preamble The cuboid lemma and Maltsev categories Marino Gran and - PowerPoint PPT Presentation

preamble

The cuboid lemma and Mal’tsev categories Marino Gran and Diana Rodelo drodelo@ualg.pt Centre for Mathematics of the University of Coimbra University of Algarve, Portugal 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 1 / 18

Contents Preliminaries Contents Preliminaries Regular Mal’tsev categories Regular Mal’tsev categories The Cuboid Lemma The Cuboid Lemma The relative context The relative context 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 2 / 18

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context Preliminaries 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 3 / 18

� � �� �� � Regular categories · C regular cat = - lex Contents - · · Preliminaries Regular categories � Relations Equivalence relations � · · Regular Mal’tsev categories The Cuboid Lemma f 1 f The relative context � A - R f B ■ ■ f 2 ■ ■ ■ � � ∃ coeq 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 4 / 18

� � � �� �� � � Regular categories · C regular cat = - lex Contents - · · Preliminaries Regular categories � Relations Equivalence relations � · · Regular Mal’tsev categories The Cuboid Lemma f 1 f The relative context � A - R f B ■ ■ f 2 ■ ■ ■ � � ∃ coeq ∀ f · C regular ⇒ A B ❉ ❉ ③ � m p � � ❉ ③ P 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 4 / 18

� �� �� � � � � Regular categories · C regular cat = - lex Contents - · · Preliminaries Regular categories � Relations Equivalence relations � · · Regular Mal’tsev categories The Cuboid Lemma f 1 f The relative context � A - R f B ■ ■ f 2 ■ ■ ■ � � ∃ coeq ∀ f · C regular ⇒ A B ❉ ❉ ③ � m p � � ❉ ③ P · C regular 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 4 / 18

Relations � � r 1 ,r 2 � � A × B · relation R from A to B R Contents � Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

Relations � � r 1 ,r 2 � � A × B · relation R from A to B R Contents � Preliminaries � � r 2 ,r 1 � Regular categories � B × A R ◦ opposite from B to A R Relations � Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

Relations � � r 1 ,r 2 � � A × B · relation R from A to B R Contents � Preliminaries � � r 2 ,r 1 � Regular categories � B × A R ◦ opposite from B to A R Relations � Equivalence relations � f = � 1 A ,f � Regular Mal’tsev f � B � A × B map A A categories � � The Cuboid Lemma The relative context � f ◦ = � f, 1 A � � B × A A 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

Relations � � r 1 ,r 2 � � A × B · relation R from A to B R Contents � Preliminaries � � r 2 ,r 1 � Regular categories � B × A R ◦ opposite from B to A R Relations � Equivalence relations � f = � 1 A ,f � Regular Mal’tsev f � B � A × B map A A categories � � The Cuboid Lemma The relative context � f ◦ = � f, 1 A � � B × A A · R ֌ A × B , S ֌ B × C SR ֌ A × C � 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

Relations � � r 1 ,r 2 � � A × B · relation R from A to B R Contents � Preliminaries � � r 2 ,r 1 � Regular categories � B × A R ◦ opposite from B to A R Relations � Equivalence relations � f = � 1 A ,f � Regular Mal’tsev f � B � A × B map A A categories � � The Cuboid Lemma The relative context � f ◦ = � f, 1 A � � B × A A · R ֌ A × B , S ֌ B × C SR ֌ A × C � · R = r 2 r ◦ R f = f ◦ f ( = f 2 f ◦ and 1 ) 1 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

Relations � � r 1 ,r 2 � � A × B · relation R from A to B R Contents � Preliminaries � � r 2 ,r 1 � Regular categories � B × A R ◦ opposite from B to A R Relations � Equivalence relations � f = � 1 A ,f � Regular Mal’tsev f � B � A × B map A A categories � � The Cuboid Lemma The relative context � f ◦ = � f, 1 A � � B × A A · R ֌ A × B , S ֌ B × C SR ֌ A × C � · R = r 2 r ◦ R f = f ◦ f ( = f 2 f ◦ and 1 ) 1 f ◦ ff ◦ = f ◦ · ff ◦ f = f and ( difunctional ) 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

Relations � � r 1 ,r 2 � � A × B · relation R from A to B R Contents � Preliminaries � � r 2 ,r 1 � Regular categories � B × A R ◦ opposite from B to A R Relations � Equivalence relations � f = � 1 A ,f � Regular Mal’tsev f � B � A × B map A A categories � � The Cuboid Lemma The relative context � f ◦ = � f, 1 A � � B × A A · R ֌ A × B , S ֌ B × C SR ֌ A × C � · R = r 2 r ◦ R f = f ◦ f ( = f 2 f ◦ and 1 ) 1 f ◦ ff ◦ = f ◦ · ff ◦ f = f and ( difunctional ) · ff ◦ = 1 B iff f regular epi 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

Relations � � r 1 ,r 2 � � A × B · relation R from A to B R Contents � Preliminaries � � r 2 ,r 1 � Regular categories � B × A R ◦ opposite from B to A R Relations � Equivalence relations � f = � 1 A ,f � Regular Mal’tsev f � B � A × B map A A categories � � The Cuboid Lemma The relative context � f ◦ = � f, 1 A � � B × A A · R ֌ A × B , S ֌ B × C SR ֌ A × C � · R = r 2 r ◦ R f = f ◦ f ( = f 2 f ◦ and 1 ) 1 relative f ◦ ff ◦ = f ◦ · ff ◦ f = f and ( difunctional ) · ff ◦ = 1 B iff f regular epi 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

Equivalence relations · R ֌ A × A is: reflexive 1 A ≤ R Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

Equivalence relations · R ֌ A × A is: reflexive 1 A ≤ R Contents Preliminaries Regular categories R ◦ ≤ R symmetric Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

Equivalence relations · R ֌ A × A is: reflexive 1 A ≤ R Contents Preliminaries Regular categories R ◦ ≤ R symmetric Relations Equivalence relations Regular Mal’tsev categories transitive RR ≤ R The Cuboid Lemma The relative context 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

Equivalence relations  · R ֌ A × A is: reflexive 1 A ≤ R Contents Preliminaries   Regular categories R ◦ ≤ R symmetric  equivalence Relations  Equivalence relations   Regular Mal’tsev  categories transitive RR ≤ R The Cuboid Lemma The relative context 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

Equivalence relations  · R ֌ A × A is: reflexive 1 A ≤ R Contents Preliminaries   Regular categories R ◦ ≤ R symmetric  equivalence Relations  Equivalence relations   Regular Mal’tsev  categories transitive RR ≤ R The Cuboid Lemma The relative context · R f effective equivalence relation 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

Equivalence relations  · R ֌ A × A is: reflexive 1 A ≤ R Contents Preliminaries   Regular categories R ◦ ≤ R symmetric  equivalence Relations  Equivalence relations   Regular Mal’tsev  categories transitive RR ≤ R The Cuboid Lemma The relative context · R f effective equivalence relation f 1 � f � � B � A · exact fork R f f 2 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

� � � Equivalence relations  · R ֌ A × A is: reflexive 1 A ≤ R Contents Preliminaries   Regular categories R ◦ ≤ R symmetric  equivalence Relations  Equivalence relations   Regular Mal’tsev  categories transitive RR ≤ R The Cuboid Lemma The relative context · R f effective equivalence relation f 1 � f � � B � A · exact fork R f f 2 f 1 f � A · R f B ❇ ❇ f 2 ⑤ ❇ ⑤ � m p � � ❇ ⑤ P 9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18