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Lawvere F.W., Schanuel S.H. — Conceptual Mathematics: A First Introduction to Categories
Lawvere F.W., Schanuel S.H. — Conceptual Mathematics: A First Introduction to Categories

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Название: Conceptual Mathematics: A First Introduction to Categories

Авторы: Lawvere F.W., Schanuel S.H.

Аннотация:

The idea of a category — a sort of mathematical universe — has brought about a remarkable unification and simplification of mathematics. Written by two of the best known names in categorical logic, this is the first book to apply categories to the most elementary mathematics.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 376

Добавлена в каталог: 10.07.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abbildung      245
Absolute value      34 140 188
Accessibility      138
Accessibility as positive property      173
action      218f 303
Analysis of sound wave      106
Arithmetic of objects      327
Arrow in graph      141
Arrow in graph, generic arrow      215
Associative law for composition      17 21
Associative law versus commutative law      25
Associative law, sum, product (objects)      220ff 28Iff
Automaton (= dynam. syst.)      137 303
Automorphism      55
Automorphism in sets = permutation      57 138 155 180
Automorphism, category of      138
Balls, spheres      120ff
Banach’s fixed point theorem      121
Base point      216 295
Binary operation      64 218 302
Bird watcher      105
bookkeeping      21 35
Brouwer      120 309
Brouwer, fixed point theorems      120ff
Brouwer, retraction theorems      122
Cancellation laws      43 44 59
Cantor, Georg      106 291 304ff
Cantor, Georg, Cantor — Bernstein theorem      106
Cantor, Georg, diagonal theorem      304 316
Cartesian closed category      315 322
Cartesian coordinates      42 87
Category      17 21
Category = of endomaps      136
Category of dynamical systems      137
Category of graphs      141 156
Category of parts      344
Category of permutations      57
Category of pointed sets      216 223 295ff
Category of smooth spaces      135
Category of topological spaces      135
Center of mass      324
Chad’s formula      75 117
Chaotic      317
Chaotic, truth values      343
Characteristic map      340
Chinese restaurant      76
Choice problem      45 71
Choice problem, examples      46 71
Choice problem, section as choice of representatives      51 100 117
Clan      163
Cofigure (dual of figure)      272
Cograph      294
Cohesive      120 135
Commuting diagram      50 201
Composition of maps      16 114
Composition of maps and combining concepts      129
Computer science      103 185
Concept, coconcept      276 284
Configuration      318
Congressional representatives      51 95
Congruent figures      67
Conic section      42
Constant map      71
Constructivist idealization      306ff
Continuous map      6 120
Contraction map      121
Contrapositive      124
Convergence to equilibrium      138
Coordinate pair      42 87
Coordinate pair, system      86
Coproduct      see Sum
Counterexample      115
Cross — section      93 105
Crystallography      180
CYCLE      140 176 183 187 271
Descartes, R.      41 322
Determination or extension problem      45 68
Determination, examples      45 47
Diagonal argument, Cantor’s      304 316
Diagram of shape G      149ff 200ff
Diagram, commuting      201
Diagram, external      15
Diagram, internal, of a map of sets      14 15
Diagram, internal, of a set      13
Differential calculus      324
Directed graph      see Graph
Disjunction      350
Disk      8 121 236
Distinguished point      295
Distributive category      223 276ff 295
Distributive law      223
Distributive law, general      278
Division problems      43
Domain of a map      14
Dot, in graph      141
Dot, in graph, naked      215
Dual      215 284
Dynamical system, discrete      137 161 303
Dynamical system, discrete, equivariant map of      182
Dynamical system, discrete, objectifying properties of      175
Eilenberg, Samuel      3
Einstein, Albert      309
Electrical engineering      227
Empty set, maps to and from      30
Empty set, maps to and from as initial set      254
Endomap      15
Endomap, automorphisms      55
Endomap, automorphisms, category of      138f
Endomap, category of      136ff
Endomap, idempotent      99ff 118
Endomap, internal diagram of      15
Endomap, involution      139
Epimorphism      53 59
Equality of maps, test for, in graphs      215 250f
Equality of maps, test for, in sets      23 115
Equality of maps, test for, indyn. syst.      215 246f
Equalizer      292
Equilibrium state      214
Equinumerous      see Isomorphic 41
Equivariant (map of dyn. syst.)      182
Euclidean algorithm      102
Euclid’s category      67
Evaluation, as composition      19
Evaluation, as composition, map      313
Exemplifying = sampling, parameterizing      83
Exponential object = map obj.      313 320ff
Exponents, laws of      324ff
Extension problem      45 see
External diagram      see Diagram
Eye of the storm      130
Factoring      102
Faithful      318
Family      82
Family of maps      303
Family tree      162
Fiber, fibering      82
Fibonacci (Leonardo of Pisa)      318
figure      83
Figure of shape      1 see
Figure, incidence of      340ff
Figure, shape of      83
Finite sets, category of      13
Fixed point      117 137
Fixed point and diagonal theorem      303
Fixed point as point of dyn. syst.      214 232f
Formulas and rules of proof      306
Fraction symbol      83 102
Free category      200 203
Full (insertion)      138 146
Function (= map)      14 22
Function space (= map object)      313
functor      167
Galileo      3 47 106 120 199 216 236 257 308 322
Gender      162 181
genealogy      162f
Generator      183 247
Goedel, Kurt      306ff
Goedel, Kurt, numbering      307
Graph as diagram shape      149 200
Graph of a map      293
Graph, irreflexive      141 189 196ff
Graph, reflexive      145 192
Gravity      309
Grothendiek, A.      352
Hair      183 187
Hamilton, William Rowan      309
Helix      240
Homeomorphic      67
Hooke, Robert      129
Idempotent endomap      54 108 117f 187
Idempotent endomap from a retract      54 100
Idempotent endomap, category of      138
Idempotent endomap, number of (in sets)      20 35
Idempotent endomap, splitting of      102
Idempotent object      289
Identity, laws      17 21 166 225
Identity, map      15 21
Identity, matrix      279
Implication      350
Incidence relations      245 249ff 258
Inclusion map      122 336 344 see
Incompleteness theorem      106 306ff
Inequality      99
Infinite sets      55 106 108
Initial object      215 216 254 280
Initial object in other categories      216 280
Initial object in sets      30 216
Initial object, uniqueness      215
Injection maps for sum      222 266ff
Injective map      52 59 146ff 267
Integers      140 187
Internal diagram      14
Intersection      349f
Inverse of a map      40
Inverse of a map, uniqueness      42 62
Invertible map (isomorphism)      40
Invertible map (isomorphism), endomap (automorphism)      55 138 155
Involution      118 139 187
Irreflexive graph      see Graph
Isomorphic      40
Isomorphism      40 6Iff
Isomorphism as coordinate system      86ff
Isomorphism, Descartes’ example      42 87
Isomorphism, reflexive, symmetric, transitive      41
Iteration      179
Jacobi, Karl      309
Klein, Felix      180
Knowledge      84
Labeling      see Sorting
laws      see Identity Associative Commutative Distributive
Laws of categories      21
Laws of exponentiation      324ff
Leibniz, Gottfried Wilhelm      129
Linear category      279ff
Logic      335ff 344ff
Logic and truth      335ff
Logic, rules of      350f
Logical operations      180 349f
Logicians      306ff
Loop, as point      232f
Mac Lane, Saunders      3
Map object (exponential)      313ff 320ff
Map object (exponential) and diagonal argument      316
Map object (exponential) and laws of exponents      314f 324ff
Map object (exponential) in graphs      33Iff
Map object (exponential) in sets      331
Map object (exponential), definition vs product      330
Map object (exponential), points of      323
Map object (exponential), transformation (for motion)      323f
Map, of sets      14 22
Map, of sets, in category      17 21
Mapification of concepts      127
Maps, number of, in dynamical syst.      182
Maps, number of, in sets      33
Mathematical universe, category as      3 17
Matrilineal      181
Matrix, identity matrix      279
Matrix, multiplication      279ff
Modal operators      343
Modeling, simulation of a theory      182
Modus ponens rule of inference      350
Momentum      318
Monic map      see Monomorphism
Monoid      166ff
Monomorphism      52 59
Monomorphism, test for, in graphs      336
Monomorphism, test for, in sets      336
Motion      3 216 236 320
Motion of bodies in space      323ff
Motion of wind (or fluid)      130ff
Motion, periodic      106
Motion, state of      318
Motion, uniform      120
Multigraph, directed irreflexive      see Graph
Multiplication of      244
Multiplication of matrices      see Matrix
Multiplication of objects (= product) of courses      7
Multiplication of objects (= product) of cycles      244
Multiplication of objects (= product) of disc and segment      8
Multiplication of objects (= product) of dynamical systems      239ff
Multiplication of objects (= product) of plane and line      4
Multiplication of objects (= product) of sentences      8
Naming (as map) in dynamical systems      176ff
Naming (as map) in sets      83
Navigation, terrestial and celestial      309
Negative of object      287
Negative properties      173 176
Newton, Isaac      199 309
Non-distributive categories      295f
Non-singular map      see Monomorphism
Numbers, natural, analog, in graphs      267
Numbers, natural, analog, in graphs, in distributive category      327
Numbers, natural, isomorphism classes of sets      39ff
Numbers, natural, monoid of      167ff
Numbers, natural, to represent states of dyn. syst.      177f
Numbers, rational      83 102
Objectification in dynamical systems      175ff
Objectification in the subjective      181
Objectification of concepts as objects, maps      127
Objective in philosophy      84
Objective, contained in subjective      84 18Iff
Observable      317
Observable, chaotic      317
Operation, unary, binary, etc.      302
Operator      14
Origin or base point      295
Paradox      306
Parameterizing      83
Parameterizing of maps      303f 313
Parameterizing of maps, weakly      306
Parity (even vs odd)      66 174
Partitioning      82
Parts of an object      335ff
Parts of an object, category of      344ff
Permutation, set-automorphism      56ff
Permutation, set-automorphism, category of      57 138ff
Philosophical algebra      129
Philosophy      84
Pick’s formula      47
plot      86
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