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Neeman A. Ч Triangulated categories
Neeman A. Ч Triangulated categories

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Ќазвание: Triangulated categories

јвтор: Neeman A.

јннотаци€:

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" Ч the "well generated triangulated categories" Ч 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.


язык: en

–убрика: ћатематика/јлгебра/“еори€ категорий/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2001

 оличество страниц: 449

ƒобавлена в каталог: 12.03.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
Abelian categories of product-preserving functors      183Ч214
Abelian categories of product-preserving functors are locally presentable      221Ч224 326
Abelian categories of product-preserving functors do not satisfy [AB5], [AB5*]      209Ч210
Abelian categories of product-preserving functors have enough projectives      212
Abelian categories of product-preserving functors may not have cogenerators      403Ч405
Abelian categories of product-preserving functors satisfy [AB3*]      186Ч187 200
Abelian categories of product-preserving functors satisfy [AB3]      196Ч200
Abelian categories of product-preserving functors satisfy [AB4*]      206
Abelian categories of product-preserving functors satisfy [AB4]      207Ч209
Abelian categories of product-preserving functors via universal homological functor      384Ч385
Abelian categories of product-preserving functors, coproducts      191
Abelian categories of product-preserving functors, definitions      185
Abelian categories of product-preserving functors, homological functors      204Ч205
Abelian categories of product-preserving functors, homological objects      224Ч229 258Ч262
Abelian categories of product-preserving functors, homological objects, as filtered colimits of representables      226Ч229
Abelian categories of product-preserving functors, homological objects, characterisation in terms of vanishing Ext      258Ч259
Abelian categories of product-preserving functors, homological objects, Embedding arbitrary objects in homological ones      259Ч262
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      345Ч361
Abelian categories-review of formalism $\beta$-filtered limits, derived functors of limit, analogy with sheaves      349Ч351
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 Ч Leffler sequences      350Ч354 359Ч361
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      346Ч348 370Ч371
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      355Ч356
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      332Ч334
Abelian categories-review of formalism $\beta$-filtered limits, localizant subcategory, characterisations      334Ч335 338Ч339
Abelian categories-review of formalism $\beta$-filtered limits, locally presentable categories      321 324Ч327
Abelian categories-review of formalism $\beta$-filtered limits, quotient by Serre subcategory      327Ч328
Abelian categories-review of formalism $\beta$-filtered limits, quotient maps and products      343Ч345
Abelian categories-review of formalism $\beta$-filtered limits, quotients      327Ч345
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]      361Ч366
Adjoints of a triangulated functor is triangulated      179
Adjoints, A(-) preserves and reflects adjoints      181Ч182
Adjoints, Bousfield localisation      288 309Ч318
Adjoints, Brown representability      286Ч287
Bousfield localisation      288 309Ч318
Bousfield localisation for homology theory E      417Ч418
Bousfield localisation is selfdual      315Ч316
Bousfield localisation, embedding the quotient      316Ч317
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      282Ч284
Brown representability for dual of E-acyclic spectra      419Ч420
Brown representability for duals of well-generated categories      303Ч306
Brown representability for E-acyclic spectra      417Ч418
Brown representability for E-local spectra      417Ч418
Brown representability for spectra      408
Brown representability for well-generated categories      285Ч286
Brown representability, adjoints      286
Brown Ч Comenetz objects      302Ч303 307
Cardinal of $T(S^n , x)$      410Ч411
Cardinal, regular      103
Cardinal, singular      103
Cofinal sequences      358
Compact generating set      274
Compact objects      130
Compact objects in quotient      138 143Ч144
Compact objects, filtrations by coproducts      371Ч378
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      371Ч378
Freyd's universal abelian category      153Ч182
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      177Ч179
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      181Ч182
Freyd's universal abelian category, A(-) is a functor, and preserves products, category $B(\delta)$ and its equivalence with $A(\delta)$      162Ч163
Freyd's universal abelian category, A(-) is a functor, and preserves products, category $C(\delta)$ and its equivalence with $A(\delta)$      167Ч169
Freyd's universal abelian category, A(-) is a functor, and preserves products, category $D(\delta)$ and its equivalence with $A(\delta)$      172Ч173
Freyd's universal abelian category, A(-) is a functor, and preserves products, coproducts in $A(\delta)$ when $\delta$ satisfies [TR5]      169Ч171
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      214Ч220
Freyd's universal abelian category, A(-) is a functor, and preserves products, subobjects and quotient objects      172Ч177
Freyd's universal abelian category, A(-) is a functor, and preserves products, universal homological functor      163Ч164
Freyd's universal abelian category, A(-) preserves and reflects adjoints      181Ч182
Freyd's universal abelian category, category $B(\delta)$ and its equivalence with $A(\delta)$      162Ч163
Freyd's universal abelian category, category $C(\delta)$ and its equivalence with $A(\delta)$      167Ч169
Freyd's universal abelian category, category $D(\delta)$ and its equivalence with $A(\delta)$      172Ч173
Freyd's universal abelian category, coproducts in $A(\delta)$ when $\delta$ satisfies [TR5]      169Ч171
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      153Ч154
Freyd's universal abelian category, relation with $\mathscr{E}_x \{T^{\alpha} \}^{op}$ , Ab      214Ч220
Freyd's universal abelian category, subobjects and quotient objects      172Ч177
Freyd's universal abelian category, universal homological functor      163Ч164
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      215Ч216
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$, Ab is the quotient by a colocalizant subcategory      216Ч218 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      289Ч290
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$, Ab, existence      214Ч215
Functor $\pi: A(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$, Ab, respects products      218
Functor categories, abelian categories of product-preserving functors      183Ч214
Functor categories, abelian categories of product-preserving functors, are locally presentable      221Ч224 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*]      209Ч210
Functor categories, abelian categories of product-preserving functors, Embedding arbitrary objects in homological ones      259Ч262
Functor categories, abelian categories of product-preserving functors, have enough projectives      212
Functor categories, abelian categories of product-preserving functors, homological functors      204Ч205
Functor categories, abelian categories of product-preserving functors, homological objects      224Ч229 258Ч262
Functor categories, abelian categories of product-preserving functors, homological objects characterised in terms of vanishing Ext      258Ч259
Functor categories, abelian categories of product-preserving functors, may not have cogenerators      403Ч405
Functor categories, abelian categories of product-preserving functors, relation with Freyd's universal abelian category      214Ч220
Functor categories, abelian categories of product-preserving functors, satisfy [AB3*]      186Ч187 200
Functor categories, abelian categories of product-preserving functors, satisfy [AB3]      196Ч200
Functor categories, abelian categories of product-preserving functors, satisfy [AB4*]      206
Functor categories, abelian categories of product-preserving functors, satisfy [AB4]      207Ч209
Functor categories, abelian categories of product-preserving functors, via universal homological functor      384Ч385
Functor, kernel of      74 99
Functor, representability      275
Functor, representability, for $\aleph_1$-perfectly generated categories      282Ч284
Functor, representability, for duals of well-generated categories      303Ч306
Functor, representability, for well-generated categories      285Ч286
Functor, triangulated      73
Generating set      205 273Ч274
Generating set a category without      438Ч441
Generating set in the dual of well-generated categories      302Ч303
Generating set, compact      274
Generating set, compactly generated categories      274
Generating set, generate category      285
Generating set, perfect      273Ч274
Generating set, well generated categories      274
Gluing data      318Ч319
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      224Ч229 258Ч262
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab, as filtered colimits of representables      226Ч229
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab, characterisation in terms of vanishing Ext      258Ч259
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab, Embedding arbitrary objects in homological ones      259Ч262
Homological functor as object in $\mathscr{E}_x \delta^{op}$ , Ab, stable under filtered colimits      225
Homological functor into abelian functor categories      204Ч205
Homological functor, definition      32
Homological functor, examples      32
Homological functor, universal      163Ч166
Homological functor, universal into [AB5*]      384Ч385
Homotopy cartesian square      52
Homotopy colimits of subsequences      68Ч70
Homotopy colimits, definition      63
Homotopy colimits, elementary properties      64Ч65
Homotopy pullback      54Ч55 183Ч184
Homotopy pushout      53Ч54
Idempotent splitting      65
Kernel of functor      74 99
Large categories      99Ч100
Limits $\beta$-filtered      321
Local object in abelian category      329
Local object in triangulated category      310
Localisation, Bousfield      288 309Ч318
Localisation, Bousfield, embedding the quotient      316Ч317
Localisation, Bousfield, existence      288 318
Localisation, Bousfield, is selfdual      315Ч316
Localisation, Bousfield, local object      310
Localisation, Bousfield, perpendicular subcategories      313
Localisation, Thomason      143Ч144
Localisation, Verdier      74Ч99 309
Localisation, Verdier, existence theorem      74Ч75
Localisation, Verdier, size of Hom-sets      99Ч100 137 318
Localising subcategory      106Ч107
Localising subcategory of small objects      126
Locally presentable categories      221Ч224 321 324Ч327
Mapping cone, definition      45
Mapping cone, [TR4]      51
Modules $p^i M$, definition      387Ч388
Modules $p^i M$, force large images      391
Modules $p^i M$, respect homomorphisms      390Ч391
Modules $p^i M$, stabilise eventually      390
Octahedral axiom      58 60
Perfect classes, definition      110Ч111
Perfect classes, maximal      120Ч121
Perfect classes, new out of old      111 116 119
Perfect classes, which are triangulated subcategories      115
Perfect generating set      273Ч274
Perfectly generated category      274
Perfectly generated category, $D(\mathbb{Q})$ is not      432Ч437
Phantom maps      219Ч220
Phantom maps and injectives in $\mathscr{E}_x \delta^{op}$, Ab      299Ч300
Phantom maps and right adjoint to $\alpha$      301
Phantom maps annihilated by homological functors into [AB5*] categories      383Ч384
Phantom maps as the kernel of $D(T) \to \mathscr{E}_x \{T^{\alpha} \}^{op}$, Ab      218Ч219
Phantom maps from coproducts of compacts vanish      369Ч370
Phantom maps in $D(\mathbb{Z})$      438Ч440
Phantom maps, definition      219
Phantom maps, existence for every $\alpha$      219Ч220
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      74Ч99 309
Quotient categories, commutative squares      85Ч86
Quotient categories, compact objects      143Ч144
Quotient categories, embedding via Bousfield localisation      316Ч317
Quotient categories, equality of morphisms      84Ч85
Quotient categories, existence      74Ч75 84
Quotient categories, isomorphisms      90 92
Quotient categories, preservation of products      107 110
Quotient categories, size of Hom-sets      99Ч100 137 318
Quotient categories, zero objects      91
Regular cardinal      103
Representability of functors      275
Representability of functors for $\aleph_1$-perfectly generated categories      282Ч284
Representability of functors for duals of well-generated categories      303Ч306
Representability of functors for well-generated categories      285Ч286
Serre subcategories      327
Singular cardinal      103
Six functors      318Ч319
Small categories      99Ч100 137
Small hom-sets      99Ч100 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      417Ч418
Spectra, Brown representability      408
Spectra, cardinal of $T(S^n, x)$      410Ч411
Spectra, E-acyclics      411Ч412
Spectra, E-acyclics, are well-generated      417Ч418
Spectra, E-acyclics, Brown representability      417Ч418
Spectra, E-acyclics, Brown representability for dual      419Ч420
Spectra, E-local      419
Spectra, E-local, are well-generated      417Ч418
Spectra, E-local, Brown representability      417Ч418
Spectra, elementary properties      407Ч408
Spectra, functor to D(R)      420
Spectra, functor to D(R), descends to $\mathscr{E}_x \{T^{\alpha} \}^{op}$, Ab      422Ч425
Spectra, functor to D(R), respects $\alpha$-compacts      420Ч421
Splitting, idempotent      65
Splitting, triangle with 0      42Ч45
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      103Ч104 106Ч107
Subcategory, localising      106Ч107
Subcategory, thick      74 99
Subcategory, thick closure      75 99 147Ч149
Subcategory, triangulated      60
Thomason localisation      143Ч144
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      163Ч166 384Ч385
Verdier localisation      74Ч99 309
Verdier localisation, existence theorem      74Ч75
Verdier localisation, size of Hom-sets      99Ч100 137 318
Well generated categories      274
Well generated categories are unions of $T^{\beta}$      285Ч286
Well generated categories, duals satisfy Brown representability      303Ч306
Well generated categories, neither $K(\mathbb{Z})$ nor $K(\mathbb{Z})^{op}$      437Ч441
Well generated categories, not both T and T op      427Ч431
Well generated categories, satisfy Brown representability      285Ч286
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