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A tour through n -permutability Diana Rodelo drodelo@ualg.pt Centre for Mathematics of the University of Coimbra University of Algarve, Portugal M. Gran, Z. Janelidze, N. Martins-Ferreira, A. Ursini, T. Van der Linden CT2015 - June 17 A tour

  1. � � � � A first answer to (Q2) · A first answer to (Q2) concerning Goursat pushouts Contents Motivation 3 -permutability · Thm. [GR–2012] C regular category. TFAE: (Goursat) A first answer to (Q1) (i) C is a Goursat cat A first answer to (Q2) The 3 × 3 Lemma - 1 α The 3 × 3 Lemma - 2 � � (ii) every pushout (1) is A C Goursat varieties 2 -permutability a Goursat pushout g f s t (1) (Mal’tsev) Star-regular categories � � D B n -permutability β CT2015 - June 17 A tour through n -permutability – 8 / 30

  2. � � � � A first answer to (Q2) · A first answer to (Q2) concerning Goursat pushouts Contents Motivation 3 -permutability · Thm. [GR–2012] C regular category. TFAE: (Goursat) A first answer to (Q1) (i) C is a Goursat cat A first answer to (Q2) The 3 × 3 Lemma - 1 α The 3 × 3 Lemma - 2 � � (ii) every pushout (1) is A C Goursat varieties 2 -permutability a Goursat pushout g f s t (1) (Mal’tsev) � Star-regular categories � � D B n -permutability β CT2015 - June 17 A tour through n -permutability – 8 / 30

  3. � � � � � � � � A first answer to (Q2) · A first answer to (Q2) concerning Goursat pushouts Contents Motivation λ � � 3 -permutability · Thm. [GR–2012] C regular category. TFAE: Eq( f ) Eq( g ) (Goursat) A first answer to (Q1) (i) C is a Goursat cat A first answer to (Q2) The 3 × 3 Lemma - 1 α The 3 × 3 Lemma - 2 � � (ii) every pushout (1) is A C Goursat varieties 2 -permutability a Goursat pushout g f s t (1) (Mal’tsev) � Star-regular categories � � D B n -permutability β CT2015 - June 17 A tour through n -permutability – 8 / 30

  4. � � � � � � � � A first answer to (Q2) · A first answer to (Q2) concerning Goursat pushouts Contents Motivation λ � � 3 -permutability · Thm. [GR–2012] C regular category. TFAE: Eq( f ) Eq( g ) (Goursat) A first answer to (Q1) (i) C is a Goursat cat A first answer to (Q2) The 3 × 3 Lemma - 1 α The 3 × 3 Lemma - 2 � � (ii) every pushout (1) is A C Goursat varieties 2 -permutability a Goursat pushout g f s t (1) (Mal’tsev) � Star-regular categories � � D B n -permutability β · Related known facts: - [Bourn–2003] regular Mal’tsev cat ⇔ every (1) is a regular pushout � � B × D C � f, α � : A - [Carboni, Kelly, Pedicchio–1993] Goursat cat ⇔ regular image of an equivalence relation is an equiv relation CT2015 - June 17 A tour through n -permutability – 8 / 30

  5. The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability (Goursat) A first answer to (Q1) A first answer to (Q2) The 3 × 3 Lemma - 1 The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 9 / 30

  6. The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability · Classical 3 × 3 Lemma Denormalised 3 × 3 Lemma vs. (Goursat) A first answer to (Q1) short exact sequences exact forks A first answer to (Q2) The 3 × 3 Lemma - 1 f � � · � · � · � · � 0 0 Eq( f ) �� · The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 9 / 30

  7. � � � � The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability · Classical 3 × 3 Lemma Denormalised 3 × 3 Lemma vs. (Goursat) A first answer to (Q1) short exact sequences exact forks A first answer to (Q2) The 3 × 3 Lemma - 1 f � � · � · � · � · � 0 0 Eq( f ) �� · The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) · Star-regular categories n -permutability α � � A C g s f (1) t � � D B β CT2015 - June 17 A tour through n -permutability – 9 / 30

  8. � � �� � �� � � � The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability · Classical 3 × 3 Lemma Denormalised 3 × 3 Lemma vs. (Goursat) A first answer to (Q1) short exact sequences exact forks A first answer to (Q2) The 3 × 3 Lemma - 1 f � � · � · � · � · � 0 0 Eq( f ) �� · The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) · Star-regular categories n -permutability α Eq( α ) � � A C g ϕ s f (1) t � � D Eq( β ) B β CT2015 - June 17 A tour through n -permutability – 9 / 30

  9. � � � � � � � � �� � �� � � � The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability · Classical 3 × 3 Lemma Denormalised 3 × 3 Lemma vs. (Goursat) A first answer to (Q1) short exact sequences exact forks A first answer to (Q2) The 3 × 3 Lemma - 1 f � � · � · � · � · � 0 0 Eq( f ) �� · The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) λ � Eq( g ) · Eq( ϕ ) �� Eq( f ) Star-regular categories n -permutability α Eq( α ) � � A C g ϕ s f (1) t � � D Eq( β ) B β CT2015 - June 17 A tour through n -permutability – 9 / 30

  10. � � �� � � �� � � � � � � � � The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability · Classical 3 × 3 Lemma Denormalised 3 × 3 Lemma vs. (Goursat) A first answer to (Q1) short exact sequences exact forks A first answer to (Q2) The 3 × 3 Lemma - 1 f � � · � · � · � · � 0 0 Eq( f ) �� · The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) λ � Eq( g ) · Eq( ϕ ) �� Eq( f ) Star-regular categories n -permutability α Eq( α ) � � A C g ϕ s f (1) t � � D Eq( β ) B β 3 columns + 2 bottom rows exact forks CT2015 - June 17 A tour through n -permutability – 9 / 30

  11. � � �� � � �� � � � � � � � � The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability · Classical 3 × 3 Lemma Denormalised 3 × 3 Lemma vs. (Goursat) A first answer to (Q1) short exact sequences exact forks A first answer to (Q2) The 3 × 3 Lemma - 1 f � � · � · � · � · � 0 0 Eq( f ) �� · The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) λ � Eq( g ) · � � Eq( ϕ ) �� Eq( f ) Star-regular categories n -permutability α Eq( α ) � � A C g ϕ s f (1) t � � D Eq( β ) B β (1) ⇒ 3 columns + 2 bottom rows exact forks top row exact fork Goursat po CT2015 - June 17 A tour through n -permutability – 9 / 30

  12. �� � � �� � � � � � � � � � � The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability · Classical 3 × 3 Lemma Denormalised 3 × 3 Lemma vs. (Goursat) A first answer to (Q1) short exact sequences exact forks A first answer to (Q2) The 3 × 3 Lemma - 1 f � � · � · � · � · � 0 0 Eq( f ) �� · The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) λ � Eq( g ) · � � Eq( ϕ ) �� Eq( f ) Star-regular categories 3 columns + middle row exact forks n -permutability α Eq( α ) � � A C g ϕ s f (1) t � � D Eq( β ) B β (1) ⇒ 3 columns + 2 bottom rows exact forks top row exact fork Goursat po CT2015 - June 17 A tour through n -permutability – 9 / 30

  13. � � � � � �� � �� � � � � � � The 3 × 3 Lemma - 1 · From Goursat pushouts to the (denormalised) 3 × 3 Lemma Contents Motivation 3 -permutability · Classical 3 × 3 Lemma Denormalised 3 × 3 Lemma vs. (Goursat) A first answer to (Q1) short exact sequences exact forks A first answer to (Q2) The 3 × 3 Lemma - 1 f � � · � · � · � · � 0 0 Eq( f ) �� · The 3 × 3 Lemma - 2 Goursat varieties 2 -permutability (Mal’tsev) λ � Eq( g ) · � � Eq( ϕ ) �� Eq( f ) Star-regular categories 3 columns + middle row exact forks n -permutability top row exact fork α Eq( α ) � � A C ⇒ g ϕ s f (1) t bottom row exact fork � � D Eq( β ) B β (1) ⇒ 3 columns + 2 bottom rows exact forks top row exact fork Goursat po CT2015 - June 17 A tour through n -permutability – 9 / 30

  14. �� �� The 3 × 3 Lemma - 2 · The 3 × 3 Lemma Contents �� Eq( f ) λ � Eq( g ) Eq( ϕ ) Motivation 3 columns + middle row exact forks 3 -permutability (Goursat) A first answer to (Q1) � � � � � � A first answer to (Q2) α � � Eq( α ) A C The 3 × 3 Lemma - 1 The 3 × 3 Lemma - 2 g f �� ϕ �� Goursat varieties 2 -permutability �� B � � D (Mal’tsev) R β Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 10 / 30

  15. �� �� The 3 × 3 Lemma - 2 · The 3 × 3 Lemma Contents �� Eq( f ) λ � Eq( g ) Eq( ϕ ) Motivation 3 columns + middle row exact forks 3 -permutability (Goursat) A first answer to (Q1) � � � � � � top row exact fork A first answer to (Q2) α � � Eq( α ) A C The 3 × 3 Lemma - 1 The 3 × 3 Lemma - 2 ⇔ g f �� ϕ �� Goursat varieties 2 -permutability �� B bottom row exact fork � � D (Mal’tsev) R β Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 10 / 30

  16. �� �� The 3 × 3 Lemma - 2 · The 3 × 3 Lemma Contents �� Eq( f ) λ � Eq( g ) Eq( ϕ ) Motivation 3 columns + middle row exact forks 3 -permutability (Goursat) A first answer to (Q1) � � � � � � top row exact fork A first answer to (Q2) α � � Eq( α ) A C The 3 × 3 Lemma - 1 The 3 × 3 Lemma - 2 ⇔ g f �� ϕ �� Goursat varieties 2 -permutability �� B bottom row exact fork � � D (Mal’tsev) R β Star-regular categories n -permutability · Known results: - [Bourn–2003] C regular Mal’tsev cat ⇒ 3 × 3 Lemma holds - [Lack–2004] C Goursat cat ⇒ 3 × 3 Lemma holds CT2015 - June 17 A tour through n -permutability – 10 / 30

  17. �� �� The 3 × 3 Lemma - 2 · The 3 × 3 Lemma Contents �� Eq( f ) λ � Eq( g ) Eq( ϕ ) Motivation 3 columns + middle row exact forks 3 -permutability (Goursat) A first answer to (Q1) � � � � � � top row exact fork A first answer to (Q2) α � � Eq( α ) A C The 3 × 3 Lemma - 1 The 3 × 3 Lemma - 2 ⇔ g f �� ϕ �� Goursat varieties 2 -permutability �� B bottom row exact fork � � D (Mal’tsev) R β Star-regular categories n -permutability · Known results: - [Bourn–2003] C regular Mal’tsev cat ⇒ 3 × 3 Lemma holds - [Lack–2004] C Goursat cat ⇒ 3 × 3 Lemma holds · Thm. [GR–2012] C regular category. TFAE: (i) C is a Goursat cat (ii) 3 × 3 Lemma holds CT2015 - June 17 A tour through n -permutability – 10 / 30

  18. �� �� The 3 × 3 Lemma - 2 · The 3 × 3 Lemma Contents �� Eq( f ) λ � Eq( g ) Eq( ϕ ) Motivation 3 columns + middle row exact forks 3 -permutability (Goursat) A first answer to (Q1) � � � � � � top row exact fork A first answer to (Q2) α � � Eq( α ) A C The 3 × 3 Lemma - 1 The 3 × 3 Lemma - 2 ⇔ g f �� ϕ �� Goursat varieties 2 -permutability �� B bottom row exact fork � � D (Mal’tsev) R β Star-regular categories n -permutability · Known results: - [Bourn–2003] C regular Mal’tsev cat ⇒ 3 × 3 Lemma holds - [Lack–2004] C Goursat cat ⇒ 3 × 3 Lemma holds · Thm. [GR–2012] C regular category. TFAE: (i) C is a Goursat cat ⇒ ⇒ (ii) 3 × 3 Lemma holds ⇔ Upper / Lower 3 × 3 Lemma / CT2015 - June 17 A tour through n -permutability – 10 / 30

  19. Goursat varieties · [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff Contents the algebraic theory T of V contains two quaternary ops p and q sth Motivation 3 -permutability (Goursat) A first answer to (Q1)  A first answer to (Q2) p ( x, y, y, z ) = x The 3 × 3 Lemma - 1  The 3 × 3 Lemma - 2 p ( x, x, y, y ) = q ( x, x, y, y ) Goursat varieties  q ( x, y, y, z ) = z 2 -permutability (Mal’tsev) Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 11 / 30

  20. � � � Goursat varieties · [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff Contents the algebraic theory T of V contains two quaternary ops p and q sth Motivation 3 -permutability (Goursat) A first answer to (Q1)  A first answer to (Q2) p ( x, y, y, z ) = x The 3 × 3 Lemma - 1  The 3 × 3 Lemma - 2 p ( x, x, y, y ) = q ( x, x, y, y ) Goursat varieties  q ( x, y, y, z ) = z 2 -permutability (Mal’tsev) Star-regular categories n -permutability λ � � Eq( ∇ 3 ) · Eq( ∇ 2 + ∇ 2 ) � � � � 1 X + ∇ 2 +1 X � � 4 X 3 X Goursat pushout ∇ 2 + ∇ 2 � i 2 + i 1 i 2 ∇ 3 � i 2 (1) 2 X � � ( X free algebra on one element) X ∇ 2 CT2015 - June 17 A tour through n -permutability – 11 / 30

  21. � � � Goursat varieties · [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff Contents the algebraic theory T of V contains two quaternary ops p and q sth Motivation 3 -permutability (Goursat) A first answer to (Q1)  A first answer to (Q2) p ( x, y, y, z ) = x The 3 × 3 Lemma - 1  The 3 × 3 Lemma - 2 p ( x, x, y, y ) = q ( x, x, y, y ) Goursat varieties  q ( x, y, y, z ) = z 2 -permutability (Mal’tsev) Star-regular categories n -permutability λ � � Eq( ∇ 3 ) � p 1 ( x, y, z ) = x · Eq( ∇ 2 + ∇ 2 ) ∋ ( p 1 , p 3 ) p 3 ( x, y, z ) = z � � � � 1 X + ∇ 2 +1 X � � 4 X 3 X Goursat pushout ∇ 2 + ∇ 2 � i 2 + i 1 i 2 ∇ 3 � i 2 (1) 2 X � � ( X free algebra on one element) X ∇ 2 CT2015 - June 17 A tour through n -permutability – 11 / 30

  22. � � � Goursat varieties · [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff Contents the algebraic theory T of V contains two quaternary ops p and q sth Motivation 3 -permutability (Goursat) A first answer to (Q1)  A first answer to (Q2) p ( x, y, y, z ) = x The 3 × 3 Lemma - 1  The 3 × 3 Lemma - 2 p ( x, x, y, y ) = q ( x, x, y, y ) Goursat varieties  q ( x, y, y, z ) = z 2 -permutability (Mal’tsev) Star-regular categories n -permutability λ � � Eq( ∇ 3 ) � p 1 ( x, y, z ) = x · ( p, q ) ∈ Eq( ∇ 2 + ∇ 2 ) ∋ ( p 1 , p 3 ) p 3 ( x, y, z ) = z � � � � 1 X + ∇ 2 +1 X � � 4 X 3 X Goursat pushout ∇ 2 + ∇ 2 � i 2 + i 1 i 2 ∇ 3 � i 2 (1) 2 X � � ( X free algebra on one element) X ∇ 2 CT2015 - June 17 A tour through n -permutability – 11 / 30

  23. � � � Goursat varieties · [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff Contents the algebraic theory T of V contains two quaternary ops p and q sth Motivation 3 -permutability (Goursat) A first answer to (Q1)  A first answer to (Q2) p ( x, y, y, z ) = x The 3 × 3 Lemma - 1  The 3 × 3 Lemma - 2 � ( p, q ) ∈ Eq( ∇ 2 + ∇ 2 ) p ( x, x, y, y ) = q ( x, x, y, y ) Goursat varieties  q ( x, y, y, z ) = z 2 -permutability (Mal’tsev) Star-regular categories n -permutability λ � � Eq( ∇ 3 ) � p 1 ( x, y, z ) = x · ( p, q ) ∈ Eq( ∇ 2 + ∇ 2 ) ∋ ( p 1 , p 3 ) p 3 ( x, y, z ) = z � � � � 1 X + ∇ 2 +1 X � � 4 X 3 X Goursat pushout ∇ 2 + ∇ 2 � i 2 + i 1 i 2 ∇ 3 � i 2 (1) 2 X � � ( X free algebra on one element) X ∇ 2 CT2015 - June 17 A tour through n -permutability – 11 / 30

  24. � � � Goursat varieties · [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff Contents the algebraic theory T of V contains two quaternary ops p and q sth Motivation 3 -permutability λ ( p, q ) = ( p 1 , p 3 ) (Goursat) A first answer to (Q1)  A first answer to (Q2) ✛ p ( x, y, y, z ) = x The 3 × 3 Lemma - 1  The 3 × 3 Lemma - 2 � ( p, q ) ∈ Eq( ∇ 2 + ∇ 2 ) p ( x, x, y, y ) = q ( x, x, y, y ) Goursat varieties  q ( x, y, y, z ) = z 2 -permutability ✛ (Mal’tsev) Star-regular categories n -permutability λ � � Eq( ∇ 3 ) � p 1 ( x, y, z ) = x · ( p, q ) ∈ Eq( ∇ 2 + ∇ 2 ) ∋ ( p 1 , p 3 ) p 3 ( x, y, z ) = z � � � � 1 X + ∇ 2 +1 X � � 4 X 3 X Goursat pushout ∇ 2 + ∇ 2 � i 2 + i 1 i 2 ∇ 3 � i 2 (1) 2 X � � ( X free algebra on one element) X ∇ 2 CT2015 - June 17 A tour through n -permutability – 11 / 30

  25. Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma 2 -permutability (Mal’tsev) The relative context Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 12 / 30

  26. Stability property - 1 · From Goursat pushouts to a stability property for regular epis Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma The relative context Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 13 / 30

  27. � � � ✤ ✤ ✤ � � � � � ✤ ✤ � ✤ � � Stability property - 1 · From Goursat pushouts to a stability property for regular epis Contents Motivation 3 -permutability · [Bourn–2003] C regular Mal’tsev cat. In: (Goursat) 2 -permutability λ � � Z × D C (Mal’tsev) Y × B A λ is a regular epi Stability property - 1 ◗ ◗ ◗ ◗ a c ◗ ◗ ◗ ◗ Stability property - 2 ◗ ◗ ◗ ◗ α � � C The Cuboid Lemma A The relative context ζ Star-regular categories � � Y Z g ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ t ❘ ❘ ❘ n -permutability ❘ ❘ ❘ f � s ❘ ❘ ❘ ❘ d ❘ ❘ b � � D B β CT2015 - June 17 A tour through n -permutability – 13 / 30

  28. � � � � � ✤ ✤ � ✤ � � � ✤ ✤ � ✤ � Stability property - 1 · From Goursat pushouts to a stability property for regular epis Contents Motivation 3 -permutability · [Bourn–2003] C regular Mal’tsev cat. In: (Goursat) 2 -permutability λ � � Z × D C (Mal’tsev) Y × B A λ is a regular epi Stability property - 1 ◗ ◗ ◗ ◗ a c ◗ ◗ ◗ ◗ Stability property - 2 ◗ ◗ ◗ ◗ α � � C The Cuboid Lemma A The relative context ζ Star-regular categories � � Y Z g ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ t ❘ ❘ ❘ n -permutability ❘ ❘ ❘ f � s ❘ ❘ ❘ ❘ d ❘ ❘ b � � D B β · Rem. - a, b, c, d arbitrary maps α, β, ζ ( ⇒ λ ) regular epis and CT2015 - June 17 A tour through n -permutability – 13 / 30

  29. � � � � � ✤ ✤ � ✤ � � � ✤ ✤ � ✤ � Stability property - 1 · From Goursat pushouts to a stability property for regular epis Contents Motivation 3 -permutability · [Bourn–2003] C regular Mal’tsev cat. In: (Goursat) 2 -permutability λ � � Z × D C (Mal’tsev) Y × B A λ is a regular epi Stability property - 1 ◗ ◗ ◗ ◗ a c ◗ ◗ ◗ ◗ Stability property - 2 ◗ ◗ ◗ ◗ α � � C The Cuboid Lemma A The relative context ζ Star-regular categories � � Y Z g ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ t ❘ ❘ ❘ n -permutability ❘ ❘ ❘ f � s ❘ ❘ ❘ ❘ d ❘ ❘ b � � D B β · Rem. - a, b, c, d arbitrary maps α, β, ζ ( ⇒ λ ) regular epis and - front face is of type (1) CT2015 - June 17 A tour through n -permutability – 13 / 30

  30. � � ✤ ✤ � � ✤ � � ✤ � ✤ � � � � ✤ Stability property - 1 · From Goursat pushouts to a stability property for regular epis Contents Motivation 3 -permutability · [Bourn–2003] C regular Mal’tsev cat. In: (Goursat) 2 -permutability λ � � Z × D C (Mal’tsev) Y × B A λ is a regular epi Stability property - 1 ◗ ◗ ◗ ◗ a c ◗ ◗ ◗ ◗ Stability property - 2 ◗ ◗ ◗ ◗ α � � C The Cuboid Lemma A The relative context ζ Star-regular categories � � Y Z g ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ t ❘ ❘ ❘ n -permutability ❘ ❘ ❘ f � s ❘ ❘ ❘ ❘ d ❘ ❘ b � � D B β · Rem. - a, b, c, d arbitrary maps α, β, ζ ( ⇒ λ ) regular epis and - front face is of type (1) - b = f ( Y × B A = Eq( f )) and d = g ( Z × D C = Eq( g )) CT2015 - June 17 A tour through n -permutability – 13 / 30

  31. � � ✤ ✤ � � ✤ � � ✤ � ✤ � � � � ✤ Stability property - 1 · From Goursat pushouts to a stability property for regular epis Contents Motivation 3 -permutability · [Bourn–2003] C regular Mal’tsev cat. In: (Goursat) 2 -permutability λ � � Z × D C (Mal’tsev) Y × B A λ is a regular epi Stability property - 1 ◗ ◗ ◗ ◗ a c ◗ ◗ ◗ ◗ Stability property - 2 ◗ ◗ ◗ ◗ α � � C The Cuboid Lemma A The relative context ζ Star-regular categories � � Y Z g ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ t ❘ ❘ ❘ n -permutability ❘ ❘ ❘ f � s ❘ ❘ ❘ ❘ d ❘ ❘ b � � D B β · Rem. - a, b, c, d arbitrary maps α, β, ζ ( ⇒ λ ) regular epis and - front face is of type (1) - b = f ( Y × B A = Eq( f )) and d = g ( Z × D C = Eq( g )) � λ regular epi means that the front face is a Goursat po CT2015 - June 17 A tour through n -permutability – 13 / 30

  32. � � � � � ✤ ✤ � ✤ � � � ✤ ✤ � ✤ � Stability property - 1 · From Goursat pushouts to a stability property for regular epis Contents Motivation 3 -permutability · [Bourn–2003] C regular Mal’tsev cat. In: (Goursat) 2 -permutability λ � � Z × D C (Mal’tsev) Y × B A λ is a regular epi Stability property - 1 ◗ ◗ ◗ ◗ a c ◗ ◗ ◗ ◗ Stability property - 2 ◗ ◗ ◗ ◗ α � � C The Cuboid Lemma A The relative context ζ Star-regular categories � � Y Z g ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ t ❘ ❘ ❘ n -permutability ❘ ❘ ❘ f � s ❘ ❘ ❘ ❘ d ❘ ❘ b � � D B β · Rem. - a, b, c, d arbitrary maps α, β, ζ ( ⇒ λ ) regular epis and - front face is of type (1) - b = f ( Y × B A = Eq( f )) and d = g ( Z × D C = Eq( g )) � λ regular epi means that the front face is a Goursat po - stability property for regular epis ⇒ Goursat pushout property CT2015 - June 17 A tour through n -permutability – 13 / 30

  33. Stability property - 2 Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma The relative context · ⇒ Mal’tsev Goursat Star-regular categories ⇒ ⇔ n -permutability ⇒ stability property Goursat po property CT2015 - June 17 A tour through n -permutability – 14 / 30

  34. Stability property - 2 · Prop. [GR–2014] C regular category. TFAE: Contents Motivation (i) C is a Mal’tsev cat 3 -permutability (Goursat) (ii) stability property holds ( λ regular epi) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma The relative context · ⇒ Mal’tsev Goursat Star-regular categories ⇔ ⇔ n -permutability ⇒ stability property Goursat po property CT2015 - June 17 A tour through n -permutability – 14 / 30

  35. Stability property - 2 · Prop. [GR–2014] C regular category. TFAE: Contents Motivation (i) C is a Mal’tsev cat 3 -permutability (Goursat) (ii) stability property holds ( λ regular epi) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma The relative context · ⇒ ⇔ 3 × 3 Lemma Mal’tsev Goursat Star-regular categories ⇔ ⇔ n -permutability ⇒ stability property Goursat po property CT2015 - June 17 A tour through n -permutability – 14 / 30

  36. Stability property - 2 · Prop. [GR–2014] C regular category. TFAE: Contents Motivation (i) C is a Mal’tsev cat 3 -permutability (Goursat) (ii) stability property holds ( λ regular epi) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma ? ⇔ The relative context · ⇒ ⇔ 3 × 3 Lemma Mal’tsev Goursat Star-regular categories ⇔ ⇔ n -permutability ⇒ stability property Goursat po property CT2015 - June 17 A tour through n -permutability – 14 / 30

  37. Stability property - 2 · Prop. [GR–2014] C regular category. TFAE: Contents Motivation (i) C is a Mal’tsev cat 3 -permutability (Goursat) (ii) stability property holds ( λ regular epi) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma ? ⇔ The relative context · ⇒ ⇔ 3 × 3 Lemma Mal’tsev Goursat Star-regular categories ⇔ ⇔ n -permutability ⇒ stability property Goursat po property · Is there a homological diagram lemma which characterises Mal’tsev cats? CT2015 - June 17 A tour through n -permutability – 14 / 30

  38. Stability property - 2 · Prop. [GR–2014] C regular category. TFAE: Contents Motivation (i) C is a Mal’tsev cat 3 -permutability (Goursat) (ii) stability property holds ( λ regular epi) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma ? ⇔ The relative context · ⇒ ⇔ 3 × 3 Lemma Mal’tsev Goursat Star-regular categories ⇔ ⇔ n -permutability ⇒ stability property Goursat po property · Is there a homological diagram lemma which characterises Mal’tsev cats? · Goursat pushouts stability property wrt wrt � kernel pairs pullbacks CT2015 - June 17 A tour through n -permutability – 14 / 30

  39. Stability property - 2 · Prop. [GR–2014] C regular category. TFAE: Contents Motivation (i) C is a Mal’tsev cat 3 -permutability (Goursat) (ii) stability property holds ( λ regular epi) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma ? ⇔ The relative context · ⇒ ⇔ 3 × 3 Lemma Mal’tsev Goursat Star-regular categories ⇔ ⇔ n -permutability ⇒ stability property Goursat po property · Is there a homological diagram lemma which characterises Mal’tsev cats? · 3 × 3 Lemma Cuboid Lemma (3-dimensional diagram) wrt wrt � kernel pairs pullbacks CT2015 - June 17 A tour through n -permutability – 14 / 30

  40. �� �� The Cuboid Lemma · The (Upper) 3 × 3 Lemma Contents �� Eq( f ) λ � Eq( g ) Eq( ϕ ) Motivation 3 columns + middle row exact forks 3 -permutability (Goursat) � � � � � � top row exact fork 2 -permutability α � � Eq( α ) A C (Mal’tsev) ( ⇐ ) ⇔ Stability property - 1 g f �� ϕ �� Stability property - 2 �� B The Cuboid Lemma bottom row exact fork � � D R The relative context β Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 15 / 30

  41. �� �� � � � � � � � � � � � � � �� The Cuboid Lemma · The (Upper) 3 × 3 Lemma Contents �� Eq( f ) λ � Eq( g ) Eq( ϕ ) Motivation 3 columns + middle row exact forks 3 -permutability (Goursat) � � � � � � top row exact fork 2 -permutability α � � Eq( α ) A C (Mal’tsev) ( ⇐ ) ⇔ Stability property - 1 g f �� ϕ �� Stability property - 2 �� B The Cuboid Lemma bottom row exact fork � � D R The relative context β Star-regular categories n -permutability · The (Upper) Cuboid Lemma λ U V W ✸ ✲ ✳ ① ✁ � ��� ✸ ✲ ✳ ① ✸ ① ✲ ✳ ① ✸ ✲ ✳ � ① � ✁ ✸ ζ ✲ ✳ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ � � Eq( ζ ) Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ ✸ ✲ ✳ ✸ ✲ ✳ ✸ ✸ ✸ ✲ ✳ a c ✲ ✳ ✸ ✸ ✲ ✳ b d ✸ ✸ ✲ ✳ � � C ✸ Eq( α ) �� A ✸ α ✲ ✳ ✸ � ✂✂✂✂ � ϕ � ①①①①① f � g � ✸ � � ✲ ✳ ✸ � � � � � D S �� B β CT2015 - June 17 A tour through n -permutability – 15 / 30

  42. �� � � � � � � � � � � � � � The Cuboid Lemma 3 diamonds are pbs Contents 2 middle rows exact forks Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma The relative context Star-regular categories n -permutability · The (Upper) Cuboid Lemma λ U V W ✸ ✲ ✳ ① ✁ � ��� ✸ ✲ ✳ ① ✸ ① ✲ ✳ ① ✸ ✲ ✳ � ① � ✁ ✸ ζ ✲ ✳ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ � � Eq( ζ ) Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ ✸ ✲ ✳ ✸ ✲ ✳ ✸ ✸ ✸ ✲ ✳ a c ✲ ✳ ✸ ✸ ✲ ✳ b d ✸ ✸ ✲ ✳ � � C ✸ Eq( α ) �� A ✸ α ✲ ✳ ✸ � ✂✂✂✂ � ϕ � ①①①①① f � g � ✸ � � ✲ ✳ ✸ � � � � � D S �� B β CT2015 - June 17 A tour through n -permutability – 15 / 30

  43. � � � � � �� � � � � � � � � The Cuboid Lemma 3 diamonds are pbs Contents 2 middle rows exact forks Motivation 3 -permutability (Goursat) top row exact fork 2 -permutability (Mal’tsev) ⇐ Stability property - 1 Stability property - 2 The Cuboid Lemma bottom row exact fork The relative context Star-regular categories n -permutability · The (Upper) Cuboid Lemma λ U V W ✸ ✲ ✳ ① ✁ � ��� ✸ ✲ ✳ ① ✸ ① ✲ ✳ ① ✸ ✲ ✳ � ① � ✁ ✸ ζ ✲ ✳ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ � � Eq( ζ ) Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ ✸ ✲ ✳ ✸ ✲ ✳ ✸ ✸ ✸ ✲ ✳ a c ✲ ✳ ✸ ✸ ✲ ✳ b d ✸ ✸ ✲ ✳ � � C ✸ Eq( α ) �� A ✸ α ✲ ✳ ✸ � ✂✂✂✂ � ϕ � ①①①①① f � g � ✸ � � ✲ ✳ ✸ � � � � � D S �� B β CT2015 - June 17 A tour through n -permutability – 15 / 30

  44. � � � � � � � � � �� � � � � The Cuboid Lemma 3 diamonds are pbs Contents 2 middle rows exact forks Motivation 3 -permutability (Goursat) top row exact fork 2 -permutability (Mal’tsev) ⇐ Stability property - 1 Stability property - 2 The Cuboid Lemma bottom row exact fork The relative context Star-regular categories n -permutability · The (Upper) Cuboid Lemma stability pp for the right cube λ U V W ✸ ✲ ✳ ① ✁ � ��� ✸ ✲ ✳ ① ✸ ① ✲ ✳ ① ✸ ✲ ✳ � ① � ✁ ✸ ζ ✲ ✳ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ � � Eq( ζ ) Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ ✸ ✲ ✳ ✸ ✲ ✳ ✸ ✸ ✸ ✲ ✳ a c ✲ ✳ ✸ ✸ ✲ ✳ b d ✸ ✸ ✲ ✳ � � C ✸ Eq( α ) �� A ✸ α ✲ ✳ ✸ � ✂✂✂✂ � ϕ � ①①①①① f � g � ✸ � � ✲ ✳ ✸ � � � � � D S �� B β CT2015 - June 17 A tour through n -permutability – 15 / 30

  45. � � � � � � � � � � �� � � � The Cuboid Lemma · Thm. [GR–2014] C regular category. TFAE: Contents Motivation (i) C is a Mal’tsev cat 3 -permutability (Goursat) (ii) (Upper) Cuboid Lemma holds 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma The relative context Star-regular categories n -permutability · The (Upper) Cuboid Lemma stability pp for the right cube λ U V W ✸ ✲ ✳ ① ✁ � ��� ✸ ✲ ✳ ① ✸ ① ✲ ✳ ① ✸ ✲ ✳ � ① � ✁ ✸ ζ ✲ ✳ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ � � Eq( ζ ) Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✸ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ ✸ ✲ ✳ ✸ ✲ ✳ ✸ ✸ ✸ ✲ ✳ a c ✲ ✳ ✸ ✸ ✲ ✳ b d ✸ ✸ ✲ ✳ � � C ✸ Eq( α ) �� A ✸ α ✲ ✳ ✸ � ✂✂✂✂ � ϕ � ①①①①① f � g � ✸ � � ✲ ✳ ✸ � � � � � D S �� B β CT2015 - June 17 A tour through n -permutability – 15 / 30

  46. The relative context · absolute context relative context [T. Janelidze–2009] � Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) Stability property - 1 Stability property - 2 The Cuboid Lemma The relative context Star-regular categories n -permutability CT2015 - June 17 A tour through n -permutability – 16 / 30

  47. The relative context · absolute context relative context [T. Janelidze–2009] � Contents Motivation 3 -permutability (Goursat) · Regular 2 -permutability (Mal’tsev) Stability property - 1 Goursat cats Mal’tsev cats Stability property - 2 The Cuboid Lemma The relative context ( 3 -permutable: RSR = SRS ) ( 2 -permutable: RS = SR ) Star-regular categories n -permutability ⇔ ⇔ Goursat po pp stability pp ⇔ 3 × 3 ⇔ Lemma Cuboid Lemma CT2015 - June 17 A tour through n -permutability – 16 / 30

  48. The relative context · absolute context relative context [T. Janelidze–2009] � Contents Motivation × s + E class of regular epis sth ... 3 -permutability (Goursat) · relative Regular 2 -permutability (Mal’tsev) Stability property - 1 Goursat cats Mal’tsev cats Stability property - 2 The Cuboid Lemma The relative context ( 3 -permutable: RSR = SRS ) ( 2 -permutable: RS = SR ) Star-regular categories n -permutability ⇔ ⇔ Goursat po pp stability pp ⇔ 3 × 3 ⇔ Lemma Cuboid Lemma CT2015 - June 17 A tour through n -permutability – 16 / 30

  49. The relative context · absolute context relative context [T. Janelidze–2009] � Contents Motivation × s + E class of regular epis sth ... 3 -permutability (Goursat) · relative Regular 2 -permutability (Mal’tsev) Stability property - 1 Goursat cats Mal’tsev cats Stability property - 2 The Cuboid Lemma The relative context ( 3 -permutable: RSR = SRS ) ( 2 -permutable: RS = SR ) Star-regular categories n -permutability ⇔ ⇔ Goursat po pp stability pp ⇔ 3 × 3 ⇔ Lemma Cuboid Lemma relative version [ Goedecke, T. Janelidze –2012] CT2015 - June 17 A tour through n -permutability – 16 / 30

  50. The relative context · absolute context relative context [T. Janelidze–2009] � Contents Motivation × s + E class of regular epis sth ... 3 -permutability (Goursat) · relative Regular 2 -permutability (Mal’tsev) Stability property - 1 relative Goursat cats Mal’tsev cats Stability property - 2 The Cuboid Lemma The relative context ( 3 -permutable: RSR = SRS ) ( 2 -permutable: RS = SR ) Star-regular categories E -relations n -permutability ⇔ ⇔ Goursat po pp stability pp ⇔ 3 × 3 ⇔ Lemma Cuboid Lemma relative version [ Goedecke, T. Janelidze –2012] CT2015 - June 17 A tour through n -permutability – 16 / 30

  51. The relative context · absolute context relative context [T. Janelidze–2009] � Contents Motivation × s + E class of regular epis sth ... 3 -permutability (Goursat) · relative Regular 2 -permutability (Mal’tsev) [Everaert, Goedecke, T. Janelidze, VdL–2013] Stability property - 1 relative Goursat cats relative Mal’tsev cats Stability property - 2 The Cuboid Lemma The relative context ( 3 -permutable: RSR = SRS ) ( 2 -permutable: RS = SR ) Star-regular categories E -relations E -relations n -permutability ⇔ ⇔ Goursat po pp stability pp ⇔ 3 × 3 ⇔ Lemma Cuboid Lemma relative version [ Goedecke, T. Janelidze –2012] CT2015 - June 17 A tour through n -permutability – 16 / 30

  52. The relative context · absolute context relative context [T. Janelidze–2009] � Contents Motivation × s + E class of regular epis sth ... 3 -permutability (Goursat) · relative Regular 2 -permutability (Mal’tsev) [Everaert, Goedecke, T. Janelidze, VdL–2013] Stability property - 1 relative Goursat cats relative Mal’tsev cats Stability property - 2 The Cuboid Lemma The relative context ( 3 -permutable: RSR = SRS ) ( 2 -permutable: RS = SR ) Star-regular categories E -relations E -relations n -permutability ⇔ ⇔ Goursat po pp stability pp ⇔ 3 × 3 ⇔ Lemma Cuboid Lemma relative version relative version [ Goedecke, T. Janelidze –2012] [GR–2014] CT2015 - June 17 A tour through n -permutability – 16 / 30

  53. Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) Star-regular categories The context Star-exact sequences Star-regular categories The 3 × 3 Lemma for star-exact sequences The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 17 / 30

  54. The context · pointed & non-pointed ( 2 -permutability and 3 -permutability) Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) Star-regular categories The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 18 / 30

  55. The context · pointed & non-pointed ( 2 -permutability and 3 -permutability) Contents Motivation [Ehresmann–1964] · C lex, N ideal: f ∈ N or g ∈ N ⇒ gf ∈ N 3 -permutability (Goursat) [Lavendhomme–1965] 2 -permutability (Mal’tsev) Star-regular categories The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 18 / 30

  56. The context · pointed & non-pointed ( 2 -permutability and 3 -permutability) Contents Motivation [Ehresmann–1964] · C lex, N ideal: f ∈ N or g ∈ N ⇒ gf ∈ N 3 -permutability (Goursat) [Lavendhomme–1965] 2 -permutability (Mal’tsev) · Ex: N = all morphisms total context � Star-regular categories N = zero morphisms pointed context � The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 18 / 30

  57. The context · pointed & non-pointed ( 2 -permutability and 3 -permutability) Contents Motivation [Ehresmann–1964] · C lex, N ideal: f ∈ N or g ∈ N ⇒ gf ∈ N 3 -permutability (Goursat) [Lavendhomme–1965] 2 -permutability (Mal’tsev) · Ex: N = all morphisms total context � Star-regular categories N = zero morphisms pointed context � The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences n f � X f � Y · N -kernel: N f sth f n f ∈ N and universal The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 18 / 30

  58. The context · pointed & non-pointed ( 2 -permutability and 3 -permutability) Contents Motivation [Ehresmann–1964] · C lex, N ideal: f ∈ N or g ∈ N ⇒ gf ∈ N 3 -permutability (Goursat) [Lavendhomme–1965] 2 -permutability (Mal’tsev) · Ex: N = all morphisms total context � Star-regular categories N = zero morphisms pointed context � The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences n f � X f � Y · N -kernel: N f sth f n f ∈ N and universal The Star-Cuboid lemma n -permutability · star: σ = ( σ 1 , σ 2 ) : S ⇒ X sth σ 1 ∈ N CT2015 - June 17 A tour through n -permutability – 18 / 30

  59. The context · pointed & non-pointed ( 2 -permutability and 3 -permutability) Contents Motivation [Ehresmann–1964] · C lex, N ideal: f ∈ N or g ∈ N ⇒ gf ∈ N 3 -permutability (Goursat) [Lavendhomme–1965] 2 -permutability (Mal’tsev) · Ex: N = all morphisms total context � Star-regular categories N = zero morphisms pointed context � The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences n f � X f � Y · N -kernel: N f sth f n f ∈ N and universal The Star-Cuboid lemma n -permutability · star: σ = ( σ 1 , σ 2 ) : S ⇒ X sth σ 1 ∈ N · C multi-pointed cat w/ kernels = C w/ ideal N and ∃ N -kernels CT2015 - June 17 A tour through n -permutability – 18 / 30

  60. The context · pointed & non-pointed ( 2 -permutability and 3 -permutability) Contents Motivation [Ehresmann–1964] · C lex, N ideal: f ∈ N or g ∈ N ⇒ gf ∈ N 3 -permutability (Goursat) [Lavendhomme–1965] 2 -permutability (Mal’tsev) · Ex: N = all morphisms total context � Star-regular categories N = zero morphisms pointed context � The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences n f � X f � Y · N -kernel: N f sth f n f ∈ N and universal The Star-Cuboid lemma n -permutability · star: σ = ( σ 1 , σ 2 ) : S ⇒ X sth σ 1 ∈ N · C multi-pointed cat w/ kernels = C w/ ideal N and ∃ N -kernels · C star-regular cat = C regular + multi-pointed w/ kernels + ( regular epi = coequaliser of a star ) CT2015 - June 17 A tour through n -permutability – 18 / 30

  61. The context · pointed & non-pointed ( 2 -permutability and 3 -permutability) Contents Motivation [Ehresmann–1964] · C lex, N ideal: f ∈ N or g ∈ N ⇒ gf ∈ N 3 -permutability (Goursat) [Lavendhomme–1965] 2 -permutability (Mal’tsev) · Ex: N = all morphisms total context � Star-regular categories N = zero morphisms pointed context � The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences n f � X f � Y · N -kernel: N f sth f n f ∈ N and universal The Star-Cuboid lemma n -permutability · star: σ = ( σ 1 , σ 2 ) : S ⇒ X sth σ 1 ∈ N · C multi-pointed cat w/ kernels = C w/ ideal N and ∃ N -kernels [GJU–2012] · C star-regular cat = C regular + multi-pointed w/ kernels + ( regular epi = coequaliser of a star ) CT2015 - June 17 A tour through n -permutability – 18 / 30

  62. Star-exact sequences σ 1 � f � Y · star-kernel: σ 2 � X sth fσ 1 = fσ 2 and universal S Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) Star-regular categories The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 19 / 30

  63. Star-exact sequences σ 1 � f � Y · star-kernel: σ 2 � X sth fσ 1 = fσ 2 and universal S Contents ∼ = Motivation 3 -permutability Eq( f ) ∗ (Goursat) 2 -permutability (Mal’tsev) Star-regular categories The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 19 / 30

  64. Star-exact sequences σ 1 � f � Y · star-kernel: σ 2 � X sth fσ 1 = fσ 2 and universal S Contents ∼ = Motivation 3 -permutability f � � Y Eq( f ) ∗ · star-exact seq: (Goursat) �� X regular epi (= coeq of star) 2 -permutability (Mal’tsev) Star-regular categories The context Star-exact sequences The 3 × 3 Lemma for star-exact sequences The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 19 / 30

  65. Star-exact sequences σ 1 � f � Y · star-kernel: σ 2 � X sth fσ 1 = fσ 2 and universal S Contents ∼ = Motivation 3 -permutability f � � Y Eq( f ) ∗ · star-exact seq: (Goursat) �� X regular epi (= coeq of star) 2 -permutability (Mal’tsev) Star-regular categories The context · Total context ( N = all morphisms) Star-exact sequences The 3 × 3 Lemma for - star = pair of parallel morphisms ( S �� X ) star-exact sequences The Star-Cuboid lemma f � � Y ) - star-exact sequence = exact fork ( Eq( f ) �� X n -permutability - star-regular cat = regular cat ( regular epis = coequalisers of their kernel pairs ) CT2015 - June 17 A tour through n -permutability – 19 / 30

  66. Star-exact sequences σ 1 � f � Y · star-kernel: σ 2 � X sth fσ 1 = fσ 2 and universal S Contents ∼ = Motivation 3 -permutability f � � Y Eq( f ) ∗ · star-exact seq: (Goursat) �� X regular epi (= coeq of star) 2 -permutability (Mal’tsev) Star-regular categories The context · Total context ( N = all morphisms) Star-exact sequences The 3 × 3 Lemma for - star = pair of parallel morphisms ( S �� X ) star-exact sequences The Star-Cuboid lemma f � � Y ) - star-exact sequence = exact fork ( Eq( f ) �� X n -permutability - star-regular cat = regular cat ( regular epis = coequalisers of their kernel pairs ) · Pointed context ( N = zero morphisms) - star = morphism ( S � X ) f � � Y ) - star-exact sequence = short exact sequence ( K k � X - star-regular cat = normal cat ( = 0 + regular + (regular epis = normal epis) ) CT2015 - June 17 A tour through n -permutability – 19 / 30

  67. Star-exact sequences σ 1 � f � Y · star-kernel: σ 2 � X sth fσ 1 = fσ 2 and universal S Contents ∼ = Motivation 3 -permutability f � � Y Eq( f ) ∗ · star-exact seq: (Goursat) �� X regular epi (= coeq of star) 2 -permutability (Mal’tsev) Star-regular categories The context · Total context ( N = all morphisms) Star-exact sequences The 3 × 3 Lemma for - star = pair of parallel morphisms ( S �� X ) star-exact sequences The Star-Cuboid lemma f � � Y ) - star-exact sequence = exact fork ( Eq( f ) exact fork �� X n -permutability - star-regular cat = regular cat ( regular epis = coequalisers of their kernel pairs ) · Pointed context ( N = zero morphisms) - star = morphism ( S � X ) f � � Y ) - star-exact sequence = short exact sequence ( K short exact sequence k � X - star-regular cat = normal cat ( = 0 + regular + (regular epis = normal epis) ) CT2015 - June 17 A tour through n -permutability – 19 / 30

  68. �� �� The 3 × 3 Lemma for star-exact sequences Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) �� Eq( f ) ∗ � Eq( g ) ∗ · Eq( ϕ ) ∗ Star-regular categories 3 cols + middle row star-exact seq The context Star-exact sequences The 3 × 3 Lemma for � � � � � � star-exact sequences α Eq( α ) ∗ � � A C The Star-Cuboid lemma n -permutability g f �� ϕ �� �� B � � D R β CT2015 - June 17 A tour through n -permutability – 20 / 30

  69. �� �� The 3 × 3 Lemma for star-exact sequences Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) �� Eq( f ) ∗ � Eq( g ) ∗ · Eq( ϕ ) ∗ Star-regular categories 3 cols + middle row star-exact seq The context Star-exact sequences The 3 × 3 Lemma for � � � � � � star-exact sequences top row star-exact seq α Eq( α ) ∗ � � A C The Star-Cuboid lemma ⇔ n -permutability g f �� ϕ �� bottom row star-exact seq �� B � � D R β CT2015 - June 17 A tour through n -permutability – 20 / 30

  70. �� �� The 3 × 3 Lemma for star-exact sequences Contents Motivation 3 -permutability (Goursat) 2 -permutability (Mal’tsev) �� Eq( f ) ∗ � Eq( g ) ∗ · Eq( ϕ ) ∗ Star-regular categories 3 cols + middle row star-exact seq The context Star-exact sequences The 3 × 3 Lemma for � � � � � � star-exact sequences top row star-exact seq α Eq( α ) ∗ � � A C The Star-Cuboid lemma ⇔ n -permutability g f �� ϕ �� bottom row star-exact seq �� B � � D R β Shape of denormalised 3 × 3 Lemma, but captures both classical and denormalised 3 × 3 Lemmas CT2015 - June 17 A tour through n -permutability – 20 / 30

  71. �� �� The 3 × 3 Lemma for star-exact sequences · Thm. [GJR–2012] C star-regular category + · · · . TFAE: Contents (i) C has symmetric saturation pp Motivation 3 -permutability (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds (Goursat) 2 -permutability (Mal’tsev) �� Eq( f ) ∗ � Eq( g ) ∗ · Eq( ϕ ) ∗ Star-regular categories 3 cols + middle row star-exact seq The context Star-exact sequences The 3 × 3 Lemma for � � � � � � star-exact sequences top row star-exact seq α Eq( α ) ∗ � � A C The Star-Cuboid lemma ⇔ n -permutability g f �� ϕ �� bottom row star-exact seq �� B � � D R β Shape of denormalised 3 × 3 Lemma, but captures both classical and denormalised 3 × 3 Lemmas CT2015 - June 17 A tour through n -permutability – 20 / 30

  72. �� �� The 3 × 3 Lemma for star-exact sequences · Thm. [GJR–2012] C star-regular category + · · · . TFAE: Contents (i) C has symmetric saturation pp ( ⇐ 3 -star-permutability [GJRU–2012]) Motivation 3 -permutability (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds (Goursat) 2 -permutability (Mal’tsev) �� Eq( f ) ∗ � Eq( g ) ∗ · Eq( ϕ ) ∗ Star-regular categories 3 cols + middle row star-exact seq The context Star-exact sequences The 3 × 3 Lemma for � � � � � � star-exact sequences top row star-exact seq α Eq( α ) ∗ � � A C The Star-Cuboid lemma ⇔ n -permutability g f �� ϕ �� bottom row star-exact seq �� B � � D R β Shape of denormalised 3 × 3 Lemma, but captures both classical and denormalised 3 × 3 Lemmas CT2015 - June 17 A tour through n -permutability – 20 / 30

  73. �� �� The 3 × 3 Lemma for star-exact sequences · Thm. [GJR–2012] C star-regular category + · · · . TFAE: Contents (i) C has symmetric saturation pp ( ⇐ 3 -star-permutability [GJRU–2012]) Motivation 3 -permutability (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds (Goursat) 2 -permutability (Mal’tsev) �� Eq( f ) � Eq( g ) Star-regular categories · Eq( ϕ ) 3 cols + middle row star-exact seq The context Star-exact sequences The 3 × 3 Lemma for � � � � � � star-exact sequences top row star-exact seq α � � Eq( α ) A C The Star-Cuboid lemma ⇔ n -permutability g f �� ϕ �� bottom row star-exact seq �� B � � D R β · Total context ( N = all morphisms) [GR–2012] � - star-exact sequence = exact fork - 3 × 3 Lemma for star-exact sequences = denormalised 3 × 3 Lemma - 3 -star-permutable categories = Goursat categories CT2015 - June 17 A tour through n -permutability – 20 / 30

  74. � � � � � � � �� The 3 × 3 Lemma for star-exact sequences · Thm. [GJR–2012] C star-regular category + · · · . TFAE: Contents (i) C has symmetric saturation pp ( ⇐ 3 -star-permutability [GJRU–2012]) Motivation 3 -permutability (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds (Goursat) 2 -permutability (Mal’tsev) � K f � K g Star-regular categories · 3 cols + middle row star-exact seq K ϕ The context ❴ ❴ ❴ Star-exact sequences The 3 × 3 Lemma for star-exact sequences top row star-exact seq K α ✤ � α The Star-Cuboid lemma � � A C ⇔ n -permutability g ϕ �� f �� bottom row star-exact seq � B � � D R β · Pointed context ( N = zero morphisms) [J–2010] � - star-exact sequence = short exact sequence - 3 × 3 Lemma for star-exact seqs = classical 3 × 3 Lemma - 3 -star-permutable cats = regular subtractive cats CT2015 - June 17 A tour through n -permutability – 20 / 30

  75. � � � � � � � �� The 3 × 3 Lemma for star-exact sequences · Thm. [GJR–2012] C star-regular category + · · · . TFAE: Contents (i) C has symmetric saturation pp ( ⇐ 3 -star-permutability [GJRU–2012]) Motivation 3 -permutability (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds (Goursat) 2 -permutability (Mal’tsev) � K f � K g Star-regular categories · 3 cols + middle row star-exact seq K ϕ The context ❴ ❴ ❴ Star-exact sequences The 3 × 3 Lemma for star-exact sequences top row star-exact seq K α ✤ � α The Star-Cuboid lemma � � A C ⇔ n -permutability g ϕ �� f �� bottom row star-exact seq � B � � D R β · Pointed context ( N = zero morphisms) [J–2010] � - star-exact sequence = short exact sequence - 3 × 3 Lemma for star-exact seqs = classical 3 × 3 Lemma - 3 -star-permutable cats = regular subtractive cats ([J–2005], [U–1994]) CT2015 - June 17 A tour through n -permutability – 20 / 30

  76. � �� � � � � � � � � � � � � The Star-Cuboid lemma λ U ∗ V ∗ W ∗ · ✹ ✲ ✳ Contents � ✇✇✇✇✇ ✂ � ✁✁✁ ✹ ✲ ✳ ✹ ✲ ✳ ✹ Motivation ✲ ✳ � ✂ ✹ ✲ ζ ✳ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Eq( ζ ) ∗ 3 -permutability � � Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ (Goursat) ✹ ✲ ✳ ✹ ✲ ✳ ✹ ✹ ✹ ✲ ✳ a c 2 -permutability ✲ ✳ ✹ ✹ ✲ ✳ b d ✹ (Mal’tsev) ✹ ✲ ✳ � � C Eq( α ) ∗ ✹ �� A Star-regular categories ✹ α ✲ ✳ ✹ � ✂✂✂✂ � The context ϕ � ✈✈✈✈✈ g � ✹ f � � � ✲ ✳ ✹ Star-exact sequences � � � The 3 × 3 Lemma for � � D S �� B star-exact sequences β The Star-Cuboid lemma n -permutability CT2015 - June 17 A tour through n -permutability – 21 / 30

  77. � �� � � � � � � � � � � � � The Star-Cuboid lemma λ U ∗ V ∗ W ∗ · ✹ ✲ ✳ Contents � ✇✇✇✇✇ ✂ � ✁✁✁ ✹ ✲ ✳ ✹ ✲ ✳ ✹ Motivation ✲ ✳ � ✂ ✹ ✲ ζ ✳ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Eq( ζ ) ∗ 3 -permutability � � Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ (Goursat) ✹ ✲ ✳ ✹ ✲ ✳ ✹ ✹ ✹ ✲ ✳ a c 2 -permutability ✲ ✳ ✹ ✹ ✲ ✳ b d ✹ (Mal’tsev) ✹ ✲ ✳ � � C Eq( α ) ∗ ✹ �� A Star-regular categories ✹ α ✲ ✳ ✹ � ✂✂✂✂ � The context ϕ � ✈✈✈✈✈ g � ✹ f � � � ✲ ✳ ✹ Star-exact sequences � � � The 3 × 3 Lemma for � � D S �� B star-exact sequences β The Star-Cuboid lemma bottom row star-exact sequence ⇒ top row star-exact sequence n -permutability CT2015 - June 17 A tour through n -permutability – 21 / 30

  78. � �� � � � � � � � � � � � � The Star-Cuboid lemma λ U ∗ V ∗ W ∗ · ✹ ✲ ✳ Contents � ✇✇✇✇✇ ✂ � ✁✁✁ ✹ ✲ ✳ ✹ ✲ ✳ ✹ Motivation ✲ ✳ � ✂ ✹ ✲ ζ ✳ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Eq( ζ ) ∗ 3 -permutability � � Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ (Goursat) ✹ ✲ ✳ ✹ ✲ ✳ ✹ ✹ ✹ ✲ ✳ a c 2 -permutability ✲ ✳ ✹ ✹ ✲ ✳ b d ✹ (Mal’tsev) ✹ ✲ ✳ � � C Eq( α ) ∗ ✹ �� A Star-regular categories ✹ α ✲ ✳ ✹ � ✂✂✂✂ � The context ϕ � ✈✈✈✈✈ g � ✹ f � � � ✲ ✳ ✹ Star-exact sequences � � � The 3 × 3 Lemma for � � D S �� B star-exact sequences β The Star-Cuboid lemma bottom row star-exact sequence ⇒ top row star-exact sequence n -permutability Shape of the Cuboid Lemma, but captures both the Cuboid Lemma and (again) the classical 3 × 3 Lemma CT2015 - June 17 A tour through n -permutability – 21 / 30

  79. � � � � � � � � � � � � � �� The Star-Cuboid lemma λ U ∗ V ∗ W ∗ · ✹ ✲ ✳ Contents � ✇✇✇✇✇ ✂ � ✁✁✁ ✹ ✲ ✳ ✹ ✲ ✳ ✹ Motivation ✲ ✳ � ✂ ✹ ✲ ζ ✳ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Eq( ζ ) ∗ 3 -permutability � � Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ (Goursat) ✹ ✲ ✳ ✹ ✲ ✳ ✹ ✹ ✹ ✲ ✳ a c 2 -permutability ✲ ✳ ✹ ✹ ✲ ✳ b d ✹ (Mal’tsev) ✹ ✲ ✳ � � C Eq( α ) ∗ ✹ �� A Star-regular categories ✹ α ✲ ✳ ✹ � ✂✂✂✂ � The context ϕ � ✈✈✈✈✈ g � ✹ f � � � ✲ ✳ ✹ Star-exact sequences � � � The 3 × 3 Lemma for � � D S �� B star-exact sequences β The Star-Cuboid lemma bottom row star-exact sequence ⇒ top row star-exact sequence n -permutability · Thm. [GR–2014] C star-regular category + · · · . TFAE: (i) C is a 2 -star-permutable cat (ii) Star-Upper Cuboid Lemma holds Shape of the Cuboid Lemma, but captures both the Cuboid Lemma and (again) the classical 3 × 3 Lemma CT2015 - June 17 A tour through n -permutability – 21 / 30

  80. � � � � � � � � � � � � � �� The Star-Cuboid lemma λ U ∗ V ∗ W ∗ · ✹ ✲ ✳ Contents � ✇✇✇✇✇ ✂ � ✁✁✁ ✹ ✲ ✳ ✹ ✲ ✳ ✹ Motivation ✲ ✳ � ✂ ✹ ✲ ζ ✳ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Eq( ζ ) ∗ 3 -permutability � � Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ (Goursat) ✹ ✲ ✳ ✹ ✲ ✳ ✹ ✹ ✹ ✲ ✳ a c 2 -permutability ✲ ✳ ✹ ✹ ✲ ✳ b d ✹ (Mal’tsev) ✹ ✲ ✳ � � C Eq( α ) ∗ ✹ �� A Star-regular categories ✹ α ✲ ✳ ✹ � ✂✂✂✂ � The context ϕ � ✈✈✈✈✈ g � ✹ f � � � ✲ ✳ ✹ Star-exact sequences � � � The 3 × 3 Lemma for � � D S �� B star-exact sequences β The Star-Cuboid lemma bottom row star-exact sequence ⇒ top row star-exact sequence n -permutability · Thm. [GR–2014] C star-regular category + · · · . TFAE: (i) C is a 2 -star-permutable cat [GJRU–2012] (ii) Star-Upper Cuboid Lemma holds Shape of the Cuboid Lemma, but captures both the Cuboid Lemma and (again) the classical 3 × 3 Lemma CT2015 - June 17 A tour through n -permutability – 21 / 30

  81. � � � � � � � � � � � � � �� The Star-Cuboid lemma λ U ∗ V ∗ W ∗ · ✹ ✲ ✳ Contents � ✇✇✇✇✇ ✂ � ✁✁✁ ✹ ✲ ✳ ✹ ✲ ✳ ✹ Motivation ✲ ✳ � ✂ ✹ ✲ ζ ✳ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Eq( ζ ) ∗ 3 -permutability � � Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ (Goursat) ✹ ✲ ✳ ✹ ✲ ✳ ✹ ✹ ✹ ✲ ✳ a c 2 -permutability ✲ ✳ ✹ ✹ ✲ ✳ b d ✹ (Mal’tsev) ✹ ✲ ✳ � � C Eq( α ) ∗ ✹ �� A Star-regular categories ✹ α ✲ ✳ ✹ � ✂✂✂✂ � The context ϕ � ✈✈✈✈✈ g � ✹ f � � � ✲ ✳ ✹ Star-exact sequences � � � The 3 × 3 Lemma for � � D S �� B star-exact sequences β The Star-Cuboid lemma bottom row star-exact sequence ⇒ top row star-exact sequence n -permutability · Thm. [GR–2014] C star-regular category + · · · . TFAE: (i) C is a 2 -star-permutable cat [GJRU–2012] (ii) Star-Upper Cuboid Lemma holds · Total cnt: C regular. C Mal’tsev iff (Upper) Cuboid Lemma holds CT2015 - June 17 A tour through n -permutability – 21 / 30

  82. � � � � � � � � � � � � � �� The Star-Cuboid lemma λ U ∗ V ∗ W ∗ · ✹ ✲ ✳ Contents � ✇✇✇✇✇ ✂ � ✁✁✁ ✹ ✲ ✳ ✹ ✲ ✳ ✹ Motivation ✲ ✳ � ✂ ✹ ✲ ζ ✳ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Eq( ζ ) ∗ 3 -permutability � � Y ✲ Z ✳ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✹ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✲ ✳ (Goursat) ✹ ✲ ✳ ✹ ✲ ✳ ✹ ✹ ✹ ✲ ✳ a c 2 -permutability ✲ ✳ ✹ ✹ ✲ ✳ b d ✹ (Mal’tsev) ✹ ✲ ✳ � � C Eq( α ) ∗ ✹ �� A Star-regular categories ✹ α ✲ ✳ ✹ � ✂✂✂✂ � The context ϕ � ✈✈✈✈✈ g � ✹ f � � � ✲ ✳ ✹ Star-exact sequences � � � The 3 × 3 Lemma for � � D S �� B star-exact sequences β The Star-Cuboid lemma bottom row star-exact sequence ⇒ top row star-exact sequence n -permutability · Thm. [GR–2014] C star-regular category + · · · . TFAE: (i) C is a 2 -star-permutable cat [GJRU–2012] (ii) Star-Upper Cuboid Lemma holds · Total cnt: C regular. C Mal’tsev iff (Upper) Cuboid Lemma holds Pointed cnt: C normal. C subtractive (Upper) Classical 3 × 3 L. iff CT2015 - June 17 A tour through n -permutability – 21 / 30

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