Conceptual mathematics: a first introduction to categories
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2009
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XII, 390 S. zahlr. graph. Darst. |
| ISBN: | 9780521894852 9780521719162 |
Staff View
MARC
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| 245 | 1 | 0 | |a Conceptual mathematics |b a first introduction to categories |c F. William Lawvere ; Stephen H. Schanuel |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2009 | |
| 300 | |a XII, 390 S. |b zahlr. graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
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Record in the Search Index
| _version_ | 1804139386925744128 |
|---|---|
| adam_text | Contents
Session 1
Preface
Organisation
of the book
Acknowledgements
Preview
Galileo and multiplication of objects
1
Introduction
2
Galileo and the flight of a bird
3
Other examples of multiplication of objects
ХШ
XV
xvii
3
3
3
7
Part I The category of sets
Article I Sets, maps, composition
13
1
Guide
20
Summary: Definition of category
21
Session
2
Sets, maps, and composition
22
1
Review of Article I
22
2
An example of different rules for a map
27
3
External diagrams
28
4
Problems on the number of maps from one set to another
29
Session
3
Composing maps and counting maps
31
Part II The algebra of composition
Article
Π
Isomorphisms
39
1
Isomorphisms
39
2
General division problems: Determination and choice
45
3
Retractions, sections, and idempotents
49
4
Isomorphisms and automorphisms
54
5
Guide
58
Summary: Special properties a map may have
59
Vlil
Contents
Session
4
Division
of maps: Isomorphisms
1
Division
of maps versus
division
of numbers
2
Inverses versus reciprocals
3
Isomorphisms as divisors
4
A small zoo of isomorphisms in other categories
60
60
61
63
64
Session
5
Division of maps: Sections and retractions
68
1
Determination problems
68
2
A special case: Constant maps
70
3
Choice problems
71
4
Two special cases of division: Sections and retractions
72
5
Stacking or sorting
74
6
Stacking in a Chinese restaurant
76
Session
6
Two general aspects or uses of maps
81
1
Sorting of the domain by a property
81
2
Naming or sampling of the codomain
82
3
Philosophical explanation of the two aspects
84
Session
7
Isomorphisms and coordinates
1
One use of isomorphisms: Coordinate systems
2
Two abuses of isomorphisms
Session
8
Pictures of a map making its features evident
Session
9
Retracts and idempotents
1
Retracts and comparisons
2
Idempotents as records of retracts
3
A puzzle
4
Three kinds of retract problems
5
Comparing infinite sets
Quiz
How to solve the quiz problems
Composition of opposed maps
Summary/quiz on pairs of Opposed maps
Summary: On the equation
ρ
°j
=
1A
Review of I-words
Test
1
86
86
89
91
99
99
100
102
103
106
108
109
114
116
117
118
119
Session
10
Brouwer s theorems
1
Balls, spheres, fixed points, and retractions
2
Digression on the
contrapositive
rule
3
Brouwer s proof
120
120
124
124
Contents ix
4 Relation
between fixed point and retraction theorems
126
5
How to understand a proof:
The objectification and mapification of concepts
127
6
The eye of the storm
130
7
Using maps to formulate guesses
131
Part III Categories of structured sets
Article III Examples of categories
135
1
The category SP of endomaps of sets
136
2
Typical applications of S^
137
3
Two subcategories of S°
138
4
Categories of endomaps
138
5 Irreflexive
graphs
141
6
Endomaps as special graphs
143
7
The simpler category sh Objects are just maps of sets
144
8
Reflexive graphs
145
9
Summary of the examples and their general significance
146
10
Retractions and injectivity
146
11
Types of structure
149
12
Guide
151
Session
11
Ascending to categories of richer structures
152
1
A category of richer structures: Endomaps of sets
152
2
Two subcategories: Idempotents and automorphisms
155
3
The category of graphs
156
Session
12
Categories of diagrams
161
1
Dynamical systems or automata
161
2
Family trees
162
3
Dynamical systems revisited
163
Session
13
Monoids
166
Session
14
Maps preserve positive properties
170
1
Positive properties versus negative properties
173
Session
15
Objectification of properties in dynamical systems
175
1
Structure-preserving maps from a cycle to another
endomap
175
2
Naming elements that have a given period by maps
176
3
Naming arbitrary elements
177
4
The philosophical role of
N 180
5
Presentations of dynamical systems
182
Contents
Session 16 Idempotents,
involutions, and graphs
1
Solving exercises on idempotents and involutions
2
Solving exercises on maps of graphs
Session
17
Some uses of graphs
1
Paths
2
Graphs as diagram shapes
3
Commuting diagrams
4
Is a diagram a map?
Test
2
187
187
189
196
196
200
201
203
204
Session
18
Review of Test
2
205
Part IV Elementary universal mapping properties
Article IV
Universal mapping properties
213
1
Terminal objects
213
2
Separating
215
3
Initial object
215
4
Products
216
5
Commutative, associative, and identity laws for
multiplication of objects
220
6
Sums
222
7
Distributive laws
222
8
Guide
223
Session
19
Terminal objects
225
Session
20
Points of an object
230
Session
21
Products in categories
236
Session
22
Universal mapping properties and incidence relations
245
1
A special property of the category of sets
245
2
A similar property in the category of endomaps
of sets
246
3
Incidence relations
249
4
Basic figure-types, singular figures, and incidence,
in the category of graphs
Session
23
More on universal mapping properties
1
A category of pairs of maps
2
How to calculate products
250
254
255
256
Contents
Xl
Session 24
Uniqueness of products and definition of sum
1
The terminal object as an identity for multiplication
2
The uniqueness theorem for products
3
Sum of two objects in a category
Session
25
Labelings and products of graphs
1
Detecting the structure of a graph by means of labelings
2
Calculating the graphs
Α χ Υ
3
The distributive law
Session
26
Distributive categories and linear categories
1
The standard map
Α χ
Bx
+
Α χ
B2
—>
A x
(fi,
+
B2)
2
Matrix multiplication in linear categories
3
Sum of maps in a linear category
4
The associative law for sums and products
Session
27
Session
28
Test
3
Test
4
Test
5
Examples of universal constructions
1
Universal constructions
2
Can objects have negatives?
3
Idempotent objects
4
Solving equations and picturing maps
The category of pointed sets
1
An example of a non-distributive category
261
261
263
265
269
270
273
275
276
276
279
279
281
284
284
287
289
292
295
295
299
300
301
Session
29
Binary operations and diagonal arguments
1
Binary operations and actions
2
Cantor s diagonal argument
Part V Higher universal mapping properties
Artide V
Map objects
1
Definition of map object
2
Distributivity
3
Map objects and the Diagonal Argument
4
Universal properties and
observables
5
Guide
Session
30
Exponentiation
1
Map objects, or function spaces
302
302
303
313
313
315
316
316
319
320
320
XU
Contents
2
A
fundamental
example of the transformation
of map objects
3
Laws of exponents
4
The distributive law in cartesian closed categories
Session
31
Map object versus product
1
Definition of map object versus definition of product
2
Calculating map objects
Article VI The
contravariant
parts functor
1
Parts and stable conditions
2
Inverse Images and Truth
Session
32
Subobject, logic, and truth
1
Subobjects
2
Truth
3
The truth value object
Session
33
Parts of an object: Toposes
1
Parts and inclusions
2
Toposes and logic
Article
ΥΠ
The Connected Components Functor
1
Connectedness versus discreteness
2
The points functor parallel to the components functor
3
The
topos
of right actions of a monoid
Session
34
Group theory and the number of types of connected objects
Session
35
Appendices
Appendix I
Constants, codiscrete objects, and many connected objects
1
Constants and codiscrete objects
2
Monoids with at least two constants
Geometery of figures and algebra of functions
1
Functors
2
Geometry of figures and algebra of functions as categories
themselves
Appendix
Π
Adjoint functors with examples from graphs and dynamical systems
Appendix
Ш
The emergence of category theory within mathematics
Appendix IV Annotated Bibliography
323
324
327
328
329
331
335
335
336
339
339
342
344
348
348
352
358
358
359
360
362
366
366
367
368
369
369
370
372
378
381
Index
385
|
| any_adam_object | 1 |
| author | Lawvere, Francis W. 1937-2023 Schanuel, Stephen H. 1933- |
| author_GND | (DE-588)108079104 (DE-588)141823712 |
| author_facet | Lawvere, Francis W. 1937-2023 Schanuel, Stephen H. 1933- |
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| dewey-full | 511.3 512/.62 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics 512 - Algebra |
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| dewey-search | 511.3 512/.62 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
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| id | DE-604.BV035680572 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:43:16Z |
| institution | BVB |
| isbn | 9780521894852 9780521719162 |
| language | English |
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| spelling | Lawvere, Francis W. 1937-2023 Verfasser (DE-588)108079104 aut Conceptual mathematics a first introduction to categories F. William Lawvere ; Stephen H. Schanuel 2. ed. Cambridge [u.a.] Cambridge Univ. Press 2009 XII, 390 S. zahlr. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Categories (Mathematics) Kategorie Mathematik (DE-588)4129930-9 gnd rswk-swf Kategorie Mathematik (DE-588)4129930-9 s DE-604 Schanuel, Stephen H. 1933- Verfasser (DE-588)141823712 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017734842&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lawvere, Francis W. 1937-2023 Schanuel, Stephen H. 1933- Conceptual mathematics a first introduction to categories Categories (Mathematics) Kategorie Mathematik (DE-588)4129930-9 gnd |
| subject_GND | (DE-588)4129930-9 |
| title | Conceptual mathematics a first introduction to categories |
| title_auth | Conceptual mathematics a first introduction to categories |
| title_exact_search | Conceptual mathematics a first introduction to categories |
| title_full | Conceptual mathematics a first introduction to categories F. William Lawvere ; Stephen H. Schanuel |
| title_fullStr | Conceptual mathematics a first introduction to categories F. William Lawvere ; Stephen H. Schanuel |
| title_full_unstemmed | Conceptual mathematics a first introduction to categories F. William Lawvere ; Stephen H. Schanuel |
| title_short | Conceptual mathematics |
| title_sort | conceptual mathematics a first introduction to categories |
| title_sub | a first introduction to categories |
| topic | Categories (Mathematics) Kategorie Mathematik (DE-588)4129930-9 gnd |
| topic_facet | Categories (Mathematics) Kategorie Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017734842&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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