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

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preamble

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9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 1 / 18

The cuboid lemma and Mal’tsev categories

Marino Gran and Diana Rodelo

drodelo@ualg.pt

Centre for Mathematics of the University of Coimbra

University of Algarve, Portugal

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Preliminaries

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 3 / 18

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Regular categories

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 4 / 18

· C regular cat =

lex
·
·
·

·

Rf

f1

f2

A

f

∃

■ ■ ■ ■ ■ B coeq

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Regular categories

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 4 / 18

· C regular cat =

lex
·
·
·

·

Rf

f1

f2

A

f

∃

■ ■ ■ ■ ■ B coeq · C regular ⇒ A

∀ f

p

❉ ❉ ❉ B P m

③

③

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Regular categories

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 4 / 18

· C regular cat =

lex
·
·
·

·

Rf

f1

f2

A

f

∃

■ ■ ■ ■ ■ B coeq · C regular ⇒ A

∀ f

p

❉ ❉ ❉ B P m

③

③ · C regular

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Relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

· relation R from A to B

R

r1,r2 A × B

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Relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

· relation R from A to B

R

r1,r2 A × B

pposite

R◦ from B to A

R

r2,r1 B × A

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Relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

· relation R from A to B

R

r1,r2 A × B

pposite

R◦ from B to A

R

r2,r1 B × A map A

f

B

A

f=1A,f A × B A f ◦=f,1A B × A

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Relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

· relation R from A to B

R

r1,r2 A × B

pposite

R◦ from B to A

R

r2,r1 B × A map A

f

B

A

f=1A,f A × B A f ◦=f,1A B × A

· R ֌ A × B, S ֌ B × C
SR ֌ A × C

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Relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

· relation R from A to B

R

r1,r2 A × B

pposite

R◦ from B to A

R

r2,r1 B × A map A

f

B

A

f=1A,f A × B A f ◦=f,1A B × A

· R ֌ A × B, S ֌ B × C
SR ֌ A × C

· R = r2r◦

1

and Rf = f ◦f ( = f2f ◦

1 )

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Relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

· relation R from A to B

R

r1,r2 A × B

pposite

R◦ from B to A

R

r2,r1 B × A map A

f

B

A

f=1A,f A × B A f ◦=f,1A B × A

· R ֌ A × B, S ֌ B × C
SR ֌ A × C

· R = r2r◦

1

and Rf = f ◦f ( = f2f ◦

1 )

· ff ◦f = f and f ◦ff ◦ = f ◦ ( difunctional )

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Relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

· relation R from A to B

R

r1,r2 A × B

pposite

R◦ from B to A

R

r2,r1 B × A map A

f

B

A

f=1A,f A × B A f ◦=f,1A B × A

· R ֌ A × B, S ֌ B × C
SR ֌ A × C

· R = r2r◦

1

and Rf = f ◦f ( = f2f ◦

1 )

· ff ◦f = f and f ◦ff ◦ = f ◦ ( difunctional ) · ff ◦ = 1B iff f regular epi

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Relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 5 / 18

· relation R from A to B

R

r1,r2 A × B

pposite

R◦ from B to A

R

r2,r1 B × A map A

f

B

A

f=1A,f A × B A f ◦=f,1A B × A

· R ֌ A × B, S ֌ B × C
SR ֌ A × C

· R = r2r◦

1

and Rf = f ◦f ( = f2f ◦

1 )

· ff ◦f = f and f ◦ff ◦ = f ◦ ( difunctional ) · ff ◦ = 1B iff f regular epi

relative

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Equivalence relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

· R ֌ A × A is: reflexive 1A ≤ R

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Equivalence relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

· R ֌ A × A is: reflexive 1A ≤ R symmetric R◦ ≤ R

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Equivalence relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

· R ֌ A × A is: reflexive 1A ≤ R symmetric R◦ ≤ R transitive RR ≤ R

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Equivalence relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

· R ֌ A × A is: reflexive 1A ≤ R symmetric R◦ ≤ R transitive RR ≤ R         equivalence

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Equivalence relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

· R ֌ A × A is: reflexive 1A ≤ R symmetric R◦ ≤ R transitive RR ≤ R         equivalence · Rf effective equivalence relation

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Equivalence relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

· R ֌ A × A is: reflexive 1A ≤ R symmetric R◦ ≤ R transitive RR ≤ R         equivalence · Rf effective equivalence relation · exact fork Rf

f1 f2

A

f B

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Equivalence relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

· R ֌ A × A is: reflexive 1A ≤ R symmetric R◦ ≤ R transitive RR ≤ R         equivalence · Rf effective equivalence relation · exact fork Rf

f1 f2

A

f B

·Rf

f1

f2

A

f

p

❇ ❇ ❇ ❇ B P m

⑤

⑤ ⑤

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Equivalence relations

Contents Preliminaries Regular categories Relations Equivalence relations Regular Mal’tsev categories The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 6 / 18

· R ֌ A × A is: reflexive 1A ≤ R symmetric R◦ ≤ R transitive RR ≤ R         equivalence · Rf effective equivalence relation · exact fork Rf

f1 f2

A

f B

·Rf

f1

f2

A

f

p

❇ ❇ ❇ ❇ B P m

⑤

⑤ ⑤ ⇒ Rf = Rp

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Regular Mal’tsev categories

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 7 / 18

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Definition

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 8 / 18

· C Mal’tsev cat =

lex
reflexive = equivalence

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Definition

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 8 / 18

· C Mal’tsev cat =

lex
reflexive = equivalence

· Exs.

Gp, Alg(T),

T w/ group op.

quasi-groups, Heyting algebras
lex + additive
(Topos)op
C Mal’tsev

⇒ C/X, X/C, Gp(C), · · · Mal’tsev

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Definition

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 8 / 18

· C Mal’tsev cat =

lex
reflexive = equivalence

· Exs.

Gp, Alg(T),

T w/ group op.

quasi-groups, Heyting algebras
lex + additive
(Topos)op
C Mal’tsev

⇒ C/X, X/C, Gp(C), · · · Mal’tsev · Prop. [ CLP, Diagram chasing in Mal’cev cats ] C

regular. TFAE:

(a) C Mal’tsev cat (b) RfRg = RgRf

( RS = SR for equivalence relations )

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Regular pushouts

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 9 / 18

· C

c g

(1)

A

f

D

d t

B,

s

g · t = 1, f · s = 1, c · t = s · d

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Regular pushouts

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 9 / 18

· C

c g

(1)

A

f

D

d t

B,

s

g · t = 1, f · s = 1, c · t = s · d

always a pushout

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Regular pushouts

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 9 / 18

· C

c g

(1)

A

f

D

d t

B,

s

g · t = 1, f · s = 1, c · t = s · d

always a pushout · regular po: g, c : C ։ D ×B A regular epi

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Regular pushouts

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 9 / 18

· C

c g

(1)

A

f

D

d t

B,

s

g · t = 1, f · s = 1, c · t = s · d

always a pushout · regular po: g, c : C ։ D ×B A regular epi ⇔ cg◦ = f ◦d ( gc◦ = d◦f )

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Regular pushouts

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 9 / 18

· C

c g

(1)

A

f

D

d t

B,

s

g · t = 1, f · s = 1, c · t = s · d

always a pushout · regular po: g, c : C ։ D ×B A regular epi ⇔ cg◦ = f ◦d ( gc◦ = d◦f ) Rc

Rd
⇒

gRc = Rd ( gc◦cg◦ = d◦d )

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Regular pushouts

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 9 / 18

· C

c g

(1)

A

f

D

d t

B,

s

g · t = 1, f · s = 1, c · t = s · d

always a pushout · regular po: g, c : C ։ D ×B A regular epi ⇔ cg◦ = f ◦d ( gc◦ = d◦f ) Rc

Rd
⇒

gRc = Rd ( gc◦cg◦ = d◦d ) relative

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Regular pushouts

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 9 / 18

· C

c g

(1)

A

f

D

d t

B,

s

g · t = 1, f · s = 1, c · t = s · d

always a pushout · regular po: g, c : C ։ D ×B A regular epi ⇔ cg◦ = f ◦d ( gc◦ = d◦f ) Rc

Rd
⇒

gRc = Rd ( gc◦cg◦ = d◦d ) relative · Prop. [ Bourn, The denormalised 3 × 3 L ] C

regular. TFAE:

(a) C Mal’tsev cat (b) (1) always regular po

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Regular pushouts

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 9 / 18

· C

c g

(1)

A

f

D

d t

B,

s

g · t = 1, f · s = 1, c · t = s · d

always a pushout · regular po: g, c : C ։ D ×B A regular epi ⇔ cg◦ = f ◦d ( gc◦ = d◦f ) Rc

Rd
⇒

gRc = Rd ( gc◦cg◦ = d◦d ) relative · Prop. [ Bourn, The denormalised 3 × 3 L ] C

regular. TFAE:

(a) C Mal’tsev cat (b) (1) always regular po Proof: calculus of relations

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Stability property

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 10 / 18

· Lem. [ Bourn, The denormalised 3 × 3 L ] C regular Mal’tsev. In: W ×D C

k

γ
◗

◗ ◗ ◗ ◗ ◗ ◗

v

Y ×B A

h

✤ ✤ ✤ ✤

α

P

P P P P P P C

g c

A

f

W

j

δ
◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Y

i

✤ ✤ ✤ ✤

β

◗

◗ ◗ ◗ D

t

d

B

s

v is a regular epi

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Stability property

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 10 / 18

· Lem. [ Bourn, The denormalised 3 × 3 L ] C regular Mal’tsev. In: W ×D C

k

γ
◗

◗ ◗ ◗ ◗ ◗ ◗

v

Y ×B A

h

✤ ✤ ✤ ✤

α

P

P P P P P P C

g c

A

f

W

j

δ
◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Y

i

✤ ✤ ✤ ✤

β

◗

◗ ◗ ◗ D

t

d

B

s

v is a regular epi

· Rem. α, β, γ, δ arbitrary maps and c, d, w, v regular epis

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Stability property

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 10 / 18

· Lem. [ Bourn, The denormalised 3 × 3 L ] C regular Mal’tsev. In: W ×D C

k

γ
◗

◗ ◗ ◗ ◗ ◗ ◗

v

Y ×B A

h

✤ ✤ ✤ ✤

α

P

P P P P P P C

g c

A

f

W

j

δ
◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Y

i

✤ ✤ ✤ ✤

β

◗

◗ ◗ ◗ D

t

d

B

s

v is a regular epi

· Rem. α, β, γ, δ arbitrary maps and c, d, w, v regular epis · Prop. C

regular. TFAE:

(a) C Mal’tsev cat (b) v is a regular epi

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Stability property

Contents Preliminaries Regular Mal’tsev categories Definition Regular pushouts Stability property The Cuboid Lemma The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 10 / 18

· Lem. [ Bourn, The denormalised 3 × 3 L ] C regular Mal’tsev. In: W ×D C

k

γ
◗

◗ ◗ ◗ ◗ ◗ ◗

v

Y ×B A

h

✤ ✤ ✤ ✤

α

P

P P P P P P C

g c

A

f

W

j

δ
◗

◗ ◗ ◗ ◗ ◗ ◗ ◗

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Y

i

✤ ✤ ✤ ✤

β

◗

◗ ◗ ◗ D

t

d

B

s

v is a regular epi

· Prop. C

regular. TFAE:

(a) C Mal’tsev cat (b) v is a regular epi C

g

❍

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

v=g,c D ×B A f ′

✤ ✤ ✤ ✤

d′

❖

❖ ❖ ❖ ❖ ❖ C

g c

A

f

D

t

■

■ ■ ■ ■ ■ ■ ■ ■ ■ D ✤ ✤ ✤ ✤

d

P

P P P D

t

d

B

s

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The Cuboid Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 11 / 18

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn)

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn)

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack)

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR)

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR)

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ ?

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Aim

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 12 / 18

· Aim. Mal’tsev ⇔ stability pp ⇔ Cuboid Lemma · Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ Cuboid Lemma

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3 × 3 Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 13 / 18

· (denormalised) 3 × 3 Lemma: R¯

g ¯ g1

¯

g2

t2
t1

Rg

g1

g2
v

Rf

f1

f2
Rc

c2

c1
¯

g

A

g

c

C

f

S

s2

s1

B

d

D upper row ⇔ lower row exact exact

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3 × 3 Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 13 / 18

· (denormalised) 3 × 3 Lemma: R¯

g ¯ g1

¯

g2

t2
t1

Rg

g1

g2
v

Rf

f1

f2
Rc

c2

c1
¯

g

A

g

c

C

f

S

s2

s1

B

d

D upper row ⇔ lower row exact exact · Gran, Rodelo: Goursat ⇔ 3 × 3 Lemma

SLIDE 53

3 × 3 Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 13 / 18

· (denormalised) 3 × 3 Lemma: R¯

g ¯ g1

¯

g2

t2
t1

Rg

g1

g2
v

Rf

f1

f2
Rc

c2

c1
¯

g

A

g

c

C

f

S

s2

s1

B

d

D upper row ⇔ lower row exact exact · Gran, Rodelo: Goursat ⇔ 3 × 3 Lemma ⇔ split 3 × 3 Lemma

SLIDE 54

3 × 3 Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 13 / 18

· (denormalised) 3 × 3 Lemma: R¯

g ¯ g1

¯

g2

t2
t1

Rg

g1

g2
v

Rf

f1

f2
Rc

c2

c1
¯

g

A

g

c

C

f

S

s2

s1

B

d

D upper row ⇔ lower row exact exact · Gran, Rodelo: Goursat ⇔ 3 × 3 Lemma ⇔ split 3 × 3 Lemma

(1)

↑ Goursat po

SLIDE 55

Stability properties

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 14 / 18

· Goursat po Rg

v

Rf C

c g

(1)

A

f

D

d t

B

s

v is a regular epi

SLIDE 56

Stability properties

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 14 / 18

· Goursat po Rg

v

Rf C

c g

(1)

A

f

D

d t

B

s

v is a regular epi

· Mal’tsev vs Goursat W ×D C

k

γ
◆

◆ ◆ ◆ ◆ ◆

v

Y ×B A

h

✤ ✤ ✤ ✤

α

▼

▼ ▼ ▼ ▼ C

g c

A

f

W

j

δ
◆

◆ ◆ ◆ ◆ ◆

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ Y

i

✤ ✤ ✤ ✤

β

▼

▼ ▼ ▼ D

t

d

B

s

Rg

g1

g2
v

Rf

f1

✤ ✤ ✤ ✤

f2

C

g c

A

f

C
g
■

■ ■ ■ ■

c

❴ ❴ ❴ ❴ ❴ ❴ A ✤ ✤ ✤ ✤

f

■

■ ■ D

t

d

B

s

split pbs

kernel pairs

SLIDE 57

Cuboid Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 15 / 18

· R¯

g ¯ g1

¯

g2

t2
t1

Rg

g1

g2
v

Rf

f1

f2
Rc

c2

c1
¯

g

A

g

c

C

f

S

s2

s1
B

d

D

SLIDE 58

Cuboid Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 15 / 18

· R¯

g ¯ g1

¯

g2

t2
t1

Rg

g1

g2
v

Rf

f1

f2
Rc

c2

c1
¯

g

A

g

c

C

f

S

s2

s1
B

d

D
·

T

¯ k

t1
t2
¯

γ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ V

k✁✁✁ v

γ
✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X

h✂✂✂ α

✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ Rw

¯ j

w1
❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

w2

❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

¯ δ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ W

j

✁

✁ ✁

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

δ

✳

✳ ✳ ✳ ✳ Y

i

✂

✂ ✂

β

✲

✲ ✲ ✲ ✲ Rc

¯ g

✁✁✁✁

c1

c2

C

g✄

✄ ✄ ✄ ✄

c

A

f✄

✄ ✄ ✄ ✄ S

¯ t

✁

✁ ✁ ✁

s1

s2

D

t

✄

✄ ✄ ✄ ✄

d

B

s

✄

✄ ✄ ✄ ✄

SLIDE 59

Cuboid Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 15 / 18

· T

¯ k

t1
t2
¯

γ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ V

k✁✁✁ v

γ
✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X

h✂✂✂ α

✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ Rw

¯ j

w1
❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

w2

❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

¯ δ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ W

j

✁

✁ ✁

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

δ

✳

✳ ✳ ✳ ✳ Y

i

✂

✂ ✂

β

✲

✲ ✲ ✲ ✲ Rc

¯ g

✁✁✁✁

c1

c2

C

g✄

✄ ✄ ✄ ✄

c

A

f✄

✄ ✄ ✄ ✄ S

¯ t

✁

✁ ✁ ✁

s1

s2

D

t

✄

✄ ✄ ✄ ✄

d

B

s

✄

✄ ✄ ✄ ✄ upper row exact ⇔ lower row exact

SLIDE 60

Cuboid Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 15 / 18

· T

¯ k

t1
t2
¯

γ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ V

k✁✁✁ v

γ
✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X

h✂✂✂ α

✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ Rw

¯ j

w1
❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

w2

❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

¯ δ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ W

j

✁

✁ ✁

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

δ

✳

✳ ✳ ✳ ✳ Y

i

✂

✂ ✂

β

✲

✲ ✲ ✲ ✲ Rc

¯ g

✁✁✁✁

c1

c2

C

g✄

✄ ✄ ✄ ✄

c

A

f✄

✄ ✄ ✄ ✄ S

¯ t

✁

✁ ✁ ✁

s1

s2

D

t

✄

✄ ✄ ✄ ✄

d

B

s

✄

✄ ✄ ✄ ✄ upper row exact ⇔ lower row exact · Rem. d regular epi, d · s1 = d · s2, v · t1 = v · t2

SLIDE 61

Cuboid Lemma

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 15 / 18

· T

¯ k

t1
t2
¯

γ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ V

k✁✁✁ v

γ
✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X

h✂✂✂ α

✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ Rw

¯ j

w1
❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

w2

❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

¯ δ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ W

j

✁

✁ ✁

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

δ

✳

✳ ✳ ✳ ✳ Y

i

✂

✂ ✂

β

✲

✲ ✲ ✲ ✲ Rc

¯ g

✁✁✁✁

c1

c2

C

g✄

✄ ✄ ✄ ✄

c

A

f✄

✄ ✄ ✄ ✄ S

¯ t

✁

✁ ✁ ✁

s1

s2

D

t

✄

✄ ✄ ✄ ✄

d

B

s

✄

✄ ✄ ✄ ✄ upper row exact ⇔ lower row exact · Rem. d regular epi, d · s1 = d · s2, v · t1 = v · t2

· lower row exact

⇒ upper row exact: v is a regular epi upper row exact ⇒ lower row exact: S = Rd

SLIDE 62

Main results

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 16 / 18

· Thm. C

regular. TFAE:

(a) C Mal’tsev cat (b) Cuboid Lemma

SLIDE 63

Main results

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 16 / 18

· Thm. C

regular. TFAE:

(a) C Mal’tsev cat (b) Cuboid Lemma T

¯ k

t1
t2
¯

γ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ V

k✁✁✁ v

γ
✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X

h✂

✂

α

✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ Rw

¯ j

w1
❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

w2

❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

¯ δ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ W

j

✁

✁ ✁

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

δ

✳

✳ ✳ ✳ ✳ Y

i

✂

✂

β

✲

✲ ✲ ✲ ✲ Rc

¯ g✁

✁ ✁ ✁

c1

c2

C

g✄✄✄✄ c

A

f✄✄✄✄

S

¯ t

✁

✁ ✁ ✁

s1

s2

D

t

✄

✄ ✄ ✄

d

B

s

✄

✄ ✄ ✄ (a) ⇒ (b) · lower row exact ⇒ upper row exact ?

stability pp: c, d, w regular epis ⇒ v regular epi

SLIDE 64

Main results

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 16 / 18

· Thm. C

regular. TFAE:

(a) C Mal’tsev cat (b) Cuboid Lemma T

¯ k

t1
t2
¯

γ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ V

k✁✁✁ v

γ
✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X

h✂

✂

α

✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ Rw

¯ j

w1
❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

w2

❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

¯ δ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ W

j

✁

✁ ✁

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

δ

✳

✳ ✳ ✳ ✳ Y

i

✂

✂

β

✲

✲ ✲ ✲ ✲ Rc

¯ g✁

✁ ✁ ✁

c1

c2

C

g✄✄✄✄ c

A

f✄✄✄✄

S

¯ t

✁

✁ ✁ ✁

s1

s2

D

t

✄

✄ ✄ ✄

d

B

s

✄

✄ ✄ ✄ (a) ⇒ (b) · lower row exact ⇒ upper row exact ?

stability pp: c, d, w regular epis ⇒ v regular epi

· upper row exact ⇒ lower row exact ?

always true!

SLIDE 65

Main results

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 16 / 18

· Thm. C

regular. TFAE:

(a) C Mal’tsev cat (b) Cuboid Lemma T

¯ k

t1
t2
¯

γ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ V

k✁✁✁ v

γ
✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X

h✂

✂

α

✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ Rw

¯ j

w1
❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

w2

❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

¯ δ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ W

j

✁

✁ ✁

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

δ

✳

✳ ✳ ✳ ✳ Y

i

✂

✂

β

✲

✲ ✲ ✲ ✲ Rc

¯ g✁

✁ ✁ ✁

c1

c2

C

g✄✄✄✄ c

A

f✄✄✄✄

S

¯ t

✁

✁ ✁ ✁

s1

s2

D

t

✄

✄ ✄ ✄

d

B

s

✄

✄ ✄ ✄ (b) ⇒ (a) · cube with stability pp ?

Rw, Rc, S = Rd, T = Rw ×Rd Rc
lower row exact ⇒ upper row exact } v regular epi

SLIDE 66

Main results

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 16 / 18

· Thm. C

regular. TFAE:

(a) C Mal’tsev cat (b) Cuboid Lemma T

¯ k

t1
t2
¯

γ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ V

k✁✁✁ v

γ
✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X

h✂

✂

α

✲

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ Rw

¯ j

w1
❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

w2

❴

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

¯ δ

✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ W

j

✁

✁ ✁

w

❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴

δ

✳

✳ ✳ ✳ ✳ Y

i

✂

✂

β

✲

✲ ✲ ✲ ✲ Rc

¯ g✁

✁ ✁ ✁

c1

c2

C

g✄✄✄✄ c

A

f✄✄✄✄

S

¯ t

✁

✁ ✁ ✁

s1

s2

D

t

✄

✄ ✄ ✄

d

B

s

✄

✄ ✄ ✄ (b) ⇒ (a) · cube with stability pp ?

Rw, Rc, S = Rd, T = Rw ×Rd Rc
lower row exact ⇒ upper row exact } v regular epi

Upper Cuboid Lemma

SLIDE 67

Main results

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma Aim 3 × 3 Lemma Stability properties Cuboid Lemma Main results The relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 16 / 18

· Thm. C

regular. TFAE:

(a) C Mal’tsev cat (b) Cuboid Lemma (c) Upper Cuboid Lemma (b) ⇒ (a) · cube with stability pp ?

Rw, Rc, S = Rd, T = Rw ×Rd Rc
lower row exact ⇒ upper row exact } v regular epi

Upper Cuboid Lemma

SLIDE 68

The relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 17 / 18

SLIDE 69

From the absolute to the relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 18 / 18

· absolute context

relative context

SLIDE 70

From the absolute to the relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 18 / 18

· absolute context

relative context

· Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ Cuboid Lemma (GR)

SLIDE 71

From the absolute to the relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 18 / 18

· absolute context

relative context

· Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ Cuboid Lemma (GR) relative (T. Janelidze)

lex + E class of regular epis sth ...

SLIDE 72

From the absolute to the relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 18 / 18

· absolute context

relative context

· Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ Cuboid Lemma (GR) relative (T. Janelidze)

lex + E class of regular epis sth ...

relative version (Goedecke, T. Janelidze)

SLIDE 73

From the absolute to the relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 18 / 18

· absolute context

relative context

· Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ Cuboid Lemma (GR) relative (T. Janelidze)

lex + E class of regular epis sth ...

relative version (Goedecke, T. Janelidze) relative

E-relations

SLIDE 74

From the absolute to the relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 18 / 18

· absolute context

relative context

· Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ Cuboid Lemma (GR) relative (T. Janelidze)

lex + E class of regular epis sth ...

relative version (Goedecke, T. Janelidze) relative

E-relations

calculus of relations

calculus of E-relations

SLIDE 75

From the absolute to the relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 18 / 18

· absolute context

relative context

· Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ Cuboid Lemma (GR) relative (T. Janelidze)

lex + E class of regular epis sth ...

relative version (Goedecke, T. Janelidze) relative

E-relations

calculus of relations

calculus of E-relations

relative

E-relations

SLIDE 76

From the absolute to the relative context

Contents Preliminaries Regular Mal’tsev categories The Cuboid Lemma The relative context From the absolute to the relative context

9 -13 July 2012 The cuboid lemma and Mal’tsev categories – 18 / 18

· absolute context

relative context

· Regular Goursat cats Mal’tsev cats

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

⇔ (1) regular po (Bourn) ⇒ 3 × 3 Lemma (Bourn) ⇒ 3 × 3 Lemma (Lack) ⇔ (1) Goursat po (GR) ⇔ 3 × 3 Lemma (GR) ⇔ Cuboid Lemma (GR) relative (T. Janelidze)

lex + E class of regular epis sth ...

relative version (Goedecke, T. Janelidze) relative

E-relations

calculus of relations

calculus of E-relations

relative

E-relations

relative version (Everaert, Goedecke

T. Janelidze, Van der Linden)