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