

/-non-degenerate, see also non-degenerate,relatively

to map

GrjH^ see also Grothendieck construction

G T K H , see also Grothendieck construction

id-bounded, see also bounded diagram,absolutely

red(f), see also reduction of a map

A, see also simplicial category

A[n, k], see also standard simplex,horn of

A[n], see also standard simplex

f li, see also over category,of a functor

14,2, see also over category

K, L, N, see also simplex category

K x N , see also simplex category,product

5A[n], see also standard simplex,boundary

of

etale space, 64, 69

absolutely bounded, see also bounded dia-

gram , absolutely

absolutely cofibrant, see also cofibrant,absolutely

additivity

of the colimit, 23

of the homotopy colimit, 36

of the etale space, 64

basic space, 66

bounded diagram, 26, 72

absolutely, 39

over a Grothendieck construction, 58

over a horn, 28

over a product, 54

over a sphere, 28

over a standard simplex, 27

relatively to map, 38

strongly, 26

Bousfield-Kan approximation, 29, 36, 56, 73

Bousfield-Kan model approximation, 61

Brown's lemma, 12, 34, 49

category

over an object, see also over category

under an object, see also under category

co-Grothendieck construction, 73

cofibrant, 6

absolutely, 44

bounded diagram, 30, 54, 59

over a sphere, 31

constant bounded diagram, 31

relatively to a map, 44

replacement, 7

of a diagram, 38

cofibration, 6

absolute, 44

acyclic, 6

of bounded diagrams, 30

over a Grothendieck construction, 59

over a product, 54

relative, 44

cofinal functor, 24, 36, 70, 83

colimit, 5, 14, 83

compatible family, 57, 85

cone

over a category, 50

over a space, 49

over a standard simplex, 50

coproduct, 10

cylinder object, 8

derived functor

left, 12

right, 71

total left, 12, 18

total right, 71

diagram

almost cubical, 27

constant, 31

indexed by a space, 22

diagram category, 5, 14

factorization

functorial, 36, 38

factorization axiom, 7, 33

fiber diagram, 24, 28, 75

fibrant, 6

replacement, 8

fibration, 6

89