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Hovey M. — Model Categories, Vol. 63
Hovey M. — Model Categories, Vol. 63

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Название: Model Categories, Vol. 63

Автор: Hovey M.

Аннотация:

[The book] starts with an account of the definitions, and a development of the homotopy theory of model categories. This is probably the first time in which the important notion of cofibrant generation has appeared in a book, and the consideration of the 2-category of model categories and Quillen adjunctions is another interesting feature. — Bulletin of the London Mathematical Society Model categories are used as a tool for inverting certain maps in a category in a controllable manner. As such, they are useful in diverse areas of mathematics. The list of such areas is continually growing. This book is a comprehensive study of the relationship between a model category and its homotopy category. The author develops the theory of model categories, giving a careful development of the main examples. One highlight of the theory is a proof that the homotopy category of any model category is naturally a closed module over the homotopy category of simplicial sets. Little is required of the reader beyond some category theory and set theory, which makes the book accessible to advanced graduate students. The book begins with the basic theory of model categories and proceeds to a careful exposition of the main examples, using the theory of cofibrantly generated model categories. It then develops the general theory more fully, showing in particular that the homotopy category of any model category is a module over the homotopy category of simplicial sets, in an appropriate sense. This leads to a simplification and generalization of the loop and suspension functors in the homotopy category of a pointed model category. The book concludes with a discussion of the stable case, where the homotopy categoryis triangulated in a strong sense and has a set of small weak generators.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 207

Добавлена в каталог: 27.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$A^{\bullet} \bigotimes K$      76
$A^{\bullet} \wedge K$      76
$C^*$ -algebras      192
$K_*$      see also k-spaces pointed
$T_*$      see also topological spaces compactly pointed
$\Delta$      see also simplicial category
$\Delta[n]$      75
$\gamma$-filtered cardinal      see also cardinal
$\lambda$-sequence      
see also lambda sequence
$\Lambda^r [n]$      75
$\partial \Delta [n]$      75
2-category      15 22 23 24—26
2-category of C-algebras      105
2-category of C-model categories      114
2-category of C-modules      104
2-category of categories      18 24
2-category of categories and adjunctions      24
2-category of central C-algebras      106
2-category of central monoidal C-algebras      115
2-category of closed monoidal categories      107
2-category of closed monoidal pre-triangulated categories      174
2-category of model categories      18 24 191
2-category of monoidal C-model categories      115
2-category of monoidal categories      103
2-category of monoidal model categories      113
2-category of pointed model categories      145
2-category of pre-triangulated categories      172
2-category of stable homotopy categories      184
2-category of stable model categories      176
2-category of symmetric C-algebras      105
2-category of symmetric monoidal C-algebras      115
2-category of symmetric monoidal categories      104
2-category of triangulated categories      176
2-category, model      191 194
2-functor      24
Acyclic chain complex      see also chain complex acyclic
Adjunction of two variables      106 116 119
Anodyne extension      79 80—83 109
Associativity isomorphism      134
Basepoint      4
Bimodules      102—103 106
Bisimplicial sets      128—129 131—132
Boundary map      87 88 156 170
Brown, Ken      see also Ken Brown’s lemma
C-algebra      104
C-algebra functor      104
C-algebra natural transformation      104
C-algebra structure      104
C-model category      101 114 118
C-module      104
C-module functor      104
C-module natural transformation      104
C-module structure      104
C-Quillen functor      114
Cardinal      28
Cardinal $\gamma$-filtered      29
Cardinality argument      45
Catad      see also 2-category of
Category of simplices of a simplicial set      75 124
Central C-algebra      105
Central C-algebra functor      106
Central monoidal C-model category      118
Ch(B)      see also chain complexes of
Ch(R)      see also chain complexes of
Chain complex, acyclic      41
Chain complexes of abelian groups      114 143—145 192
Chain complexes of comodules      28 60—72 112 114 176 188 194
Chain complexes of modules      27 40 41—48 111—112 114 176 183 188 194
Chain homotopy      43
Chain map      40
Closed $T_1$ inclusion      49
Closed monoidal category      106
Closed monoidal functor      106
Closed monoidal structure      106
Closed symmetric monoidal category      63 77
Cofiber      147
Cofiber sequence      147 151 156 157—165 169 177—182
Cofibrant      4
Cofibrant replacement functor      5
Cofibration      3
Cofibration, generating      34
Cofibration, trivial      3
Cofibration, trivial, generating      34
Cofinality of a cardinal      29
Cogroup object      151
Cogroup structure      150 151
Cogroup structure on $\Sigma X$      151
Cohomology functor      184 195
Colimit, homotopy      185 195
Colimit, sequential      185 195
Commutative monoid      192—193
Comodule      61
Comodule, cofree      63
Comodule, injective      63—65
Comodule, simple      61
Comodules      102 105
Comonad      193
Compactly closed      58
Compactly open      58
Correspondence of homotopies      150
Cosimplicial frame      127 128 130—139 144 167—169
Cosimplicial frames, map of      127
Cosimplicial identities      73
Cosimplicial object      73 76
Cosmall      34
Cotriple      193
Cube lemma      126 153 158 161 166 168
CW -complex      51
Cylinder object      8 9 127 152—153
Deformation retract      53
Degeneracy map      73
Degeneracy of a simplex      74
Degree function      124
Derived adjunction      18
Derived functor, total left      16 17—22
Derived functor, total right      16
Diagonal functor      128—129 131—132
Dimension of a simplex      73
Direct category      119 120 121—124
Dual model category      4
Duality 2-functor      24 107 115 118 122 126 127 170 173
Dwyer, Bill      1 119
Equivalence in a 2-category      24
Exact adjunction      172 181
F $\Box$ g      81 108
Face map      73
Face of a simplex      74
Fiber      148
Fiber homotopy equivalence      89 90—91
Fiber sequence      147 151 156 157—165 169 177—182
Fibrant      4
Fibrant replacement functor      5
Fibration      3
Fibration, Kan      79
Fibration, locally trivial      89 92—93 95—97
Fibration, minimal      89 91 92—95 97
Fibration, trivial      3
finite      29 187
Framing      119 123 127 128—129 131—145
Franke, Jens      195
Frobenius ring      36 37—40 71—72 112
Frobenius ring, Noetherian      39
Function complex      128
Function complex of simplicial sets      77
Functor, monoidal      see also monoidal functor
Functorial factorization      2 15 16 28
Geometric realization      77 78—81 85 95—99 102 114 118
Geometric realization, preserves fibrations      97
Geometric realization, preserves finite limits      80
Geometric realization, preserves products      77
Goerss, Paul      193
Group object      151
Group ring      36
Group structure      151
Group structure on $\Omega X$      151
Hirschhorn, Phil, x      1
Homology of a chain complex      41
Homotopic maps      9
Homotopy category      7
Homotopy equivalence      9 11
Homotopy groups of a chain complex of comodules      65
Homotopy groups of a fibrant simplicial set      83 85 86—89 97—99
Homotopy groups of a topological space      50
Homotopy of continuous maps      50
Homotopy of simplicial maps      86
Homotopy of vertices      84
Hopf algebra      36 60 112
Hopkins, Mike      193
Horizontal composition in a 2-category      23
Horizontal composition of natural transformations      18 23
Hurewicz cofibration      191
Hurewicz fibration      96 192
I-cell      see also relative I-cell complex
I-cell complex      31
I-cofibration      30
I-fibration      30
I-injective      30
I-projective      30
Inclusion of topological spaces      49
Injective model structure      44 112
Inverse category      119 120 121—124
Johnson, Mark      x
K      see also k-spaces
k-small      29
k-space      58
k-spaces      77 80 98 106 111 114 118
k-spaces, pointed      111 114 118 176
Kan, Dan      x 1 3 119 194
Kelley space      see also k-space
Ken Brown’s lemma      6 11—12 14 131—132 136
Lambda sequence      28
Latching space      120 122 124
Left exact      172
Left homotopy      9
Left homotopy between right homotopies      149
Left lifting property      3
Lewis, Gaunce      x
Limit, homotopy      175 185 186 195
Linear extension      120
Localizing subcategory      184 195
Locally trivial fibration      see also fibration locally
Loop functor      148 149—174 176—177 179—183
Mapping cylinder      165 167
Mapping cylinder, double      165
Matching space      120 122 124
May, Peter      184 193
McClure, Jim      192
Minimal fibration      see also fibration minimal
MOD      see also 2-category of
Model category      3
Model category, cofibrantly generated      27 34 35—36 108—109 123 183 185 187
Model category, fibrantly generated      34
Model category, finitely generated      34 175 184 187—190
Model category, pointed      4 14 15 21 26 36 115 144—145 147—174 185—190 195
Model category, simplicial      101 114 119 128 136 138—139 142—145 194
Model category, simplicial, pointed      115
Model category, stable      175 176 185 194
Model structure      3
Model structure, product      4 7 14 19
Module over a monoidal category      see also C-module
Modules over a Frobenius ring      27
Monad      193
Monoidal C-model category      115 118
Monoidal C-Quillen functor      115
Monoidal category      101 102
Monoidal functor      102
Monoidal model category      101 109 110—119 140—145
Monoidal model category, pointed      145 173—174
Monoidal model category, stable      176
Monoidal model category, symmetric      110—118
Monoidal natural transformation      103
Monoidal Quillen adjunction      113
Monoidal Quillen functor      113 141
Monoidal structure      102
Natural transformation, monoidal      see also monoidal natural transformation
Natural transformation, total derived      16 18 22
Non-degenerate, simplex      74
Octahedral axiom      160—161 170 177
Operads      193
Ordinal      28 120
p-related vertices      91 93
Palmieri, John      x
Paracompact      96
Path component of a simplicial set      85
Path object      8 9
Pre-triangulated category      147 169 170 171—174 183 194
Pre-triangulated category, closed central monoidal      173
Pre-triangulated category, closed monoidal      173 174
Pre-triangulated category, closed symmetric monoidal      173
Pre-triangulation      169 170
Pseudo-2-functor      18 22 24 101 115 117—118 136 138 140—141 143 145 173—174 184 195
Pseudo-2-functor, homotopy      25
Pseudo-2-functor, natural isomorphism of      137
Pushout product      108 109
q      see also cofibrant replacement functor
Quillen adjunction      14 15—22 35 71 131 136—139 163 191
Quillen adjunction of two variables      107
Quillen bifunctor      108 109—116
Quillen equivalence      13 19 20—22 26 48 98 191
Quillen functor      35 48 123 128 163—165
Quillen functor, left      13 14—19
Quillen functor, monoidal      see also monoidal Quillen functor
Quillen functor, right      14 15—19
Quillen, Daniel      1—195
R      see also fibrant replacement functor
Reedy category      119 123 124 125—127 195
Reedy model structure      126 127 130 133 144 167
Reedy scheme      195
reflect      21
Relative I-cell complex      30 31—36
Retract argument      5
Rezk, Charles, x      191 193
Right exact      172
Right homotopy      9
Right homotopy between right homotopies      149
Right lifting property      3
Right proper      57
S-modules      113 176 184
Schwede, Stefan      49 109 194
Serre fibration      51 96 97
Shipley, Brooke      x
Sierpinski space      49
Simplicial category      73 119 123—125
Simplicial frame      127 128 130—133 144 163
Simplicial frames, map of      127
Simplicial identities      74
Simplicial object      73
Simplicial set, finite      74
Simplicial sets      73 74—101 109 114 118 119 128—145 188 192
Simplicial sets, pointed      98 100 110 114 118 176 183 185—187
Singular functor      77 98
Small      29 175 183 187—190
Small object argument      28 32 187
Smith, Jeff      x 107 109
SSET      see also simplicial sets
SSet$_*$      see also simplicial sets pointed
Stable category of modules      37
Stable equivalence of modules      36
Stable homotopy category      195
Stable homotopy category, algebraic      175 184
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