Algebraic theories : a categorical introduction to general algebra

269 Pages·2011·3.05 MB·English

by  Jiří Adámek

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CAMBRIDGE TRACTS IN MATHEMATICS
GeneralEditors
B. BOLLOBA´ S, W. FULTON, A. KATOK, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
184 AlgebraicTheories

Algebraic Theories
A Categorical Introduction to General Algebra
J. ADA´ MEK
TechnischeUniversita¨tCaroloWilhelminazuBraunschweig,
Germany
J. ROSICKY´
MasarykovaUniverzitavBrneˇ,CzechRepublic
E. M. VITALE
Universite´CatholiquedeLouvain,Belgium
WithaForewordbyF. W. LAWVERE
cambridge university press
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(cid:1)C J.Ada´mek,J.Rosicky´,andE.M.Vitale2011
Foreword(cid:1)C F.W.Lawvere2011
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Ada´mek,Jiˇr´ı,dr.
Algebraictheories:acategoricalintroductiontogeneralalgebra/J.Ada´mek,
J.Rosicky´,E.M.Vitale.
p. cm.–(Cambridgetractsinmathematics;184)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-11922-1(hardback)
1.Categories(Mathematics) 2.Algebraiclogic. I.Rosicky´,Jiˇr´ı. II.Vitale,E.M.
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Contents
Foreword pageix
F.W.Lawvere
Preface xv
PART I: ABSTRACT ALGEBRAIC CATEGORIES
0 Preliminaries 3
1 Algebraictheoriesandalgebraiccategories 10
2 Siftedandfilteredcolimits 21
3 Reflexivecoequalizers 30
4 Algebraiccategoriesasfreecompletions 38
5 Propertiesofalgebras 46
6 Acharacterizationofalgebraiccategories 54
7 Fromfilteredtosifted 65
8 Canonicaltheories 74
9 Algebraicfunctors 80
10 Birkhoff’svarietytheorem 89
PART II: CONCRETE ALGEBRAIC CATEGORIES
11 One-sortedalgebraiccategories 103
12 Algebrasforanendofunctor 117
vii
viii Contents
13 Equationalcategoriesof(cid:1)-algebras 127
14 S-sortedalgebraiccategories 139
PART III: SPECIAL TOPICS
15 Moritaequivalence 153
16 Freeexactcategories 163
17 Exactcompletionandreflexive-coequalizercompletion 182
18 Finitarylocalizationsofalgebraiccategories 195
Postscript 204
AppendixA Monads 207
AppendixB Abeliancategories 227
AppendixC Moreaboutdualitiesforone-sortedalgebraic
categories 232
References 241
Listofsymbols 245
Index 247