A tour through n -permutability Diana Rodelo drodelo@ualg.pt - - PowerPoint PPT Presentation

CT2015 - June 17 A tour through n-permutability – 1 / 30

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

Contents

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 2 / 30

Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

Motivation

Contents Motivation Mal’tsev vs. Goursat categories Examples 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 3 / 30

Mal’tsev vs. Goursat categories

Contents Motivation Mal’tsev vs. Goursat categories Examples 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 4 / 30

· n-permutability, n 2: RSR · · ·

• = SRS · · ·
• , ∀R, S on same obj

n n

Mal’tsev vs. Goursat categories

Contents Motivation Mal’tsev vs. Goursat categories Examples 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 4 / 30

· n-permutability, n 2: RSR · · ·

• = SRS · · ·
• , ∀R, S on same obj

n n · 2-permutable (=Mal’tsev):

• n = 2

RS = SR (regular)

• easiest case
• widely studied “popular”
• reflexive = equivalence (non-regular)

Mal’tsev vs. Goursat categories

Contents Motivation Mal’tsev vs. Goursat categories Examples 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 4 / 30

· n-permutability, n 2: RSR · · ·

• = SRS · · ·
• , ∀R, S on same obj

n n · 2-permutable (=Mal’tsev):

• n = 2

RS = SR (regular)

• easiest case
• widely studied “popular”
• reflexive = equivalence (non-regular)

❄ weaker · 3-permutable (=Goursat):

• n = 3

RSR = SRS (regular)

• not so easy case
• less studied less “popular”

Mal’tsev vs. Goursat categories

Contents Motivation Mal’tsev vs. Goursat categories Examples 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 4 / 30

· n-permutability, n 2: RSR · · ·

• = SRS · · ·
• , ∀R, S on same obj

n n · 2-permutable (=Mal’tsev):

• n = 2

RS = SR (regular)

• easiest case
• widely studied “popular”
• reflexive = equivalence (non-regular)

❄ weaker · 3-permutable (=Goursat):

• n = 3

RSR = SRS (regular)

• not so easy case
• less studied less “popular”

· Natural questions: (Q1) Which pps known for Mal’tsev cats still hold for Goursat cats? (Q2) Which are the typical properties of Goursat categories?

Examples

Contents Motivation Mal’tsev vs. Goursat categories Examples 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 5 / 30

· Mal’tsev cats:

• Gp, Alg(T),

T w/ group op.

• quasi-groups, Heyting algebras
• lex + additive
• (Topos)op

Examples

Contents Motivation Mal’tsev vs. Goursat categories Examples 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 5 / 30

· Mal’tsev cats:

• Gp, Alg(T),

T w/ group op.

• quasi-groups, Heyting algebras
• lex + additive
• (Topos)op

· Goursat cats:

• Malt’sev cats
• implication algebras (non Mal’tsev)
• right-complemented semigroups (non Mal’tsev)

Examples

Contents Motivation Mal’tsev vs. Goursat categories Examples 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability

CT2015 - June 17 A tour through n-permutability – 5 / 30

· Mal’tsev cats:

• Gp, Alg(T),

T w/ group op.

• quasi-groups, Heyting algebras
• lex + additive
• (Topos)op

· Goursat cats:

• Malt’sev cats
• implication algebras (non Mal’tsev)
• right-complemented semigroups (non Mal’tsev)

· n-permutable cats:

• (n − 1)-permutable cats
• ∃ examples of non (n − 1)-permutable cats

(n 3)

3-permutability (Goursat)

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 – 6 / 30

A first answer to (Q1)

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 – 7 / 30

· A first answer to (Q1) concerning reflectiveness of internal structures

A first answer to (Q1)

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 – 7 / 30

· A first answer to (Q1) concerning reflectiveness of internal structures · [Bourn–2004] C finitely cocomplete regular Mal’tsev category ⇒ Gpd(C) reflective subcategory of Rg(C)

A first answer to (Q1)

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 – 7 / 30

· A first answer to (Q1) concerning reflectiveness of internal structures · [Bourn–2004] C finitely cocomplete regular Mal’tsev category ⇒ Gpd(C) reflective subcategory of Rg(C)

• consequence of good properties of commutators
• Rg:

X1

d

• c

X0

e

• Gpd:

X1/[Eq(d), Eq(c)]

X0

A first answer to (Q1)

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 – 7 / 30

· A first answer to (Q1) concerning reflectiveness of internal structures · [Bourn–2004] C finitely cocomplete regular Mal’tsev category ⇒ Gpd(C) reflective subcategory of Rg(C)

• consequence of good properties of commutators
• Rg:

X1

d

• c

X0

e

• Gpd:

X1/[Eq(d), Eq(c)]

X0

• · Generalisation to the Goursat case:
• regular quotients won’t work ( ⇒ reflexive relation = equiv relation)
• different construction inspired by [Bourn–2003]

A first answer to (Q1)

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 – 7 / 30

· A first answer to (Q1) concerning reflectiveness of internal structures · [Bourn–2004] C finitely cocomplete regular Mal’tsev category ⇒ Gpd(C) reflective subcategory of Rg(C)

• consequence of good properties of commutators
• Rg:

X1

d

• c

X0

e

• Gpd:

X1/[Eq(d), Eq(c)]

X0

• · Generalisation to the Goursat case:
• regular quotients won’t work ( ⇒ reflexive relation = equiv relation)
• different construction inspired by [Bourn–2003]

Mal’tsev

A first answer to (Q1)

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 – 7 / 30

· A first answer to (Q1) concerning reflectiveness of internal structures · [Bourn–2004] C finitely cocomplete regular Mal’tsev category ⇒ Gpd(C) reflective subcategory of Rg(C)

• consequence of good properties of commutators
• Rg:

X1

d

• c

X0

e

• Gpd:

X1/[Eq(d), Eq(c)]

X0

• · Generalisation to the Goursat case:
• regular quotients won’t work ( ⇒ reflexive relation = equiv relation)
• different construction inspired by [Bourn–2003]

Mal’tsev · Prop. [GR–2008] C Goursat category with coequalisers ⇒

• Gpd(C) reflective subcategory of Rg(C)
• Gpd(C) is a Goursat category

A first answer to (Q2)

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 – 8 / 30

· A first answer to (Q2) concerning Goursat pushouts

A first answer to (Q2)

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 – 8 / 30

· A first answer to (Q2) concerning Goursat pushouts · Thm. [GR–2012] C regular category. TFAE: (i) C is a Goursat cat (ii) every pushout (1) is a Goursat pushout A

f

• α

(1)

C

g

• B

s

• β

D

t

A first answer to (Q2)

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 – 8 / 30

· A first answer to (Q2) concerning Goursat pushouts · Thm. [GR–2012] C regular category. TFAE: (i) C is a Goursat cat (ii) every pushout (1) is a Goursat pushout A

f

• α

(1)

C

g

• B

s

• β

D

t

A first answer to (Q2)

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 – 8 / 30

· A first answer to (Q2) concerning Goursat pushouts · Thm. [GR–2012] C regular category. TFAE: (i) C is a Goursat cat (ii) every pushout (1) is a Goursat pushout A

f

• α

(1)

C

g

• B

s

• β

D

t

• Eq(f)
• Eq(g)
• λ

A first answer to (Q2)

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 – 8 / 30

· A first answer to (Q2) concerning Goursat pushouts · Thm. [GR–2012] C regular category. TFAE: (i) C is a Goursat cat (ii) every pushout (1) is a Goursat pushout A

f

• α

(1)

C

g

• B

s

• β

D

t

• Eq(f)
• Eq(g)
• λ

· Related known facts:

• [Bourn–2003] regular Mal’tsev cat ⇔ every (1) is a regular pushout

f, α : A

B ×D C

• [Carboni, Kelly, Pedicchio–1993] Goursat cat ⇔ regular image of an

equivalence relation is an equiv relation

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma · Classical 3 × 3 Lemma vs. Denormalised 3 × 3 Lemma short exact sequences exact forks · · ·

Eq(f)

·

f ·

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma · Classical 3 × 3 Lemma vs. Denormalised 3 × 3 Lemma short exact sequences exact forks · · ·

Eq(f)

·

f ·

· A

f

• α

(1)

C

g

• B

s

• β

D

t

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma · Classical 3 × 3 Lemma vs. Denormalised 3 × 3 Lemma short exact sequences exact forks · · ·

Eq(f)

·

f ·

· A

f

• α

(1)

C

g

• B

s

• β

D

t

• Eq(α)
• ϕ
• Eq(β)

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma · Classical 3 × 3 Lemma vs. Denormalised 3 × 3 Lemma short exact sequences exact forks · · ·

Eq(f)

·

f ·

· A

f

• α

(1)

C

g

• B

s

• β

D

t

• Eq(α)
• ϕ
• Eq(β)
• Eq(ϕ)
• Eq(f)
• λ Eq(g)

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma · Classical 3 × 3 Lemma vs. Denormalised 3 × 3 Lemma short exact sequences exact forks · · ·

Eq(f)

·

f ·

· A

f

• α

(1)

C

g

• B

s

• β

D

t

• Eq(α)
• ϕ
• Eq(β)
• Eq(ϕ)
• Eq(f)
• λ Eq(g)
• 3 columns + 2 bottom rows exact forks

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma · Classical 3 × 3 Lemma vs. Denormalised 3 × 3 Lemma short exact sequences exact forks · · ·

Eq(f)

·

f ·

· A

f

• α

(1)

C

g

• B

s

• β

D

t

• Eq(α)
• ϕ
• Eq(β)
• Eq(ϕ)
• Eq(f)
• λ Eq(g)
• 3 columns + 2 bottom rows exact forks

(1)

⇒

Goursat po

top row exact fork

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma · Classical 3 × 3 Lemma vs. Denormalised 3 × 3 Lemma short exact sequences exact forks · · ·

Eq(f)

·

f ·

· A

f

• α

(1)

C

g

• B

s

• β

D

t

• Eq(α)
• ϕ
• Eq(β)
• Eq(ϕ)
• Eq(f)
• λ Eq(g)
• 3 columns + 2 bottom rows exact forks

(1)

⇒

Goursat po

top row exact fork 3 columns + middle row exact forks

The 3 × 3 Lemma - 1

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

· From Goursat pushouts to the (denormalised) 3 × 3 Lemma · Classical 3 × 3 Lemma vs. Denormalised 3 × 3 Lemma short exact sequences exact forks · · ·

Eq(f)

·

f ·

· A

f

• α

(1)

C

g

• B

s

• β

D

t

• Eq(α)
• ϕ
• Eq(β)
• Eq(ϕ)
• Eq(f)
• λ Eq(g)
• 3 columns + 2 bottom rows exact forks

(1)

⇒

Goursat po

top row exact fork 3 columns + middle row exact forks top row exact fork ⇒ bottom row exact fork

The 3 × 3 Lemma - 2

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 – 10 / 30

· The 3 × 3 Lemma Eq(ϕ) Eq(f)

λ Eq(g)

Eq(α)

• ϕ

A

α f

C

g

• R

B

β

D 3 columns + middle row exact forks

The 3 × 3 Lemma - 2

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 – 10 / 30

· The 3 × 3 Lemma Eq(ϕ) Eq(f)

λ Eq(g)

Eq(α)

• ϕ

A

α f

C

g

• R

B

β

D 3 columns + middle row exact forks top row exact fork ⇔ bottom row exact fork

The 3 × 3 Lemma - 2

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 – 10 / 30

· The 3 × 3 Lemma Eq(ϕ) Eq(f)

λ Eq(g)

Eq(α)

• ϕ

A

α f

C

g

• R

B

β

D 3 columns + middle row exact forks top row exact fork ⇔ bottom row exact fork · Known results:

• [Bourn–2003] C regular Mal’tsev cat ⇒ 3 × 3 Lemma holds
• [Lack–2004] C Goursat cat ⇒ 3 × 3 Lemma holds

The 3 × 3 Lemma - 2

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 – 10 / 30

· The 3 × 3 Lemma Eq(ϕ) Eq(f)

λ Eq(g)

Eq(α)

• ϕ

A

α f

C

g

• R

B

β

D 3 columns + middle row exact forks top row exact fork ⇔ bottom row exact fork · 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

The 3 × 3 Lemma - 2

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 – 10 / 30

· The 3 × 3 Lemma Eq(ϕ) Eq(f)

λ Eq(g)

Eq(α)

• ϕ

A

α f

C

g

• R

B

β

D 3 columns + middle row exact forks top row exact fork ⇔ bottom row exact fork · 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 ⇒ / ⇒

Goursat varieties

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 – 11 / 30

· [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff the algebraic theory T

• f V contains two quaternary ops p and q sth

   p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z

Goursat varieties

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 – 11 / 30

· [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff the algebraic theory T

• f V contains two quaternary ops p and q sth

   p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z · Eq(∇2 + ∇2)

λ Eq(∇3)

4X

∇2+∇2 1X+∇2+1X (1)

3X

∇3

2X

i2+i1

• ∇2

i2

• X

i2

• Goursat pushout

(X free algebra on one element)

Goursat varieties

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 – 11 / 30

· [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff the algebraic theory T

• f V contains two quaternary ops p and q sth

   p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z · Eq(∇2 + ∇2)

λ Eq(∇3)

4X

∇2+∇2 1X+∇2+1X (1)

3X

∇3

2X

i2+i1

• ∇2

i2

• X

i2

• Goursat pushout

(X free algebra on one element) ∋ (p1, p3)

• p1(x, y, z) = x

p3(x, y, z) = z

Goursat varieties

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 – 11 / 30

· [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff the algebraic theory T

• f V contains two quaternary ops p and q sth

   p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z · Eq(∇2 + ∇2)

λ Eq(∇3)

4X

∇2+∇2 1X+∇2+1X (1)

3X

∇3

2X

i2+i1

• ∇2

i2

• X

i2

• Goursat pushout

(X free algebra on one element) ∋ (p1, p3)

• p1(x, y, z) = x

p3(x, y, z) = z

(p, q) ∈

Goursat varieties

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 – 11 / 30

· [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff the algebraic theory T

• f V contains two quaternary ops p and q sth

   p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z · Eq(∇2 + ∇2)

λ Eq(∇3)

4X

∇2+∇2 1X+∇2+1X (1)

3X

∇3

2X

i2+i1

• ∇2

i2

• X

i2

• Goursat pushout

(X free algebra on one element) ∋ (p1, p3)

• p1(x, y, z) = x

p3(x, y, z) = z

(p, q) ∈ (p, q) ∈ Eq(∇2 + ∇2)

Goursat varieties

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 – 11 / 30

· [Hagemann, Mitschke–1973] V Goursat variety of universal algebras iff the algebraic theory T

• f V contains two quaternary ops p and q sth

   p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z · Eq(∇2 + ∇2)

λ Eq(∇3)

4X

∇2+∇2 1X+∇2+1X (1)

3X

∇3

2X

i2+i1

• ∇2

i2

• X

i2

• Goursat pushout

(X free algebra on one element) ∋ (p1, p3)

• p1(x, y, z) = x

p3(x, y, z) = z

(p, q) ∈ (p, q) ∈ Eq(∇2 + ∇2) ✛ ✛ λ(p, q) = (p1, p3)

2-permutability (Mal’tsev)

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 – 12 / 30

Stability property - 1

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

· From Goursat pushouts to a stability property for regular epis

Stability property - 1

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

· From Goursat pushouts to a stability property for regular epis · [Bourn–2003] C regular Mal’tsev cat. In: Y ×B A

• a
• ◗

◗ ◗ ◗ ◗ ◗

λ

Z ×D C ✤ ✤ ✤

c

• ◗

◗ ◗ ◗ ◗ ◗ A

f α

C

g

• Y
• b
• ❘

❘ ❘ ❘ ❘ ❘ ❘ ❘

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❘

❘ ❘ ❘ B

s

• β

D

t

• λ is a regular epi

Stability property - 1

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

· From Goursat pushouts to a stability property for regular epis · [Bourn–2003] C regular Mal’tsev cat. In: Y ×B A

• a
• ◗

◗ ◗ ◗ ◗ ◗

λ

Z ×D C ✤ ✤ ✤

c

• ◗

◗ ◗ ◗ ◗ ◗ A

f α

C

g

• Y
• b
• ❘

❘ ❘ ❘ ❘ ❘ ❘ ❘

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❘

❘ ❘ ❘ B

s

• β

D

t

• λ is a regular epi

· Rem. - a, b, c, d arbitrary maps and α, β, ζ (⇒ λ) regular epis

Stability property - 1

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

· From Goursat pushouts to a stability property for regular epis · [Bourn–2003] C regular Mal’tsev cat. In: Y ×B A

• a
• ◗

◗ ◗ ◗ ◗ ◗

λ

Z ×D C ✤ ✤ ✤

c

• ◗

◗ ◗ ◗ ◗ ◗ A

f α

C

g

• Y
• b
• ❘

❘ ❘ ❘ ❘ ❘ ❘ ❘

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❘

❘ ❘ ❘ B

s

• β

D

t

• λ is a regular epi

· Rem. - a, b, c, d arbitrary maps and α, β, ζ (⇒ λ) regular epis

• front face is of type (1)

Stability property - 1

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

· From Goursat pushouts to a stability property for regular epis · [Bourn–2003] C regular Mal’tsev cat. In: Y ×B A

• a
• ◗

◗ ◗ ◗ ◗ ◗

λ

Z ×D C ✤ ✤ ✤

c

• ◗

◗ ◗ ◗ ◗ ◗ A

f α

C

g

• Y
• b
• ❘

❘ ❘ ❘ ❘ ❘ ❘ ❘

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❘

❘ ❘ ❘ B

s

• β

D

t

• λ is a regular epi

· Rem. - a, b, c, d arbitrary maps and α, β, ζ (⇒ λ) regular epis

• front face is of type (1)
• b = f (Y ×B A = Eq(f)) and d = g (Z ×D C = Eq(g))

Stability property - 1

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

· From Goursat pushouts to a stability property for regular epis · [Bourn–2003] C regular Mal’tsev cat. In: Y ×B A

• a
• ◗

◗ ◗ ◗ ◗ ◗

λ

Z ×D C ✤ ✤ ✤

c

• ◗

◗ ◗ ◗ ◗ ◗ A

f α

C

g

• Y
• b
• ❘

❘ ❘ ❘ ❘ ❘ ❘ ❘

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❘

❘ ❘ ❘ B

s

• β

D

t

• λ is a regular epi

· Rem. - a, b, c, d arbitrary maps and α, β, ζ (⇒ λ) regular epis

• 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 - 1

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

· From Goursat pushouts to a stability property for regular epis · [Bourn–2003] C regular Mal’tsev cat. In: Y ×B A

• a
• ◗

◗ ◗ ◗ ◗ ◗

λ

Z ×D C ✤ ✤ ✤

c

• ◗

◗ ◗ ◗ ◗ ◗ A

f α

C

g

• Y
• b
• ❘

❘ ❘ ❘ ❘ ❘ ❘ ❘

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❘

❘ ❘ ❘ B

s

• β

D

t

• λ is a regular epi

· Rem. - a, b, c, d arbitrary maps and α, β, ζ (⇒ λ) regular epis

• 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

Stability property - 2

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 – 14 / 30

· Mal’tsev ⇒ Goursat ⇒ ⇔ stability property ⇒ Goursat po property

Stability property - 2

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 – 14 / 30

· Mal’tsev ⇒ Goursat ⇔ stability property ⇒ Goursat po property · Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) stability property holds ( λ regular epi) ⇔

Stability property - 2

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 – 14 / 30

· Mal’tsev ⇒ Goursat ⇔ stability property ⇒ Goursat po property · Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) stability property holds ( λ regular epi) ⇔ ⇔ 3 × 3 Lemma

Stability property - 2

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 – 14 / 30

· Mal’tsev ⇒ Goursat ⇔ stability property ⇒ Goursat po property · Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) stability property holds ( λ regular epi) ⇔ ⇔ 3 × 3 Lemma

? ⇔

Stability property - 2

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 – 14 / 30

· Mal’tsev ⇒ Goursat ⇔ stability property ⇒ Goursat po property · Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) stability property holds ( λ regular epi) ⇔ ⇔ 3 × 3 Lemma

? ⇔

· Is there a homological diagram lemma which characterises Mal’tsev cats?

Stability property - 2

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 – 14 / 30

· Mal’tsev ⇒ Goursat ⇔ stability property ⇒ Goursat po property · Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) stability property holds ( λ regular epi) ⇔ ⇔ 3 × 3 Lemma

? ⇔

· Is there a homological diagram lemma which characterises Mal’tsev cats? · Goursat pushouts stability property wrt

• wrt

kernel pairs pullbacks

Stability property - 2

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 – 14 / 30

· Mal’tsev ⇒ Goursat ⇔ stability property ⇒ Goursat po property · Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) stability property holds ( λ regular epi) ⇔ ⇔ 3 × 3 Lemma

? ⇔

· Is there a homological diagram lemma which characterises Mal’tsev cats? · wrt

• wrt

kernel pairs pullbacks 3 × 3 Lemma Cuboid Lemma (3-dimensional diagram)

The Cuboid Lemma

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 – 15 / 30

· The (Upper) 3 × 3 Lemma Eq(ϕ) Eq(f)

λ Eq(g)

Eq(α)

• ϕ

A

α f

C

g

• R

B

β

D 3 columns + middle row exact forks top row exact fork ⇐ ⇔ bottom row exact fork

( )

The Cuboid Lemma

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 – 15 / 30

· The (Upper) 3 × 3 Lemma Eq(ϕ) Eq(f)

λ Eq(g)

Eq(α)

• ϕ

A

α f

C

g

• R

B

β

D 3 columns + middle row exact forks top row exact fork ⇐ ⇔ bottom row exact fork

( )

· The (Upper) Cuboid Lemma U

• ①

① ① ① ①

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ V

• ✁

✁

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W

• c
• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)

ϕ

• ①①①①①

A

f

✂✂✂✂

α

C

g

• S

B

β

D

The Cuboid Lemma

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 – 15 / 30

· The (Upper) Cuboid Lemma U

• ①

① ① ① ①

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ V

• ✁

✁

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W

• c
• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)

ϕ

• ①①①①①

A

f

✂✂✂✂

α

C

g

• S

B

β

D 3 diamonds are pbs 2 middle rows exact forks

The Cuboid Lemma

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 – 15 / 30

· The (Upper) Cuboid Lemma U

• ①

① ① ① ①

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ V

• ✁

✁

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W

• c
• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)

ϕ

• ①①①①①

A

f

✂✂✂✂

α

C

g

• S

B

β

D 3 diamonds are pbs 2 middle rows exact forks top row exact fork ⇐ bottom row exact fork

The Cuboid Lemma

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 – 15 / 30

· The (Upper) Cuboid Lemma U

• ①

① ① ① ①

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ V

• ✁

✁

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W

• c
• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)

ϕ

• ①①①①①

A

f

✂✂✂✂

α

C

g

• S

B

β

D 3 diamonds are pbs 2 middle rows exact forks top row exact fork ⇐ bottom row exact fork stability pp for the right cube

The Cuboid Lemma

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 – 15 / 30

· The (Upper) Cuboid Lemma U

• ①

① ① ① ①

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ V

• ✁

✁

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W

• c
• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✸

✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)

ϕ

• ①①①①①

A

f

✂✂✂✂

α

C

g

• S

B

β

D stability pp for the right cube · Thm. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) (Upper) Cuboid Lemma holds

The relative context

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

· absolute context

• relative context

[T. Janelidze–2009]

The relative context

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

· absolute context

• relative context

[T. Janelidze–2009] · Regular Goursat cats Mal’tsev cats

(3-permutable: RSR = SRS) (2-permutable: RS = SR)

⇔ Goursat po pp ⇔ stability pp ⇔ 3 × 3 Lemma ⇔ Cuboid Lemma

The relative context

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

· absolute context

• relative context

[T. Janelidze–2009] · Regular Goursat cats Mal’tsev cats

(3-permutable: RSR = SRS) (2-permutable: RS = SR)

⇔ Goursat po pp ⇔ stability pp ⇔ 3 × 3 Lemma ⇔ Cuboid Lemma relative

×s + E class of regular epis sth ...

The relative context

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

· absolute context

• relative context

[T. Janelidze–2009] · Regular Goursat cats Mal’tsev cats

(3-permutable: RSR = SRS) (2-permutable: RS = SR)

⇔ Goursat po pp ⇔ stability pp ⇔ 3 × 3 Lemma ⇔ Cuboid Lemma relative

×s + E class of regular epis sth ...

relative version [Goedecke, T. Janelidze–2012]

The relative context

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

· absolute context

• relative context

[T. Janelidze–2009] · Regular Goursat cats Mal’tsev cats

(3-permutable: RSR = SRS) (2-permutable: RS = SR)

⇔ Goursat po pp ⇔ stability pp ⇔ 3 × 3 Lemma ⇔ Cuboid Lemma relative

×s + E class of regular epis sth ...

relative version [Goedecke, T. Janelidze–2012] relative

E-relations

The relative context

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

· absolute context

• relative context

[T. Janelidze–2009] · Regular Goursat cats Mal’tsev cats

(3-permutable: RSR = SRS) (2-permutable: RS = SR)

⇔ Goursat po pp ⇔ stability pp ⇔ 3 × 3 Lemma ⇔ Cuboid Lemma relative

×s + E class of regular epis sth ...

relative version [Goedecke, T. Janelidze–2012] relative

E-relations

[Everaert, Goedecke, T. Janelidze, VdL–2013]

relative

E-relations

The relative context

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

· absolute context

• relative context

[T. Janelidze–2009] · Regular Goursat cats Mal’tsev cats

(3-permutable: RSR = SRS) (2-permutable: RS = SR)

⇔ Goursat po pp ⇔ stability pp ⇔ 3 × 3 Lemma ⇔ Cuboid Lemma relative

×s + E class of regular epis sth ...

relative version [Goedecke, T. Janelidze–2012] relative

E-relations

[Everaert, Goedecke, T. Janelidze, VdL–2013]

relative

E-relations

relative version

[GR–2014]

Star-regular categories

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 – 17 / 30

The context

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

· pointed & non-pointed (2-permutability and 3-permutability)

The context

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

· pointed & non-pointed (2-permutability and 3-permutability) · C lex, N ideal: f ∈ N

• r g ∈ N

⇒ gf ∈ N

[Ehresmann–1964] [Lavendhomme–1965]

The context

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

· pointed & non-pointed (2-permutability and 3-permutability) · C lex, N ideal: f ∈ N

• r g ∈ N

⇒ gf ∈ N

[Ehresmann–1964] [Lavendhomme–1965]

· Ex: N = all morphisms

• total context

N = zero morphisms

• pointed context

The context

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

· pointed & non-pointed (2-permutability and 3-permutability) · C lex, N ideal: f ∈ N

• r g ∈ N

⇒ gf ∈ N

[Ehresmann–1964] [Lavendhomme–1965]

· Ex: N = all morphisms

• total context

N = zero morphisms

• pointed context

· N -kernel: Nf

nf X f

Y sth fnf ∈ N and universal

The context

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

· pointed & non-pointed (2-permutability and 3-permutability) · C lex, N ideal: f ∈ N

• r g ∈ N

⇒ gf ∈ N

[Ehresmann–1964] [Lavendhomme–1965]

· Ex: N = all morphisms

• total context

N = zero morphisms

• pointed context

· N -kernel: Nf

nf X f

Y sth fnf ∈ N and universal · star: σ = (σ1, σ2) : S ⇒ X sth σ1 ∈ N

The context

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

· pointed & non-pointed (2-permutability and 3-permutability) · C lex, N ideal: f ∈ N

• r g ∈ N

⇒ gf ∈ N

[Ehresmann–1964] [Lavendhomme–1965]

· Ex: N = all morphisms

• total context

N = zero morphisms

• pointed context

· N -kernel: Nf

nf X f

Y sth fnf ∈ N and universal · star: σ = (σ1, σ2) : S ⇒ X sth σ1 ∈ N · C multi-pointed cat w/ kernels = C w/ ideal N and ∃ N -kernels

The context

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

· pointed & non-pointed (2-permutability and 3-permutability) · C lex, N ideal: f ∈ N

• r g ∈ N

⇒ gf ∈ N

[Ehresmann–1964] [Lavendhomme–1965]

· Ex: N = all morphisms

• total context

N = zero morphisms

• pointed context

· N -kernel: Nf

nf X f

Y sth fnf ∈ N and universal · star: σ = (σ1, σ2) : S ⇒ X sth σ1 ∈ 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 )

The context

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

· pointed & non-pointed (2-permutability and 3-permutability) · C lex, N ideal: f ∈ N

• r g ∈ N

⇒ gf ∈ N

[Ehresmann–1964] [Lavendhomme–1965]

· Ex: N = all morphisms

• total context

N = zero morphisms

• pointed context

· N -kernel: Nf

nf X f

Y sth fnf ∈ N and universal · star: σ = (σ1, σ2) : S ⇒ X sth σ1 ∈ 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 ) [GJU–2012]

Star-exact sequences

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

· star-kernel: S

σ1 σ2 X f

Y sth fσ1 = fσ2 and universal

Star-exact sequences

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

· star-kernel: S

σ1 σ2 X f

Y sth fσ1 = fσ2 and universal ∼ = Eq(f)∗

Star-exact sequences

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

· star-kernel: S

σ1 σ2 X f

Y sth fσ1 = fσ2 and universal ∼ = Eq(f)∗ X

f Y

regular epi (= coeq of star) · star-exact seq:

Star-exact sequences

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

· star-kernel: S

σ1 σ2 X f

Y sth fσ1 = fσ2 and universal ∼ = Eq(f)∗ X

f Y

regular epi (= coeq of star) · star-exact seq: · Total context (N = all morphisms)

• star = pair of parallel morphisms ( S

X )

• star-exact sequence = exact fork ( Eq(f)

X

f Y )

• star-regular cat = regular cat (regular epis = coequalisers of their kernel pairs)

Star-exact sequences

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

· star-kernel: S

σ1 σ2 X f

Y sth fσ1 = fσ2 and universal ∼ = Eq(f)∗ X

f Y

regular epi (= coeq of star) · star-exact seq: · Total context (N = all morphisms)

• star = pair of parallel morphisms ( S

X )

• star-exact sequence = exact fork ( Eq(f)

X

f Y )

• star-regular cat = regular cat (regular epis = coequalisers of their kernel pairs)

· Pointed context (N = zero morphisms)

• star = morphism ( S

X )

• star-exact sequence = short exact sequence ( K

k X f Y )

• star-regular cat = normal cat (= 0 + regular + (regular epis = normal epis))

Star-exact sequences

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

· star-kernel: S

σ1 σ2 X f

Y sth fσ1 = fσ2 and universal ∼ = Eq(f)∗ X

f Y

regular epi (= coeq of star) · star-exact seq: · Total context (N = all morphisms)

• star = pair of parallel morphisms ( S

X )

• star-exact sequence = exact fork ( Eq(f)

X

f Y )

• star-regular cat = regular cat (regular epis = coequalisers of their kernel pairs)

· Pointed context (N = zero morphisms)

• star = morphism ( S

X )

• star-exact sequence = short exact sequence ( K

k X f Y )

• star-regular cat = normal cat (= 0 + regular + (regular epis = normal epis))

exact fork short exact sequence

The 3 × 3 Lemma for star-exact sequences

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 – 20 / 30

· Eq(ϕ)∗ Eq(f)∗ Eq(g)∗ Eq(α)∗

• ϕ

A

α f

C

g

• R

B

β

D 3 cols + middle row star-exact seq

The 3 × 3 Lemma for star-exact sequences

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 – 20 / 30

· Eq(ϕ)∗ Eq(f)∗ Eq(g)∗ Eq(α)∗

• ϕ

A

α f

C

g

• R

B

β

D 3 cols + middle row star-exact seq top row star-exact seq ⇔ bottom row star-exact seq

The 3 × 3 Lemma for star-exact sequences

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 – 20 / 30

· Eq(ϕ)∗ Eq(f)∗ Eq(g)∗ Eq(α)∗

• ϕ

A

α f

C

g

• R

B

β

D 3 cols + middle row star-exact seq top row star-exact seq ⇔ bottom row star-exact seq Shape of denormalised 3 × 3 Lemma, but captures both classical and denormalised 3 × 3 Lemmas

The 3 × 3 Lemma for star-exact sequences

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 – 20 / 30

· Eq(ϕ)∗ Eq(f)∗ Eq(g)∗ Eq(α)∗

• ϕ

A

α f

C

g

• R

B

β

D 3 cols + middle row star-exact seq top row star-exact seq ⇔ bottom row star-exact seq Shape of denormalised 3 × 3 Lemma, but captures both classical and denormalised 3 × 3 Lemmas · Thm. [GJR–2012] C star-regular category + · · ·. TFAE: (i) C has symmetric saturation pp (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds

The 3 × 3 Lemma for star-exact sequences

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 – 20 / 30

· Eq(ϕ)∗ Eq(f)∗ Eq(g)∗ Eq(α)∗

• ϕ

A

α f

C

g

• R

B

β

D 3 cols + middle row star-exact seq top row star-exact seq ⇔ bottom row star-exact seq Shape of denormalised 3 × 3 Lemma, but captures both classical and denormalised 3 × 3 Lemmas · Thm. [GJR–2012] C star-regular category + · · ·. TFAE: (i) C has symmetric saturation pp (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds

(⇐ 3-star-permutability [GJRU–2012])

The 3 × 3 Lemma for star-exact sequences

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 – 20 / 30

· Eq(ϕ) Eq(f) Eq(g) Eq(α)

• ϕ

A

α f

C

g

• R

B

β

D 3 cols + middle row star-exact seq top row star-exact seq ⇔ bottom row star-exact seq · Thm. [GJR–2012] C star-regular category + · · ·. TFAE: (i) C has symmetric saturation pp (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds

(⇐ 3-star-permutability [GJRU–2012])

· 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

The 3 × 3 Lemma for star-exact sequences

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 – 20 / 30

3 cols + middle row star-exact seq top row star-exact seq ⇔ bottom row star-exact seq · Thm. [GJR–2012] C star-regular category + · · ·. TFAE: (i) C has symmetric saturation pp (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds

(⇐ 3-star-permutability [GJRU–2012])

· Kϕ ❴

• Kf

❴

• Kg

❴

• Kα ✤
• ϕ

A

α f

C

g

• R

B

β

D · 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

The 3 × 3 Lemma for star-exact sequences

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 – 20 / 30

3 cols + middle row star-exact seq top row star-exact seq ⇔ bottom row star-exact seq · Thm. [GJR–2012] C star-regular category + · · ·. TFAE: (i) C has symmetric saturation pp (ii) (Upper/Lower) 3 × 3 Lemma for star-exact sequences holds

(⇐ 3-star-permutability [GJRU–2012])

· Kϕ ❴

• Kf

❴

• Kg

❴

• Kα ✤
• ϕ

A

α f

C

g

• R

B

β

D · 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])

The Star-Cuboid lemma

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 – 21 / 30

· U ∗

• ✇✇✇✇✇
• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ V ∗

• ✂

✂

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W ∗

• ✁✁✁

c

• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)∗

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)∗

ϕ

• ✈✈✈✈✈

A

f

✂✂✂✂

α

C

g

• S

B

β

D

The Star-Cuboid lemma

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 – 21 / 30

· U ∗

• ✇✇✇✇✇
• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ V ∗

• ✂

✂

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W ∗

• ✁✁✁

c

• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)∗

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)∗

ϕ

• ✈✈✈✈✈

A

f

✂✂✂✂

α

C

g

• S

B

β

D bottom row star-exact sequence ⇒ top row star-exact sequence

The Star-Cuboid lemma

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 – 21 / 30

· U ∗

• ✇✇✇✇✇
• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ V ∗

• ✂

✂

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W ∗

• ✁✁✁

c

• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)∗

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)∗

ϕ

• ✈✈✈✈✈

A

f

✂✂✂✂

α

C

g

• S

B

β

D bottom row star-exact sequence ⇒ top row star-exact sequence Shape of the Cuboid Lemma, but captures both the Cuboid Lemma and (again) the classical 3 × 3 Lemma

The Star-Cuboid lemma

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 – 21 / 30

· U ∗

• ✇✇✇✇✇
• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ V ∗

• ✂

✂

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W ∗

• ✁✁✁

c

• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)∗

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)∗

ϕ

• ✈✈✈✈✈

A

f

✂✂✂✂

α

C

g

• S

B

β

D bottom row star-exact sequence ⇒ top row star-exact sequence Shape of the Cuboid Lemma, but captures both the Cuboid Lemma and (again) the classical 3 × 3 Lemma · Thm. [GR–2014] C star-regular category + · · ·. TFAE: (i) C is a 2-star-permutable cat (ii) Star-Upper Cuboid Lemma holds

The Star-Cuboid lemma

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 – 21 / 30

· U ∗

• ✇✇✇✇✇
• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ V ∗

• ✂

✂

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W ∗

• ✁✁✁

c

• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)∗

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)∗

ϕ

• ✈✈✈✈✈

A

f

✂✂✂✂

α

C

g

• S

B

β

D bottom row star-exact sequence ⇒ top row star-exact sequence Shape of the Cuboid Lemma, but captures both the Cuboid Lemma and (again) the classical 3 × 3 Lemma · Thm. [GR–2014] C star-regular category + · · ·. TFAE: (i) C is a 2-star-permutable cat (ii) Star-Upper Cuboid Lemma holds

[GJRU–2012]

The Star-Cuboid lemma

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 – 21 / 30

· U ∗

• ✇✇✇✇✇
• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ V ∗

• ✂

✂

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W ∗

• ✁✁✁

c

• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)∗

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)∗

ϕ

• ✈✈✈✈✈

A

f

✂✂✂✂

α

C

g

• S

B

β

D bottom row star-exact sequence ⇒ top row star-exact sequence · Thm. [GR–2014] C star-regular category + · · ·. TFAE: (i) C is a 2-star-permutable cat (ii) Star-Upper Cuboid Lemma holds

[GJRU–2012]

· Total cnt: C regular. C Mal’tsev iff (Upper) Cuboid Lemma holds

The Star-Cuboid lemma

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 – 21 / 30

· U ∗

• ✇✇✇✇✇
• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ V ∗

• ✂

✂

λ

• a
• ✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ W ∗

• ✁✁✁

c

• ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Eq(ζ)∗

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✹

✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ Y

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

b

• ✲

✲ ✲ ✲ ✲ Z

d

• ✳

✳ ✳ ✳ ✳ Eq(α)∗

ϕ

• ✈✈✈✈✈

A

f

✂✂✂✂

α

C

g

• S

B

β

D bottom row star-exact sequence ⇒ top row star-exact sequence · Thm. [GR–2014] C star-regular category + · · ·. TFAE: (i) C is a 2-star-permutable cat (ii) Star-Upper Cuboid Lemma holds

[GJRU–2012]

· Total cnt: C regular. C Mal’tsev iff (Upper) Cuboid Lemma holds Pointed cnt: C normal. C subtractive iff (Upper) Classical 3 × 3 L.

n-permutability

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 22 / 30

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

·

f · f−1

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

·

f · f−1

• · [G. Janelidze–1990] V Mal’tsev variety ⇒ ∃ ! m and U, V

are isos

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

·

f · f−1

• · [G. Janelidze–1990] V Mal’tsev variety ⇒ ∃ ! m and U, V

are isos · [Carboni, Pedicchio, Pirovano–1992] V Mal’tsev category ⇒ ∃ ! m, U, V are isos and W is full

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

·

f · f−1

• · [G. Janelidze–1990] V Mal’tsev variety ⇒ ∃ ! m and U, V

are isos · [Carboni, Pedicchio, Pirovano–1992] V Mal’tsev category ⇒ ∃ ! m, U, V are isos and W is full · [R–2012] V Goursat var ⇒ ∃ ! m , U, V are isos

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

·

f · f−1

• · [G. Janelidze–1990] V Mal’tsev variety ⇒ ∃ ! m and U, V

are isos · [Carboni, Pedicchio, Pirovano–1992] V Mal’tsev category ⇒ ∃ ! m, U, V are isos and W is full · [R–2012] V Goursat var ⇒ ∃ ! m , U, V are isos (W full - new)

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

·

f · f−1

• · [G. Janelidze–1990] V Mal’tsev variety ⇒ ∃ ! m and U, V

are isos · [Carboni, Pedicchio, Pirovano–1992] V Mal’tsev category ⇒ ∃ ! m, U, V are isos and W is full · [R–2012] V Goursat var ⇒ ∃ ! m , U, V are isos (W full - new) · [R–2012] V n-permutable variety ⇒ ∃ ! m, U is an iso Rmg + ·

f g◦f

• ·

g ·

= Cat

[ ]

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

·

f · f−1

• · [G. Janelidze–1990] V Mal’tsev variety ⇒ ∃ ! m and U, V

are isos · [Carboni, Pedicchio, Pirovano–1992] V Mal’tsev category ⇒ ∃ ! m, U, V are isos and W is full · [R–2012] V Goursat var ⇒ ∃ ! m , U, V are isos (W full - new) · [R–2012] V n-permutable variety ⇒ ∃ ! m, U is an iso Rmg + ·

f g◦f

• ·

g ·

= Cat

[ ]

• Cat(V)

full Rg(V)

• new

Internal structures in n-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 23 / 30

· Gpd(V)

U

Cat(V)

V

Rmg(V)

W

Rg(V)

groupoids categories reflexive mult graphs reflexive graphs

∃ m

f ◦ 1 = f = 1 ◦ f m assoc. + ·

f g◦f

• ·

g ·

·

f · f−1

• · [G. Janelidze–1990] V Mal’tsev variety ⇒ ∃ ! m and U, V

are isos · [Carboni, Pedicchio, Pirovano–1992] V Mal’tsev category ⇒ ∃ ! m, U, V are isos and W is full · [R–2012] V Goursat var ⇒ ∃ ! m , U, V are isos (W full - new) · [R–2012] V n-permutable variety ⇒ ∃ ! m, U is an iso Rmg + ·

f g◦f

• ·

g ·

= Cat

[ ]

• Cat(V)

full Rg(V)

• new

n-permutable cats ?

Hagemann’s theorem for varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 24 / 30

· Hagemann’s thm. V variety of universal algebras. TFAE: (i) V is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1

Hagemann’s theorem for varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 24 / 30

· Hagemann’s thm. V variety of universal algebras. TFAE: (i) V is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [Carboni, Kelly, Pedicchio–1993] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) R reflexive ⇒ R◦ R (iii) R reflexive ⇒ RR R (R symmetric) (R transitive)

Hagemann’s theorem for varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 24 / 30

· Hagemann’s thm. V variety of universal algebras. TFAE: (i) V is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [Carboni, Kelly, Pedicchio–1993] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) R reflexive ⇒ R◦ R (iii) R reflexive ⇒ RR R (R symmetric) (R transitive) (general symmetry) (general transitivity)

Hagemann’s theorem for varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 24 / 30

· Hagemann’s thm. V variety of universal algebras. TFAE: (i) V is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [Carboni, Kelly, Pedicchio–1993] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) R reflexive ⇒ R◦ R (iii) R reflexive ⇒ RR R (R symmetric) (R transitive) (general symmetry) (general transitivity) · Rem. (non-regular) Mal’tsev categories are also defined by the property: reflexive relations are symmetric/transitive/equivalence relations

Hagemann’s theorem for varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 24 / 30

· Hagemann’s thm. V variety of universal algebras. TFAE: (i) V is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [Carboni, Kelly, Pedicchio–1993] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) R reflexive ⇒ R◦ R (iii) R reflexive ⇒ RR R (R symmetric) (R transitive) (general symmetry) (general transitivity) · Rem. (non-regular) Mal’tsev categories are also defined by the property: reflexive relations are symmetric/transitive/equivalence relations · Hagemann’s theorem extends to categories for n = 2 Aim: extend it for all n 3

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach)

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach) · Ex. Goursat context (n = 3): R◦ RR

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach) · Ex. Goursat context (n = 3): R◦ RR [Hagemann, Mitschke–1973]    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach) · Ex. Goursat context (n = 3): R◦ RR [Hagemann, Mitschke–1973]    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y · x y y x α x x y α y

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach) · Ex. Goursat context (n = 3): R◦ RR [Hagemann, Mitschke–1973]    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y · x y y x α x x y α y R

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach) · Ex. Goursat context (n = 3): R◦ RR [Hagemann, Mitschke–1973]    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y · x y y x α x x y α y R

• reflexive

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach) · Ex. Goursat context (n = 3): R◦ RR [Hagemann, Mitschke–1973]    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y · x y y x α x x y α y R

• reflexive
• y R x ⇒ x R α, α R y

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach) · Ex. Goursat context (n = 3): R◦ RR [Hagemann, Mitschke–1973]    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y · x y y x α x x y α y R

• reflexive
• y R x ⇒ x R α, α R y

x R◦ y ⇒ x RR y

Hagemann’s theorem for categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 25 / 30

· [JRVdL–2014] Hagemann’s thm. C regular category. TFAE: (i) C is n-permutable (ii) R reflexive ⇒ R◦ Rn−1 (iii) R reflexive ⇒ Rn Rn−1 · [J–2006 · · · 2009] Closedness pps for internal relations (matrix approach) · Ex. Goursat context (n = 3): R◦ RR [Hagemann, Mitschke–1973]    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y · x y y x α x x y α y R

• reflexive
• y R x ⇒ x R α, α R y

x R◦ y ⇒ x RR y R◦ RR

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories?

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories? Yes - by Hagemann’s theorem

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories? Yes - by Hagemann’s theorem · Thm. [M-FRVdL–2014] C regular category. TFAE: (i) reflexive + transitive ⇒ symmetric (ii) Cat(C) ∼ = Gpd(C)

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories? Yes - by Hagemann’s theorem · Thm. [M-FRVdL–2014] C regular category. TFAE: (i) reflexive + transitive ⇒ symmetric (ii) Cat(C) ∼ = Gpd(C) These conditions hold when C is n-permutable, for some n 2

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories? Yes - by Hagemann’s theorem · Thm. [M-FRVdL–2014] C regular category. TFAE: (i) reflexive + transitive ⇒ symmetric (ii) Cat(C) ∼ = Gpd(C) These conditions hold when C is n-permutable, for some n 2 (U is an iso)

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories? Yes - by Hagemann’s theorem · Thm. [M-FRVdL–2014] C regular category. TFAE: (i) reflexive + transitive ⇒ symmetric (ii) Cat(C) ∼ = Gpd(C) These conditions hold when C is n-permutable, for some n 2 (U is an iso) R◦

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories? Yes - by Hagemann’s theorem · Thm. [M-FRVdL–2014] C regular category. TFAE: (i) reflexive + transitive ⇒ symmetric (ii) Cat(C) ∼ = Gpd(C) These conditions hold when C is n-permutable, for some n 2 (U is an iso) R◦

• reflexive

Hagemann’s thm

Rn−1

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories? Yes - by Hagemann’s theorem · Thm. [M-FRVdL–2014] C regular category. TFAE: (i) reflexive + transitive ⇒ symmetric (ii) Cat(C) ∼ = Gpd(C) These conditions hold when C is n-permutable, for some n 2 (U is an iso) R◦

• reflexive

Hagemann’s thm

Rn−1 =

transitive

R

Preorders

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 26 / 30

· Mal’tsev categories are sth: reflexive ⇒ symmetric ⇒ transitive ⇒ equivalence · Question: Is there any such result for n-permutable categories? Yes - by Hagemann’s theorem · Thm. [M-FRVdL–2014] C regular category. TFAE: (i) reflexive + transitive ⇒ symmetric (ii) Cat(C) ∼ = Gpd(C) These conditions hold when C is n-permutable, for some n 2 (U is an iso) R◦

• reflexive

Hagemann’s thm

Rn−1 =

transitive

R

]

symmetry

Stability property for Goursat categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 27 / 30

· Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) λ is a regular epi in Y ×B A

• a

❙ ❙ ❙ ❙ ❙ ❙

λ

Z ×D C ✤ ✤ ✤

c

• ❙

❙ ❙ ❙ ❙ ❙ A

f α

C

g

• Y
• b
• ❙

❙ ❙ ❙ ❙ ❙ ❙ ❙

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❙

❙ ❙ ❙ B

s

• β

D

t

Stability property for Goursat categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 27 / 30

· Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) λ is a regular epi in Y ×B A

• a

❙ ❙ ❙ ❙ ❙ ❙

λ

Z ×D C ✤ ✤ ✤

c

• ❙

❙ ❙ ❙ ❙ ❙ A

f α

C

g

• Y
• b
• ❙

❙ ❙ ❙ ❙ ❙ ❙ ❙

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❙

❙ ❙ ❙ B

s

• β

D

t

• · Prop. [GR–in prep] C regular category. TFAE:

(i) C is a Goursat cat (ii) λ is a regular epi in Y ×B A

• a
• ❙

❙ ❙ ❙ ❙ ❙

λ

Z ×D C ✤ ✤ ✤

c

• ❙

❙ ❙ ❙ ❙ ❙ A ❙❙❙❙❙❙

f α

C ❙❙❙❙❙❙

g

• Y
• b
• ❙

❙ ❙ ❙ ❙ ❙ ❙ ❙

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❙

❙ ❙ ❙ B ❙❙❙❙❙❙❙❙

s

• β

D ❙ ❙ ❙ ❙

t

Stability property for Goursat categories

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 27 / 30

· Prop. [GR–2014] C regular category. TFAE: (i) C is a Mal’tsev cat (ii) λ is a regular epi in Y ×B A

• a

❙ ❙ ❙ ❙ ❙ ❙

λ

Z ×D C ✤ ✤ ✤

c

• ❙

❙ ❙ ❙ ❙ ❙ A

f α

C

g

• Y
• b
• ❙

❙ ❙ ❙ ❙ ❙ ❙ ❙

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❙

❙ ❙ ❙ B

s

• β

D

t

• · Prop. [GR–in prep] C regular category. TFAE:

(i) C is a Goursat cat (ii) λ is a regular epi in Y ×B A

• a
• ❙

❙ ❙ ❙ ❙ ❙

λ

Z ×D C ✤ ✤ ✤

c

• ❙

❙ ❙ ❙ ❙ ❙ A ❙❙❙❙❙❙

f α

C ❙❙❙❙❙❙

g

• Y
• b
• ❙

❙ ❙ ❙ ❙ ❙ ❙ ❙

ζ

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Z ✤ ✤ ✤

d

• ❙

❙ ❙ ❙ B ❙❙❙❙❙❙❙❙

s

• β

D ❙ ❙ ❙ ❙

t

• ⇒

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y    p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y    p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z

Eq(∇2 + ∇2)

λ Eq(∇3)

4X

∇2+∇2 1X +∇2+1X (1)

3X

∇3

2X

i2+i1

• ∇2

i2

• X

i2

• Goursat pushout

(p1, p3) ∈ (p, q) ∈

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y    p(x, y, y, z) = x p(x, x, y, y) = q(x, x, y, y) q(x, y, y, z) = z

Eq(∇2 + ∇2)

λ Eq(∇3)

4X

∇2+∇2 1X +∇2+1X (1)

3X

∇3

2X

i2+i1

• ∇2

i2

• X

i2

• Goursat pushout

(p1, p3) ∈ (p, q) ∈

P

• ❏

❏ ❏ ❏ ❏ ❏ ❏

λ

Eq(∇) ✤ ✤ ✤ ✤

• ◆

◆ ◆ ◆ ◆ ◆ 3X ❏❏❏❏❏❏❏

1+∇

• ∇+1

2X ◆◆◆◆◆◆

∇

• 3X
• ∇+1
• ❑

❑ ❑ ❑ ❑ ❑

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 2X ✤ ✤ ✤ ✤

∇

• ❖

❖ ❖ ❖ 2X ❑❑❑❑❑❑

• ∇

X

i2

❖ ❖ ❖ ❖

i1

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y P

• ❏

❏ ❏ ❏ ❏ ❏ ❏

λ

Eq(∇) ✤ ✤ ✤ ✤

• ◆

◆ ◆ ◆ ◆ ◆ 3X ❏❏❏❏❏❏❏

1+∇

• ∇+1

2X ◆◆◆◆◆◆

∇

• 3X
• ∇+1
• ❑

❑ ❑ ❑ ❑ ❑

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 2X ✤ ✤ ✤ ✤

∇

• ❖

❖ ❖ ❖ 2X ❑❑❑❑❑❑

• ∇

X

i2

❖ ❖ ❖ ❖

i1

• (p1, p2)

∈

• p1(x, y) = x

p2(x, y) = y

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y P

• ❏

❏ ❏ ❏ ❏ ❏ ❏

λ

Eq(∇) ✤ ✤ ✤ ✤

• ◆

◆ ◆ ◆ ◆ ◆ 3X ❏❏❏❏❏❏❏

1+∇

• ∇+1

2X ◆◆◆◆◆◆

∇

• 3X
• ∇+1
• ❑

❑ ❑ ❑ ❑ ❑

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 2X ✤ ✤ ✤ ✤

∇

• ❖

❖ ❖ ❖ 2X ❑❑❑❑❑❑

• ∇

X

i2

❖ ❖ ❖ ❖

i1

• (p1, p2)

∈

• p1(x, y) = x

p2(x, y) = y (r, s) ∈

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y P

• ❏

❏ ❏ ❏ ❏ ❏ ❏

λ

Eq(∇) ✤ ✤ ✤ ✤

• ◆

◆ ◆ ◆ ◆ ◆ 3X ❏❏❏❏❏❏❏

1+∇

• ∇+1

2X ◆◆◆◆◆◆

∇

• 3X
• ∇+1
• ❑

❑ ❑ ❑ ❑ ❑

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 2X ✤ ✤ ✤ ✤

∇

• ❖

❖ ❖ ❖ 2X ❑❑❑❑❑❑

• ∇

X

i2

❖ ❖ ❖ ❖

i1

• (p1, p2)

∈

• p1(x, y) = x

p2(x, y) = y (r, s) ∈ (r, s) ∈ P

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y P

• ❏

❏ ❏ ❏ ❏ ❏ ❏

λ

Eq(∇) ✤ ✤ ✤ ✤

• ◆

◆ ◆ ◆ ◆ ◆ 3X ❏❏❏❏❏❏❏

1+∇

• ∇+1

2X ◆◆◆◆◆◆

∇

• 3X
• ∇+1
• ❑

❑ ❑ ❑ ❑ ❑

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 2X ✤ ✤ ✤ ✤

∇

• ❖

❖ ❖ ❖ 2X ❑❑❑❑❑❑

• ∇

X

i2

❖ ❖ ❖ ❖

i1

• (p1, p2)

∈

• p1(x, y) = x

p2(x, y) = y (r, s) ∈ (r, s) ∈ P ✛ ✛ λ(r, s) = (p1, p2)

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y P

• ❏

❏ ❏ ❏ ❏ ❏ ❏

λ

Eq(∇) ✤ ✤ ✤ ✤

• ◆

◆ ◆ ◆ ◆ ◆ 3X ❏❏❏❏❏❏❏

1+∇

• ∇+1

2X ◆◆◆◆◆◆

∇

• 3X
• ∇+1
• ❑

❑ ❑ ❑ ❑ ❑

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 2X ✤ ✤ ✤ ✤

∇

• ❖

❖ ❖ ❖ 2X ❑❑❑❑❑❑

• ∇

X

i2

❖ ❖ ❖ ❖

i1

• (p1, p2)

∈

• p1(x, y) = x

p2(x, y) = y (r, s) ∈ (r, s) ∈ P ✛ ✛ λ(r, s) = (p1, p2) · What about 4-permutable varieties?

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y P

• ❏

❏ ❏ ❏ ❏ ❏ ❏

λ

Eq(∇) ✤ ✤ ✤ ✤

• ◆

◆ ◆ ◆ ◆ ◆ 3X ❏❏❏❏❏❏❏

1+∇

• ∇+1

2X ◆◆◆◆◆◆

∇

• 3X
• ∇+1
• ❑

❑ ❑ ❑ ❑ ❑

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 2X ✤ ✤ ✤ ✤

∇

• ❖

❖ ❖ ❖ 2X ❑❑❑❑❑❑

• ∇

X

i2

❖ ❖ ❖ ❖

i1

• (p1, p2)

∈

• p1(x, y) = x

p2(x, y) = y (r, s) ∈ (r, s) ∈ P ✛ ✛ λ(r, s) = (p1, p2) · What about 4-permutable varieties?

         w1(x, y, y) = x w1(x, x, y) = w2(x, y, y) w2(x, x, y) = w3(x, y, y) w3(x, x, y) = y

Goursat varieties - revisited

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 28 / 30

·    r(x, y, y) = x r(x, x, y) = s(x, y, y) s(x, x, y) = y P

• ❏

❏ ❏ ❏ ❏ ❏ ❏

λ

Eq(∇) ✤ ✤ ✤ ✤

• ◆

◆ ◆ ◆ ◆ ◆ 3X ❏❏❏❏❏❏❏

1+∇

• ∇+1

2X ◆◆◆◆◆◆

∇

• 3X
• ∇+1
• ❑

❑ ❑ ❑ ❑ ❑

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 2X ✤ ✤ ✤ ✤

∇

• ❖

❖ ❖ ❖ 2X ❑❑❑❑❑❑

• ∇

X

i2

❖ ❖ ❖ ❖

i1

• (p1, p2)

∈

• p1(x, y) = x

p2(x, y) = y (r, s) ∈ (r, s) ∈ P ✛ ✛ λ(r, s) = (p1, p2) · What about 4-permutable varieties?

         w1(x, y, y) = x w1(x, x, y) = w2(x, y, y) w2(x, x, y) = w3(x, y, y) w3(x, x, y) = y

right face → the same left face → more pbs

4-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 29 / 30

P2

π12

✝ ✝ ✝ ✝

π23

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ Eq(∇) ✞ ✞ ✞ ✞ ✞ ✞

• ✼

✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ P

π1

✟✟✟✟✟

π2

• ❂

❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂

• ✝

✝ ✝ ✝ P

π1

✟✟✟✟✟

π2

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❇❇❇❇❇❇❇❇❇ 3X

∇+1

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✟

✟ ✟ ✟ ✟ 2X

∇

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ✞

✞ ✞ ✞ ✞ ✞ 3X

1+∇

✞✞✞✞✞

∇+1

• ❀

❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

• ✟

✟ ✟ ✟ ✟ ❂❂❂❂❂❂❂❂❂❂ 3X

1+∇

✟✟✟✟✟✟

∇+1

❁❁❁❁❁❁❁❁❁❁ 2X

∇

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ X

∇

❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪

• ✞

✞ ✞ ✞ ✞ ❁❁❁❁❁❁❁❁❁❁ X

∇

• ✟

✟ ✟ ✟ ✟ ✟ ❀❀❀❀❀❀❀❀❀❀ X,

i2

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

i1

• ⑥

⑥ ⑥ ⑥ ⑥ ⑥ ⑥

4-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 29 / 30

P2

π12

✝ ✝ ✝ ✝

π23

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ Eq(∇) ✞ ✞ ✞ ✞ ✞ ✞

• ✼

✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ P

π1

✟✟✟✟✟

π2

• ❂

❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂

• ✝

✝ ✝ ✝ P

π1

✟✟✟✟✟

π2

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❇❇❇❇❇❇❇❇❇ 3X

∇+1

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✟

✟ ✟ ✟ ✟ 2X

∇

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ✞

✞ ✞ ✞ ✞ ✞ 3X

1+∇

✞✞✞✞✞

∇+1

• ❀

❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

• ✟

✟ ✟ ✟ ✟ ❂❂❂❂❂❂❂❂❂❂ 3X

1+∇

✟✟✟✟✟✟

∇+1

❁❁❁❁❁❁❁❁❁❁ 2X

∇

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ X

∇

❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪

• ✞

✞ ✞ ✞ ✞ ❁❁❁❁❁❁❁❁❁❁ X

∇

• ✟

✟ ✟ ✟ ✟ ✟ ❀❀❀❀❀❀❀❀❀❀ X,

i2

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

i1

• ⑥

⑥ ⑥ ⑥ ⑥ ⑥ ⑥

4-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 29 / 30

P2

π12

✝ ✝ ✝ ✝

π23

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

λ

Eq(∇) ✞ ✞ ✞ ✞ ✞ ✞

• ✼

✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ P

π1

✟✟✟✟✟

π2

• ❂

❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂

• ✝

✝ ✝ ✝ P

π1

✟✟✟✟✟

π2

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❇❇❇❇❇❇❇❇❇ 3X

∇+1

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✟

✟ ✟ ✟ ✟ 2X

∇

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ✞

✞ ✞ ✞ ✞ ✞ 3X

1+∇

✞✞✞✞✞

∇+1

• ❀

❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

• ✟

✟ ✟ ✟ ✟ ❂❂❂❂❂❂❂❂❂❂ 3X

1+∇

✟✟✟✟✟✟

∇+1

❁❁❁❁❁❁❁❁❁❁ 2X

∇

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ X

∇

❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪

• ✞

✞ ✞ ✞ ✞ ❁❁❁❁❁❁❁❁❁❁ X

∇

• ✟

✟ ✟ ✟ ✟ ✟ ❀❀❀❀❀❀❀❀❀❀ X,

i2

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

i1

• ⑥

⑥ ⑥ ⑥ ⑥ ⑥ ⑥

4-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 29 / 30

P2

π12

✝ ✝ ✝ ✝

π23

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

λ

Eq(∇) ✞ ✞ ✞ ✞ ✞ ✞

• ✼

✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ P

π1

✟✟✟✟✟

π2

• ❂

❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂

• ✝

✝ ✝ ✝ P

π1

✟✟✟✟✟

π2

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❇❇❇❇❇❇❇❇❇ 3X

∇+1

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✟

✟ ✟ ✟ ✟ 2X

∇

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ✞

✞ ✞ ✞ ✞ ✞ 3X

1+∇

✞✞✞✞✞

∇+1

• ❀

❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

• ✟

✟ ✟ ✟ ✟ ❂❂❂❂❂❂❂❂❂❂ 3X

1+∇

✟✟✟✟✟✟

∇+1

❁❁❁❁❁❁❁❁❁❁ 2X

∇

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ X

∇

❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪

• ✞

✞ ✞ ✞ ✞ ❁❁❁❁❁❁❁❁❁❁ X

∇

• ✟

✟ ✟ ✟ ✟ ✟ ❀❀❀❀❀❀❀❀❀❀ X,

i2

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

i1

• ⑥

⑥ ⑥ ⑥ ⑥ ⑥ ⑥

• surjective

4-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 29 / 30

P2

π12

✝ ✝ ✝ ✝

π23

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

λ

Eq(∇) ✞ ✞ ✞ ✞ ✞ ✞

• ✼

✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ P

π1

✟✟✟✟✟

π2

• ❂

❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂

• ✝

✝ ✝ ✝ P

π1

✟✟✟✟✟

π2

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❇❇❇❇❇❇❇❇❇ 3X

∇+1

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✟

✟ ✟ ✟ ✟ 2X

∇

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ✞

✞ ✞ ✞ ✞ ✞ 3X

1+∇

✞✞✞✞✞

∇+1

• ❀

❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

• ✟

✟ ✟ ✟ ✟ ❂❂❂❂❂❂❂❂❂❂ 3X

1+∇

✟✟✟✟✟✟

∇+1

❁❁❁❁❁❁❁❁❁❁ 2X

∇

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ X

∇

❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪

• ✞

✞ ✞ ✞ ✞ ❁❁❁❁❁❁❁❁❁❁ X

∇

• ✟

✟ ✟ ✟ ✟ ✟ ❀❀❀❀❀❀❀❀❀❀ X,

i2

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

i1

• ⑥

⑥ ⑥ ⑥ ⑥ ⑥ ⑥

• surjective

(p1, p2) ∈

4-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 29 / 30

P2

π12

✝ ✝ ✝ ✝

π23

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

λ

Eq(∇) ✞ ✞ ✞ ✞ ✞ ✞

• ✼

✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ P

π1

✟✟✟✟✟

π2

• ❂

❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂

• ✝

✝ ✝ ✝ P

π1

✟✟✟✟✟

π2

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❇❇❇❇❇❇❇❇❇ 3X

∇+1

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✟

✟ ✟ ✟ ✟ 2X

∇

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ✞

✞ ✞ ✞ ✞ ✞ 3X

1+∇

✞✞✞✞✞

∇+1

• ❀

❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

• ✟

✟ ✟ ✟ ✟ ❂❂❂❂❂❂❂❂❂❂ 3X

1+∇

✟✟✟✟✟✟

∇+1

❁❁❁❁❁❁❁❁❁❁ 2X

∇

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ X

∇

❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪

• ✞

✞ ✞ ✞ ✞ ❁❁❁❁❁❁❁❁❁❁ X

∇

• ✟

✟ ✟ ✟ ✟ ✟ ❀❀❀❀❀❀❀❀❀❀ X,

i2

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

i1

• ⑥

⑥ ⑥ ⑥ ⑥ ⑥ ⑥

• surjective

(p1, p2) ∈ (w1, w2, w3) ∈

4-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 29 / 30

P2

π12

✝ ✝ ✝ ✝

π23

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

λ

Eq(∇) ✞ ✞ ✞ ✞ ✞ ✞

• ✼

✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ P

π1

✟✟✟✟✟

π2

• ❂

❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂

• ✝

✝ ✝ ✝ P

π1

✟✟✟✟✟

π2

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❇❇❇❇❇❇❇❇❇ 3X

∇+1

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✟

✟ ✟ ✟ ✟ 2X

∇

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ✞

✞ ✞ ✞ ✞ ✞ 3X

1+∇

✞✞✞✞✞

∇+1

• ❀

❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

• ✟

✟ ✟ ✟ ✟ ❂❂❂❂❂❂❂❂❂❂ 3X

1+∇

✟✟✟✟✟✟

∇+1

❁❁❁❁❁❁❁❁❁❁ 2X

∇

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ X

∇

❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪

• ✞

✞ ✞ ✞ ✞ ❁❁❁❁❁❁❁❁❁❁ X

∇

• ✟

✟ ✟ ✟ ✟ ✟ ❀❀❀❀❀❀❀❀❀❀ X,

i2

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

i1

• ⑥

⑥ ⑥ ⑥ ⑥ ⑥ ⑥

• surjective

(p1, p2) ∈ (w1, w2, w3) ∈ (w1, w2) ∈ P ⇒ w1(x, x, y) = w2(x, y, y) (w2, w3) ∈ P ⇒ w2(x, x, y) = w3(x, y, y)

4-permutable varieties

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 29 / 30

P2

π12

✝ ✝ ✝ ✝

π23

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

λ

Eq(∇) ✞ ✞ ✞ ✞ ✞ ✞

• ✼

✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ P

π1

✟✟✟✟✟

π2

• ❂

❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂

• ✝

✝ ✝ ✝ P

π1

✟✟✟✟✟

π2

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❇❇❇❇❇❇❇❇❇ 3X

∇+1

• ❁

❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁ ❁

1+∇

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

• ✟

✟ ✟ ✟ ✟ 2X

∇

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ✞

✞ ✞ ✞ ✞ ✞ 3X

1+∇

✞✞✞✞✞

∇+1

• ❀

❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀

• ✟

✟ ✟ ✟ ✟ ❂❂❂❂❂❂❂❂❂❂ 3X

1+∇

✟✟✟✟✟✟

∇+1

❁❁❁❁❁❁❁❁❁❁ 2X

∇

⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ X

∇

❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪ ❪

• ✞

✞ ✞ ✞ ✞ ❁❁❁❁❁❁❁❁❁❁ X

∇

• ✟

✟ ✟ ✟ ✟ ✟ ❀❀❀❀❀❀❀❀❀❀ X,

i2

❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

i1

• ⑥

⑥ ⑥ ⑥ ⑥ ⑥ ⑥

• surjective

(p1, p2) ∈ (w1, w2, w3) ∈ (w1, w2) ∈ P ⇒ w1(x, x, y) = w2(x, y, y) (w2, w3) ∈ P ⇒ w2(x, x, y) = w3(x, y, y) λ(w1, w2, w3) = (p1, p2) ⇒ w1(x, y, y) = x and w3(x, x, y) = y

Work in progress

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 30 / 30

· Stability property for n-permutable categories:

• right side → the same
• left side → stacked pullbacks

Work in progress

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 30 / 30

· Stability property for n-permutable categories:

• right side → the same
• left side → stacked pullbacks

· Prop. [Jacqmin,RVdL–in prep] C regular category with binary coproducts. TFAE: (i) C is n-permutable (ii) stability property holds

Work in progress

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 30 / 30

· Stability property for n-permutable categories:

• right side → the same
• left side → stacked pullbacks

· Prop. [Jacqmin,RVdL–in prep] C regular category with binary coproducts. TFAE: (i) C is n-permutable (ii) stability property holds · Avoid coproducts [Jacqmin,J–in prep]

Work in progress

Contents Motivation 3-permutability (Goursat) 2-permutability (Mal’tsev) Star-regular categories n-permutability Internal structures in n-permutable varieties Hagemann’s theorem for varieties Hagemann’s theorem for categories Preorders Stability property for Goursat categories Goursat varieties - revisited 4-permutable varieties Work in progress

CT2015 - June 17 A tour through n-permutability – 30 / 30

· Stability property for n-permutable categories:

• right side → the same
• left side → stacked pullbacks

· Prop. [Jacqmin,RVdL–in prep] C regular category with binary coproducts. TFAE: (i) C is n-permutable (ii) stability property holds · Avoid coproducts [Jacqmin,J–in prep]

END