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Moerdijk I., Reyes G.E. — Models for smooth infinitesimal analysis
Moerdijk I., Reyes G.E. — Models for smooth infinitesimal analysis



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Название: Models for smooth infinitesimal analysis

Авторы: Moerdijk I., Reyes G.E.

Аннотация:

The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 399

Добавлена в каталог: 05.06.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$C^{\infty}$-homomorphism      16
$C^{\infty}$-ring      15
$\Delta$-continuous      321 323
(differential) complex      136
(generalized) Kock — Lawvere, axiom      184 228 296 182
Acceleration field      213
Accessible functions      321
Accessible natural numbers      299
Accessible reals      318
Action-preserving parallel, transport of rays      214
Affine connection      189 197
Affine connection on $C^{\infty}(M, N)$      231
Affine structure      204
Alternation      135
Ambrose — Palais — Singer theorem      229
Ambrose — Palais — Singer theorem for function spaces      231
Archimedean      88
Associated sheaf functor      104
Axiom of bounded search      298
Axiom of finite choice      253
Axiomatic system for smooth, infinitesimal analysis      294
Barycentric subdivision      151
Basel topos      286
Bockstein homomorphism      144
Borel’s theorem      18
Boundary operator      135 148
Chain map      136
Chain-connected      127
Characteristic function      20
Christoffel symbols of the, second kind      236
Closed n-forms      137
Closed sublocus      60
Coherent formula      297
Coherent induction in B and Z      344
Coherent induction scheme      298
Compactness      121
Comparison theorem for      231
Comparison theorem for De Rham, cohomology in Z      281
Conformal Gauss — Bonnet      234 192
Connected      128
Connection      189
Connection form      215
Connection map      198
Constant sheaves      105
continuous      123 270
Convolution      323
Covariant derivative of Y along, X      199
Covering families      98 241 288
Curvature form      214
Curvature of the Ehresmann, connection      224
De Rham cohomology      137
de Rham complex      134 137
De Rham’s theorem in Z      280
De Rham’s theorem with smooth, parameters      170
De Rham’s theorem with, parameters      173
De Rham’s theorem with, parameters for Cech, cohomology      174
De Rham’s theorem, classical version      162 388
Decidable      124 306
Decidable induction      306
Degeneracy      135
Degree      280
Derivative      202
Dirac distribution      323
Dirac function      285
Directional derivative      200
Distribution with compact, support      94
Distributions      323
Ehresmann connections      219 220
Ehresmann jets      35
End-extension      307
Equivalence of predistributions, evaluation      78
Evaluation map      74
Exact n-forms      137
Exponentials      65
Extension Principle      322
Extension principle in B and Z      332
Extension property      146
Exterior differentiation map      136
External distributions      336
Fermat axiom      304
Fi-separable      279
Fiberwise i?-module structure      185
finite      252
Finite cardinals      307
Finite extensions of loci      339
Finite good cover      280
Finite n-cube      138
Finite sets      307
Finitely generated $C^{\infty}$-rings      21
Finitely presented $C^{\infty}$-rings      24
Finitely presented type term      297
Finitely presented type, recursion      298
First-order infinitesimal      75 77
Flat ideals      49
Flow equation      199
Formal $C^{\infty}$-varietie      8 58
Formal dual      58
Formally integrate      200
Formally real ring      40
Fourier integral representation of the Dirac functional      325
Frechet topology      46
Free $C^{\infty}$-ring      17
Free s-group      258
Gauss curvature      234
Gauss — Bonnet      215
Gauss — Bonnet theorem      213
Generic element      81 254
Generic infinitely large      252 328
Geodesic curvature      216
Geodesic curve      213
Geodesic spray      212
Germ      17
Germ determined      44
Good cover of M      163
Grothendieck topology      99
Hadamard’s lemma      305
Henselian      40
Holds in (is valid in, is true in)      87
Homogeneity      135
Homomorphism of $C^{\infty}$-rings      16
Homotopy invariance      143
Horizontal component      223
Horizontal vectors      198
Infinitesimal      77
Infinitesimal curve      78
Infinitesimal deformation of the, identity      208
Infinitesimal flow      208
Infinitesimal loci      64
Infinitesimal n-cube on M      134
Infinitesimal spaces      110 185
Infinitesimal version of      389
Inhabited      307
Integral curve      213
Integration axiom      83 111 297
Internal manifold      182
Internal metric      123
Internal partitions of unity      124
Internal topology      261
Inverse function theorem      297
Invertible infinitesimal      239
Iterated tangent bundle      203
Jacobi identity      187
Jets      35
Kock — Lawvere Axiom      80 110
Koszul’s law      201
Krull-topology      38
Kth-order infinitesimal      77
Lebesgue numbers      92 250
Li e-bracket      187
Lie algebra      187
Lie monoid (group)      187 208
Lipschitz condition      312
Local $C^{\infty}$-ring      31
Local ring      31 87
Local version of Gauss — Bonnet      218 234
Locally closed      20
Locally finite      47
Loci      58
Locus of invertible infinitesimals      65
L’Hopital’s Rule      304
M-colimit      185
Mayer — Vietoris sequence      144 153
Metric spaces      309
Microlinear space      182 185
N-form on M      134
Nakayama lemma      36
Natural numbers object      88 244
Near-point determined      44
Nilpotent infinitesimals      239
Open refinement property      298 310
Open refinement theorem      130
Open subloci      61
Order topology      88 120 246
Ordered local ring in the ex, tended sense      303
Ostrand’s theorem      258
Parallel transport      190
Peano axioms      298
PoincarS Lemma      140 150
Point determined      44
Pointed $C^{\infty}$-ring      32
Pointfinite cover      176
Powersheaf      104
Predistributions      322
Principal fiber bundle      219
Rays (oriented)      213
Real closed ring      40
Regular s-chain      234
Regular values      284
Residue field      31
Riemann Sums      330
Riemann — Christoffel tensor      235
Ring of dual numbers      19
S — Archimedean      252
S — Lindelof      274
S-chain connected      275
S-compactness      252
S-countable      264
S-countable, s-partition of unity      271
s-finite      252
S-finite refinement      266
S-group      257
S-local ring      258
S-topological space      253
Separably real closed      40
Sheafification functor      104
Singular g-chains      148
Singular g-simplex      148
Singular homology      282
Site      100
Smooth integers      252 390
Smooth natural numbers      252 289
Smooth Zariski topos      241 243
Sochozki’s formula      329
Spray      192
Spray on $C^{\infty}(M, V)$      231
Square root of      8 330
Standard natural numbers      252
Stokes’ Theorem      139
Stokes’ theorem for, infinitesimal n-chains      136
Strong difference      207
Subcanonical topology      101
Subfunctor      104
Subordinate to      124
Subsheaf      104
Symmetric connection      189
Taylor’s Formula      303
Tensor product      322
Test functions      322
Theories of variables types      295
Tietze extension theorem      62
Torsion      211
Torsion-free      189 211
Torsor structure      204
Transfer principle      337
Translation-space structure      204
Transversal pullback      30 121
Twist-Map      203
Vector bundle over M      195
Vector field      199
Velocity field      212
Vertical vectors      197
Weil algebra      20 35
Whitney’s spectral theorem      46
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