On the 3 × 3 lemma
Diana Rodelo
Marino Gran
& Zurab Janelidze
AIM: (denormalized) 3 × 3 lemma for star-regular cats
PLAN:
1. (Denormalized) 3 × 3 lemma and known results
2. Star-regular cats and basic notions
3. The (upper/lower and middle) 3 × 3 lemmas for starregular cats and necessary conditions
4. The equivalence of the upper and lower 3 × 3 lemmas
5. The middle 3 × 3 lemma
CatAlg 2011, 28th of September of 2011
1. (Denormalized) 3 × 3 lemma and known results
· Classical (abelian cats)
•_
/
_
•_
/
•
/
•
•
/
•
_
3 cols ses
/
•
•
/
•_
_
+ middle row ses
3 × 3 lemma: upper row ses ⇐⇒ lower row ses
upper 3 × 3 lemma: upper row ses ⇐= lower row ses
lower 3 × 3 lemma: upper row ses =⇒ lower row ses
+ middle row null
middle 3 × 3 lemma: upper and lower rows ses =⇒ middle row ses
· Known results:
[ZJ, The pointed subobject fnct, 3 × 3 lemmas and subtractivity of spans, 2010]
normal cat (pointed, regular cat sth regular epis = normal epis)
- upper 3×3 lemma ⇐⇒ lower 3×3 lemma ⇐⇒ subtractivity
- middle 3 × 3 lemma ⇐⇒ protomodularity
2
1. (Denormalized) 3 × 3 lemma and known results
· Denormalized (non-pointed context)
ses → exact fork
•
//
•
/
•
•
// •
/
•
_
_
// /
•
•
3 cols ef
_
•
+ middle row ef
3 × 3 lemma: upper row ef ⇐⇒ lower row ef
upper 3 × 3 lemma: upper row ef ⇐= lower row ef
lower 3 × 3 lemma: upper row ef =⇒ lower row ef
(+ middle row equal)
middle 3 × 3 lemma: upper and lower rows ef =⇒ middle row ef
· Known results:
[DB, The denormalized 3 × 3 lemma, 2001]
- 3 × 3 lemma holds for regular Mal’tsev cats
[SL, The 3-by-3 lemma for regular Goursat categories, 2004]
- 3 × 3 lemma holds for regular Goursat cats
[MG & DR, A new characterisation of Goursat categories, 2010]
regular cat
- upper 3 × 3 lemma ⇐⇒ lower 3 × 3 lemma ⇐⇒ Goursat
- middle 3 × 3 lemma always true
3
2. Star-regular cats and basic results
normal cats
regular cats
,
r
star regular cats
[MG, ZJ, AU, A good theory of ideals in a regular multi-pointed cat, 2011]
C lex
& N ideal
· N -kernel (of f ):
· star:
(f ∈ N or g ∈ N ⇒ gf ∈ N )
k /
X
K
σ = [σ1 , σ2 ]
S
Def. C star-regular:
f /
Y
σ1 ∈N /
/
σ2
fk ∈ N
& k univ.
X
- C regular
- ∀ map f , ∃ N -kernel of f
- regular epi = coequalizer of a star
pointed context: N = class of null ms (C pointed)
- N -kernel = kernel
- star = morphism
- star-regular cat = normal cat
total context: N = class of all ms
- N -kernel = identity
- star = pair of parallel ms
- star-regular cat = regular cat
4
2. Star-regular cats and basic results
C lex
& N ideal
· N -kernel (of f ):
· star:
(f ∈ N or g ∈ N ⇒ gf ∈ N )
σ = [σ1 , σ2 ]
· star-pullback:
σ1 ∈N /
/
σ2
S
σ
S
//
X
g
fk ∈ N
& k univ.
X
f σi = τi g & (g, σ) univ.
f
T
· constellation:
f /
Y
/k / X
K
τ
//
β
Y
H
//
E
α
εi βj = ϕj αi
ε
F
ϕ
//
X
· universal cstl. (over (ε, ϕ)):
· star-kernel (of f ):
K
· star-exact sequence:
κ∗f
K
//
cstl. sth (α, β) univ.
f /
X
κ∗f
//
X
f κ1 = f κ2 & κ∗f univ.
Y
f ,2
Y
f = coeq(κ∗f )
pointed context
total context
star-pb = pb
star-pb = joint pb
cstl. = comm. square
ex. of cstl.: double equiv.
universal cstl. = pb
ex. of universal cstl.: H = F E
star-kernel = kernel
star-kernel = kernel pair
star-es = ses
star-es = exact fork
5
2. Star-regular cats and basic results
P
e
λ
//
X
c
f
2
1
K
κ
//
Y
d
/
C
m
/Q
Lem 1. C regular w/ N -kernels
(a) κ = κ∗d , m mono.
1 star-pb iff λ = κ∗c
(b) c = coeq(λ), e epi.
2
po iff d = coeq(κ)
Rem. C star-regular
f regular epi ⇒ f = coeq(κ∗f )
Lem 2. C star-regular, κ = κ∗d , c regular epi.
3-out-of-2 pp for:
(a)
m mono.
(b)
λ = κ∗c
(c)
1 star-pb
6
3. The 3×3 lemmas for star-regular cats & nec conds
· C star-regular cat
· star-es
look like denormalized 3 × 3 lemmas
generalizes (de)normalized 3 × 3 lemmas
· 3 × 3 diagram
H
κ∗a
F
a
β
ϕ
//
δ
/G
_
W
3 cols star es
κ∗g
κ∗e
X
e
//
b
E
1
_
D
//
f
/
2
Y
d
/
_
g
Z
upper 3 × 3 lemma: middle + lower rows star-es
=⇒ upper row star-es
lower 3 × 3 lemma: upper + middle rows star-es =⇒ lower row star-es
+ f ϕ1 = f ϕ2
middle 3 × 3 lemma: upper + lower rows star-es =⇒ middle row star-es
· Necessary conditions:
all cols and rows star-es
- 1 universal constellation & 2 po
3 × 3 diagram + middle row star-es
- 2 po ⇐⇒ d = coeq(δ)
+ C has enough trivial objs
(“harmless”)
- 1 universal constellation ⇐⇒ β = κ∗b ⇐⇒ δ monic star
7
3. The 3×3 lemmas for star-regular cats & nec conds
Def. X N -trivial
iff 1X ∈ N (∀ X
/
·, ·
/
X ∈ N)
pointed context
total context
X N -trivial ⇒ X = 0
all objs are N -trivial
Rem. 1.
⇓
2.
gf ∈ N , f regular epi ⇒ g ∈ N
N -trivial objs closed for quotients
Def. C has enough trivial objects:
- N closed ideal:
·
∈N
=
/
·
!
·
N trivial
- N -trivial objs closed for subobjs
- X N -trivial =⇒ X × X N -trivial
Prop. C regular w/ N -kernels. TFAE:
(a) C has enough trivial objs
(b)
R
(c)
N
(d)
H
ρ1 ∈N /
/
ρ2 ∈N
n
/
X monic bi-star =⇒ R N -trivial
S
β
//
(s1 ,s2 )
/
/
X ×X
E
α
sth s1 n, s2 n ∈ N
=⇒ β star
ε monic star
F
ϕ
//
X
star
8
=⇒ n ∈ N
3. The 3×3 lemmas for star-regular cats & nec conds
Thm. C star-regular w/ enough trivial objs.
H
κ∗a
//
κ∗f
δ
//
X
/
G
κ∗e
e
_
D
//
b
E
1
F
a
β
W
γ monic star
f
/Y
g
2
/
d
Z
TFAE:
(a)
1
universal constellation
(b) β = κ∗b
(c) δ monic star
Lem.
ε /
/
- E
- E
X
ε=κ∗e
//
X
star iff
e
/
W
E×E
ε×ε /
/
iff E × E
X ×X
ε×ε=κ∗e×e
star
//
e×e
X ×X
/
W ×W
Prop.
H
β
//
⇐⇒
E
α
F
ϕ
(β1 ,β2 )
/E
×E
α
ε monic star
H
ε×ε
// X
F
universal constellation
/
(ϕ1 ,ϕ2 )
X ×X
star-pb
upper 3 × 3 lemma: middle + lower rows star-es
=⇒ b regular epi
lower 3 × 3 lemma: upper + middle rows star-es =⇒ δ = κ∗d
9
4. The equivalence of upper and lower 3 × 3 lemmas
Aim. upper 3 × 3 lemma
⇐⇒ lower 3 × 3 lemma
⇐⇒ subtractivity
Goursat
)
w
symmetric
saturation
property
Def. regular diamond
e
x ;
W
d &
X
Z
f
%
Y
y = g
- left saturated: e(κ∗f ) = κ∗d
- right saturated: f (κ∗e ) = κ∗g
- saturated: left saturated + right saturated
Def. C has symmetric saturation property:
left saturated = right saturated = saturated
Thm. C star-regular w/ enough trivial objs. TFAE:
(a) upper 3 × 3 lemma holds
(b) lower 3 × 3 lemma holds
(c) C has symmetric saturation pp
pointed context
total context
symm. sat. pp = subtractivity
symm. sat. pp = Goursat pp
10
5. The middle 3 × 3 lemmas
Aim. middle 3 × 3 lemma
⇐⇒ protomodularity
always true
)
v
short
five
lemma
short five lemma.
F
a
κ∗f
D
κ∗d
//
//
X
f ,2
e
W
d
,2
rows star-es
Y
g
Z
a and g isos ⇒ e iso
pointed context
total context
short 5 lemma = classical one
short 5 lemma always true
Lem. C star-regular
short 5 lemma =⇒ short 5 lemma for regular epis
Thm. C star-regular w/ enough trivial objs
short 5 lemma ⇐⇒ short 5 lemma for regular epis
Prop. C star-regular w/ enough trivial objs
short 5 lemma =⇒ middle 3 × 3 lemma
⇐=
extra condition
11
5. The middle 3 × 3 lemmas
Def. f saturating
KX
N ker
b ,2
X
X
f
∀f
/
N ker
Y
total context
X
X
/
0
b regular epi
/Y
f
pointed context
0
KY
Y
X
f ,2
Y
,2
Y
f
regular epi
Thm. C star-regular w/ enough trivial objs
&
saturating regular epis
TFAE:
(a) The middle 3 × 3 lemma holds
(b) The short five lemma holds
(c) The short five lemma for regular epis holds
12