Neeman A. — Triangulated categories :: Электронная библиотека попечительского совета мехмата МГУ

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Neeman A. — Triangulated categories

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Название: Triangulated categories

Автор: Neeman A.

Аннотация:

The first two chapters of this book offer a modern, self-contained exposition of the elementary theory of triangulated categories and their quotients. The simple, elegant presentation of these known results makes these chapters eminently suitable as a text for graduate students. The remainder of the book is devoted to new research, providing, among other material, some remarkable improvements on Brown's classical representability theorem. In addition, the author introduces a class of triangulated categories" — the "well generated triangulated categories" — and studies their properties. This exercise is particularly worthwhile in that many examples of triangulated categories are well generated, and the book proves several powerful theorems for this broad class. These chapters will interest researchers in the fields of algebra, algebraic geometry, homotopy theory, and mathematical physics.

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Рубрика: Математика/Алгебра/Теория категорий/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 449

Добавлена в каталог: 12.03.2005

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Предметный указатель

Abelian categories of product-preserving functors      183—214
Abelian categories of product-preserving functors are locally presentable      221—224 326
Abelian categories of product-preserving functors do not satisfy [AB5], [AB5*]      209—210
Abelian categories of product-preserving functors have enough projectives      212
Abelian categories of product-preserving functors may not have cogenerators      403—405
Abelian categories of product-preserving functors satisfy [AB3*]      186—187 200
Abelian categories of product-preserving functors satisfy [AB3]      196—200
Abelian categories of product-preserving functors satisfy [AB4*]      206
Abelian categories of product-preserving functors satisfy [AB4]      207—209
Abelian categories of product-preserving functors via universal homological functor      384—385
Abelian categories of product-preserving functors, coproducts      191
Abelian categories of product-preserving functors, definitions      185
Abelian categories of product-preserving functors, homological functors      204—205
Abelian categories of product-preserving functors, homological objects      224—229 258—262
Abelian categories of product-preserving functors, homological objects, as filtered colimits of representables      226—229
Abelian categories of product-preserving functors, homological objects, characterisation in terms of vanishing Ext      258—259
Abelian categories of product-preserving functors, homological objects, Embedding arbitrary objects in homological ones      259—262
Abelian categories of product-preserving functors, homological objects, stable under filtered colimits      225
Abelian categories-review of formalism $\beta$ -filtered limits      321
Abelian categories-review of formalism $\beta$ -filtered limits, definition of [AB4.5( $\alpha$ )]      354
Abelian categories-review of formalism $\beta$ -filtered limits, definition of [AB5 $^\alpha$ ]      378
Abelian categories-review of formalism $\beta$ -filtered limits, derived functors of limit      345—361
Abelian categories-review of formalism $\beta$ -filtered limits, derived functors of limit, analogy with sheaves      349—351
Abelian categories-review of formalism $\beta$ -filtered limits, derived functors of limit, cofinal sequences      358
Abelian categories-review of formalism $\beta$ -filtered limits, derived functors of limit, flabby sequences      350
Abelian categories-review of formalism $\beta$ -filtered limits, derived functors of limit, Mittag — Leffler sequences      350—354 359—361
Abelian categories-review of formalism $\beta$ -filtered limits, derived functors of limit, sequences of length $\gamma$       348
Abelian categories-review of formalism $\beta$ -filtered limits, derived functors of limit, via canonical resolution      346—348 370—371
Abelian categories-review of formalism $\beta$ -filtered limits, derived functors of limit, via injectives      356
Abelian categories-review of formalism $\beta$ -filtered limits, injectives in functor categories      355—356
Abelian categories-review of formalism $\beta$ -filtered limits, local object      329
Abelian categories-review of formalism $\beta$ -filtered limits, localizant subcategory      328
Abelian categories-review of formalism $\beta$ -filtered limits, localizant subcategory, basic properties      332—334
Abelian categories-review of formalism $\beta$ -filtered limits, localizant subcategory, characterisations      334—335 338—339
Abelian categories-review of formalism $\beta$ -filtered limits, locally presentable categories      321 324—327
Abelian categories-review of formalism $\beta$ -filtered limits, quotient by Serre subcategory      327—328
Abelian categories-review of formalism $\beta$ -filtered limits, quotient maps and products      343—345
Abelian categories-review of formalism $\beta$ -filtered limits, quotients      327—345
Abelian categories-review of formalism $\beta$ -filtered limits, Serre subcategories      327
Abelian categories-review of formalism $\beta$ -filtered limits, [AB3* ( $\alpha$ )] and [AB4* ( $\alpha$ )]      346
Abelian categories-review of formalism $\beta$ -filtered limits, [AB3* ( $\alpha$ )] in functor categories      355
Abelian categories-review of formalism $\beta$ -filtered limits, [AB4] does not imply [AB4.5]      361—366
Adjoints of a triangulated functor is triangulated      179
Adjoints, A(-) preserves and reflects adjoints      181—182
Adjoints, Bousfield localisation      288 309—318
Adjoints, Brown representability      286—287
Bousfield localisation      288 309—318
Bousfield localisation for homology theory E      417—418
Bousfield localisation is selfdual      315—316
Bousfield localisation, embedding the quotient      316—317
Bousfield localisation, existence      288 318
Bousfield localisation, local object      310
Bousfield localisation, perpendicular subcategories      313
Brown representability      275
Brown representability for $\aleph_1$ -perfectly generated categories      282—284
Brown representability for dual of E-acyclic spectra      419—420
Brown representability for duals of well-generated categories      303—306
Brown representability for E-acyclic spectra      417—418
Brown representability for E-local spectra      417—418
Brown representability for spectra      408
Brown representability for well-generated categories      285—286
Brown representability, adjoints      286
Brown — Comenetz objects      302—303 307
Cardinal of       410—411
Cardinal, regular      103
Cardinal, singular      103
Cofinal sequences      358
Compact generating set      274
Compact objects      130
Compact objects in quotient      138 143—144
Compact objects, filtrations by coproducts      371—378
Compact objects, generators for      140
Compact objects, subcategory of      129
Compact objects, subcategory of, inclusion relations      129
Compact objects, subcategory of, is localising      130
Compactly generated categories      274
Existence of products      288
Filtrations by coproducts of compact objects      371—378
Freyd's universal abelian category      153—182
Freyd's universal abelian category, $A(\delta)$ is a Frobenius category      169
Freyd's universal abelian category, $A(\delta)$ is an abelian subcategory closed under extensions      161
Freyd's universal abelian category, A(-) is a functor, and preserves products      177—179
Freyd's universal abelian category, A(-) is a functor, and preserves products, $A(\delta)$ is a Frobenius category      169
Freyd's universal abelian category, A(-) is a functor, and preserves products, $A(\delta)$ is an abelian subcategory closed under extensions      161
Freyd's universal abelian category, A(-) is a functor, and preserves products, A(-) preserves and reflects adjoints      181—182
Freyd's universal abelian category, A(-) is a functor, and preserves products, category $B(\delta)$ and its equivalence with $A(\delta)$       162—163
Freyd's universal abelian category, A(-) is a functor, and preserves products, category $C(\delta)$ and its equivalence with $A(\delta)$       167—169
Freyd's universal abelian category, A(-) is a functor, and preserves products, category $D(\delta)$ and its equivalence with $A(\delta)$       172—173
Freyd's universal abelian category, A(-) is a functor, and preserves products, coproducts in $A(\delta)$ when $\delta$ satisfies [TR5]      169—171
Freyd's universal abelian category, A(-) is a functor, and preserves products, definition of $A(\delta)$       154
Freyd's universal abelian category, A(-) is a functor, and preserves products, example of non-well-powered      394
Freyd's universal abelian category, A(-) is a functor, and preserves products, functors in $A(\delta)$ preserve products      154
Freyd's universal abelian category, A(-) is a functor, and preserves products, relation with $\mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab      214—220
Freyd's universal abelian category, A(-) is a functor, and preserves products, subobjects and quotient objects      172—177
Freyd's universal abelian category, A(-) is a functor, and preserves products, universal homological functor      163—164
Freyd's universal abelian category, A(-) preserves and reflects adjoints      181—182
Freyd's universal abelian category, category $B(\delta)$ and its equivalence with $A(\delta)$       162—163
Freyd's universal abelian category, category $C(\delta)$ and its equivalence with $A(\delta)$       167—169
Freyd's universal abelian category, category $D(\delta)$ and its equivalence with $A(\delta)$       172—173
Freyd's universal abelian category, coproducts in $A(\delta)$ when $\delta$ satisfies [TR5]      169—171
Freyd's universal abelian category, definition of $A(\delta)$       154
Freyd's universal abelian category, example of non-well-powered      394
Freyd's universal abelian category, functors in $A(\delta)$ preserve products      154
Freyd's universal abelian category, have enough projectives      153—154
Freyd's universal abelian category, relation with $\mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab      214—220
Freyd's universal abelian category, subobjects and quotient objects      172—177
Freyd's universal abelian category, universal homological functor      163—164
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab is exact and respects coproducts      215
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab is restriction      215—216
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab is the quotient by a colocalizant subcategory      216—218 290
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab is the quotient by a localizant subcategory in the presence of injectives      289—290
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab, existence      214—215
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab, respects products      218
Functor categories, abelian categories of product-preserving functors      183—214
Functor categories, abelian categories of product-preserving functors, are locally presentable      221—224 326
Functor categories, abelian categories of product-preserving functors, coproducts      191
Functor categories, abelian categories of product-preserving functors, definitions      185
Functor categories, abelian categories of product-preserving functors, do not satisfy [AB5], [AB5*]      209—210
Functor categories, abelian categories of product-preserving functors, Embedding arbitrary objects in homological ones      259—262
Functor categories, abelian categories of product-preserving functors, have enough projectives      212
Functor categories, abelian categories of product-preserving functors, homological functors      204—205
Functor categories, abelian categories of product-preserving functors, homological objects      224—229 258—262
Functor categories, abelian categories of product-preserving functors, homological objects characterised in terms of vanishing Ext      258—259
Functor categories, abelian categories of product-preserving functors, may not have cogenerators      403—405
Functor categories, abelian categories of product-preserving functors, relation with Freyd's universal abelian category      214—220
Functor categories, abelian categories of product-preserving functors, satisfy [AB3*]      186—187 200
Functor categories, abelian categories of product-preserving functors, satisfy [AB3]      196—200
Functor categories, abelian categories of product-preserving functors, satisfy [AB4*]      206
Functor categories, abelian categories of product-preserving functors, satisfy [AB4]      207—209
Functor categories, abelian categories of product-preserving functors, via universal homological functor      384—385
Functor, kernel of      74 99
Functor, representability      275
Functor, representability, for $\aleph_1$ -perfectly generated categories      282—284
Functor, representability, for duals of well-generated categories      303—306
Functor, representability, for well-generated categories      285—286
Functor, triangulated      73
Generating set      205 273—274
Generating set a category without      438—441
Generating set in the dual of well-generated categories      302—303
Generating set, compact      274
Generating set, compactly generated categories      274
Generating set, generate category      285
Generating set, perfect      273—274
Generating set, well generated categories      274

Gluing data      318—319
Good morhism of triangles      52
Good object in a subcategory      113
Grothendieck's duality theorem      306
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab      224—229 258—262
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab, as filtered colimits of representables      226—229
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab, characterisation in terms of vanishing Ext      258—259
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab, Embedding arbitrary objects in homological ones      259—262
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab, stable under filtered colimits      225
Homological functor into abelian functor categories      204—205
Homological functor, definition      32
Homological functor, examples      32
Homological functor, universal      163—166
Homological functor, universal into [AB5*]      384—385
Homotopy cartesian square      52
Homotopy colimits of subsequences      68—70
Homotopy colimits, definition      63
Homotopy colimits, elementary properties      64—65
Homotopy pullback      54—55 183—184
Homotopy pushout      53—54
Idempotent splitting      65
Kernel of functor      74 99
Large categories      99—100
Limits $\beta$ -filtered      321
Local object in abelian category      329
Local object in triangulated category      310
Localisation, Bousfield      288 309—318
Localisation, Bousfield, embedding the quotient      316—317
Localisation, Bousfield, existence      288 318
Localisation, Bousfield, is selfdual      315—316
Localisation, Bousfield, local object      310
Localisation, Bousfield, perpendicular subcategories      313
Localisation, Thomason      143—144
Localisation, Verdier      74—99 309
Localisation, Verdier, existence theorem      74—75
Localisation, Verdier, size of Hom-sets      99—100 137 318
Localising subcategory      106—107
Localising subcategory of small objects      126
Locally presentable categories      221—224 321 324—327
Mapping cone, definition      45
Mapping cone, [TR4]      51
Modules , definition      387—388
Modules , force large images      391
Modules , respect homomorphisms      390—391
Modules , stabilise eventually      390
Octahedral axiom      58 60
Perfect classes, definition      110—111
Perfect classes, maximal      120—121
Perfect classes, new out of old      111 116 119
Perfect classes, which are triangulated subcategories      115
Perfect generating set      273—274
Perfectly generated category      274
Perfectly generated category, $D(\mathbb{Q})$ is not      432—437
Phantom maps      219—220
Phantom maps and injectives in $\mathscr{E}_x \delta^{op}$ , Ab      299—300
Phantom maps and right adjoint to $\alpha$       301
Phantom maps annihilated by homological functors into [AB5*] categories      383—384
Phantom maps as the kernel of $D(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab      218—219
Phantom maps from coproducts of compacts vanish      369—370
Phantom maps in $D(\mathbb{Z})$       438—440
Phantom maps, definition      219
Phantom maps, existence for every $\alpha$       219—220
Pretriangle, definition      33
Pretriangle, examples      34
Pretriangle, new out of old      34 49
Pretriangulated category      29 70
Pretriangulated category, definition      29
Products of triangles are triangles      37
Products, existence of      288
Quotient categories      74—99 309
Quotient categories, commutative squares      85—86
Quotient categories, compact objects      143—144
Quotient categories, embedding via Bousfield localisation      316—317
Quotient categories, equality of morphisms      84—85
Quotient categories, existence      74—75 84
Quotient categories, isomorphisms      90 92
Quotient categories, preservation of products      107 110
Quotient categories, size of Hom-sets      99—100 137 318
Quotient categories, zero objects      91
Regular cardinal      103
Representability of functors      275
Representability of functors for $\aleph_1$ -perfectly generated categories      282—284
Representability of functors for duals of well-generated categories      303—306
Representability of functors for well-generated categories      285—286
Serre subcategories      327
Singular cardinal      103
Six functors      318—319
Small categories      99—100 137
Small hom-sets      99—100 137 318
Small object, definition      123
Small object, subcategory of      124
Small object, subcategory of, is localising      126
Small object, subcategory of, is triangulated      124
Spectra, $\mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab does not have a cogenerator      425
Spectra, Bousfield localisation for homology      417—418
Spectra, Brown representability      408
Spectra, cardinal of       410—411
Spectra, E-acyclics      411—412
Spectra, E-acyclics, are well-generated      417—418
Spectra, E-acyclics, Brown representability      417—418
Spectra, E-acyclics, Brown representability for dual      419—420
Spectra, E-local      419
Spectra, E-local, are well-generated      417—418
Spectra, E-local, Brown representability      417—418
Spectra, elementary properties      407—408
Spectra, functor to D(R)      420
Spectra, functor to D(R), descends to $\mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab      422—425
Spectra, functor to D(R), respects $\alpha$ -compacts      420—421
Splitting, idempotent      65
Splitting, triangle with 0      42—45
Subcategory of compact objects      129 130
Subcategory of compact objects, inclusion relations      129
Subcategory of compact objects, is localising      130
Subcategory of small objects      124
Subcategory of small objects is localising      126
Subcategory of small objects is triangulated      124
Subcategory, generated by a set      103—104 106—107
Subcategory, localising      106—107
Subcategory, thick      74 99
Subcategory, thick closure      75 99 147—149
Subcategory, triangulated      60
Thomason localisation      143—144
Thomason localisation, applied to finding $T^{\alpha}$       409
TR0      29
TR1      29
TR2      29
TR3      30
TR4, equivalent formulations      51 60
TR5, dual      63
TR5, statement      63
Triangles, contractible      47 48
Triangles, distinguished      29
Triangles, products of      37
Triangles, summands of      38
Triangulated subcategories, which are perfect classes      115
Universal homological functors      163—166 384—385
Verdier localisation      74—99 309
Verdier localisation, existence theorem      74—75
Verdier localisation, size of Hom-sets      99—100 137 318
Well generated categories      274
Well generated categories are unions of $T^{\beta}$       285—286
Well generated categories, duals satisfy Brown representability      303—306
Well generated categories, neither $K(\mathbb{Z})$ nor $K(\mathbb{Z})^{op}$       437—441
Well generated categories, not both T and T op      427—431
Well generated categories, satisfy Brown representability      285—286

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