Categories for the working mathematician:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2010
|
| Edition: | 2. ed. |
| Series: | Graduate texts in mathematics
5 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 314 S. graph. Darst. |
| ISBN: | 9781441931238 9781475747218 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804148644558929920 |
|---|---|
| adam_text | Contents Preface to the Second Edition .................................................................... v Preface to the First Edition .......................................................................... vii Introduction .................................................................................................... I. Categories, Functors, and Natural Transformations 1. 2, 3. 4. 5. 6. 7. 8. ............... 1 7 Axioms for Categories....................................................................... 7 Categories ............................................................................................. 10 Functors................................................................................................ 13 Natural Transformations .................................................................... 16 Monies, Epis, and Zeros........................................................................19 Foundations ......................................................................................... 21 Large Categories .................................................................................. 24 Horn-Sets................................................................................................ 27 II. Constructions on Categories................................................................ 31 1. 2. 3. 4. 5. 6. 7. 8. Duality................................................................................................... 31 Contravariance and Opposites............................................................ 33 Products of
Categories........................................................................... 36 Functor Categories .............................................................................. 40 The Category of All Categories ......................................................... 42 Comma Categories .............................................................................. 45 Graphs and Free Categories................................................................ 48 Quotient Categories.............................................................................. 51 III. Universals and Limits........................................................................... 55 1. 2. 3. 4. Universal Arrows.................................................................................. 55 The Yoneda Lemma ........................................................................... 59 Coproducts and Colimits .................................................................... 62 Products and Limits.............................................................................. 68 ix
Contents x 5. Categories with FiniteProducts .......................................................... 72 6. Groups in Categories ........................................................................ 75 7. Colimits of RepresentableFunctors ................................................... 76 IV. Adjoints 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. .................................................................................................... 79 Adjunctions........................................................................................ 79 Examples of Adjoints.......................................................................... 86 Reflective Subcategories ................................................................... 90 Equivalence of Categories ............................................................... 92 Adjoints for Preorders...................................................................... 95 Cartesian Closed Categories ........................................................... 97 Transformations of Adjoints ........................................................... 99 Composition of Adjoints........................................................... 103 Subsets and Characteristic Functions ................... .1................ 105 Categories Like Sets.................................................................... 106 V. Limits ................................................................................................ 109 Creation of Limits........................................................................... Limits by Products
and Equalizers.............................................. Limits with Parameters ................................................................ Preservation of Limits.................................................................... Adjoints on Limits ....................................................................... Freyd’s Adjoint Functor Theorem.............................................. Subobjects and Generators............................................................. The Special Adjoint Functor Theorem....................................... Adjoints in Topology.................................................................... 109 112 115 116 118 120 126 128 132 VI. Monads and Algebras.................................................................... 137 Monads in a Category.................................................................... Algebras for a Monad.................................................................... The Comparison with Algebras .................................................. Words and Free Semigroups......................................................... Free Algebras for a Monad ......................................................... Split Coequalizers........................................................................... Beck’s Theorem............................................................................... Algebras Are Г-Algebras ............................................................. Compact Hausdorff Spaces ......................................................... 137 139 142 144 147 149
151 156 157 VII. Monoids............................................................................................. 161 1. Monoidal Categories .................................................................... 2. Coherence ...................................................................................... 161 165 1. 2. 3. 4. 5. 6. 7. 8. 9. 1. 2. 3. 4. 5. 6. 7. 8. 9.
xi Contents Monoids......................................................................................... Actions............................................................................................. The Simplicial Category................................................................ Monads and Homology................................................................ Closed Categories........................................................................... Compactly Generated Spaces ...................................................... Loops and Suspensions ................................................................ 170 174 175 180 184 185 188 VIII. Abelian Categories................................................................... 191 Kernels and Cokernels.................................................................... Additive Categories....................................................................... Abelian Categories ....................................................................... Diagram Lemmas........................................................................... 191 194 198 202 3. 4. 5. 6. 7. 8. 9. 1. 2. 3. 4. IX. Special Limits 1. 2. 3. 4. 5. 6. 7. 8. .......................................................................... 211 Filtered Limits .............................................................................. Interchange of Limits.................................................................... Final Functors ............................................................................... Diagonal
Naturality....................................................................... Ends................................................................................................ Coends............................................................................................. Ends with Parameters.................................................................... Iterated Ends and Limits ............................................................. 211 214 217 218 222 226 228 230 X. Kan Extensions 1. 2. 3. 4. 5. 6. 7. ....................................................................... 233 Adjoints and Limits....................................................................... Weak Universality ....................................................................... The Kan Extension....................................................................... Kan Extensions as Coends............................................................. Pointwise Kan Extensions............................................................. Density............................................................................................. All Concepts Are Kan Extensions ............................................... 233 235 236 240 243 245 248 XI. Symmetry and Braiding in Monoidal Categories 1. 2. 3. 4. 5. 6. ................ 251 Symmetric Monoidal Categories .................................................. Monoidal Functors..................... Strict Monoidal Categories ......................................................... The Braid Groups B„ and the Braid
Category............................. Braided Coherence ........................................................................ Perspectives...................................................................................... 251 255 257 260 263 266
Contents xii XII. Structures in Categories.......................................................... 267 Internal Categories ....................................................................... The Nerve of a Category ............................................................. 2-Categories .................................................................................. Operations in 2-Categories............................................................. Single-Set Categories .................................................................... Bicategories..................................................................................... Examples of Bicategories ............................................................. Crossed Modules and Categories in Grp .................................... 267 270 272 276 279 281 283 285 ................................................................... 289 Table of Standard Categories: Objects and Arrows....................... 293 Table of Terminology ...................................................................... 295 Bibliography...................................................................................... 297 Index................................................................................................... 303 1. 2. 3. 4. 5. 6. 7. 8. Appendix. Foundations
|
| any_adam_object | 1 |
| author | Mac Lane, Saunders 1909-2005 |
| author_GND | (DE-588)118575953 |
| author_facet | Mac Lane, Saunders 1909-2005 |
| author_role | aut |
| author_sort | Mac Lane, Saunders 1909-2005 |
| author_variant | l s m ls lsm |
| building | Verbundindex |
| bvnumber | BV039751984 |
| classification_rvk | SK 320 |
| classification_tum | MAT 180f |
| ctrlnum | (OCoLC)772909196 (DE-599)BVBBV039751984 |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV039751984 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:10:25Z |
| institution | BVB |
| isbn | 9781441931238 9781475747218 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024599418 |
| oclc_num | 772909196 |
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| owner_facet | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-824 DE-739 DE-706 |
| physical | XII, 314 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Mac Lane, Saunders 1909-2005 Verfasser (DE-588)118575953 aut Categories for the working mathematician Saunders Mac Lane 2. ed. New York [u.a.] Springer 2010 XII, 314 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 5 Categories (Mathematics) Algebra (DE-588)4001156-2 gnd rswk-swf Kategorie Mathematik (DE-588)4129930-9 gnd rswk-swf Mathematiker (DE-588)4037945-0 gnd rswk-swf Kategorie Mathematik (DE-588)4129930-9 s Mathematiker (DE-588)4037945-0 s 1\p DE-604 Algebra (DE-588)4001156-2 s 2\p DE-604 Graduate texts in mathematics 5 (DE-604)BV000000067 5 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024599418&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Mac Lane, Saunders 1909-2005 Categories for the working mathematician Graduate texts in mathematics Categories (Mathematics) Algebra (DE-588)4001156-2 gnd Kategorie Mathematik (DE-588)4129930-9 gnd Mathematiker (DE-588)4037945-0 gnd |
| subject_GND | (DE-588)4001156-2 (DE-588)4129930-9 (DE-588)4037945-0 |
| title | Categories for the working mathematician |
| title_auth | Categories for the working mathematician |
| title_exact_search | Categories for the working mathematician |
| title_full | Categories for the working mathematician Saunders Mac Lane |
| title_fullStr | Categories for the working mathematician Saunders Mac Lane |
| title_full_unstemmed | Categories for the working mathematician Saunders Mac Lane |
| title_short | Categories for the working mathematician |
| title_sort | categories for the working mathematician |
| topic | Categories (Mathematics) Algebra (DE-588)4001156-2 gnd Kategorie Mathematik (DE-588)4129930-9 gnd Mathematiker (DE-588)4037945-0 gnd |
| topic_facet | Categories (Mathematics) Algebra Kategorie Mathematik Mathematiker |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024599418&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
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