D IAGRAMMES

C LAUDE H ENRY
C HRISTIAN L AIR
Atlas des Figures
Diagrammes, tome 59-60 (2008), p. 1-36
<http://www.numdam.org/item?id=DIA_2008__59-60__A2_0>

© Université Paris 7, UER math., 2008, tous droits réservés.
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DIAGRAMMES

VOLUMES 59 + 60, FASCICULE 2, 2008

SUR
CERTAINES SOUS-CATEGORIES NON PLEINES
DES
CATEGORIES DE MODELES

Fascicule 2 :

Atlas des Figures

Claude Henry
(maxchen@free.fr)

et
Christian Lair
(lairchrist@aol.com)

A.M.S. Sub. Class. : 18 C 10, 18 C 30, 18 C 35

Compl( E , j )

F

E

.....

¬j

j

¬ ( E, j )

.....

E

Figure 1

1

.....
.

..... I .....

I

...

XI

...

X

...

...

M

. . . . . RI . . . . .

pI , X

...

PI , X

∈ Fl( M )

Figure 2

2

...

M, M '

RI

rI
M

pI , X
r 'I , X

PI , X

∈ Fl(M )
∈ Fl(M ' )
Résol. 1

∀M ∈ Ob( M ' ) = Ob( M ) ∀I ∈ I ∀rI : RI → M ∈ Fl( M )
∃ X ∈ X I ∃ r 'I , X : PI , X → M ∈ Fl( M ' )
r 'I , X .pI , X = rI

Figure 3

3

M, M '

RI

rI
M

m

m.rI

N

∈ Fl(M )
∉ Fl(M ' )
Résol. 2

∀M ∈ Ob( M ) ∀N ∈ Ob( M ) ∀m : M → N ∈ Fl( M )
∀I ∈ I ∀rI : RI → M ∈ Fl( M )
rI : RI → M ∉ Fl( M ' ) ⇒ m.rI ∉ Fl( M ' )

Figure 4

4

M, M '

RI

r 'I
M

m

m.r 'I

N

∈ Fl(M )

∈ Fl(M ' )

Résol. 3

∀M ∈ Ob( M ) ∀N ∈ Ob( M ) ∀m : M → N ∈ Fl( M )
[∀I ∈ I ∀r ' I : RI → M ∈ Fl( M ' ) m . r 'I : RI → N ∈ Fl( M ' )]
⇒
[m : M → N ∈ Fl( M ' )]

Figure 5

5

...

Y

Y1 . . .

M

Q

s
M

qY1

qY2

m

sY2

QY2

N

QY1

tY1

Equil. 1

∀M ∈ Ob( M ) ∀N ∈ Ob( M ) ∀m : M → N ∈ Fl( M )
∀q = (qY : Q → QY ∈ Fl( M )) ∈ q ∀s : Q → M ∈ Fl( M )
( ∃Y1 ∈ Y ∃tY1 : QY1 → N ∈ Fl( M ) m . s = tY1 . qY1 )
⇒
( ∃Y2 ∈ Y ∃sY2 : QY2 → M ∈ Fl( M ) s = sY2 . qY2 )

Figure 6

6

Hj

...

H ...

. . . R " j,H . . .
gH

M"

M"
Ens

∆
f j, H

g
R " j, H ( F j )
M " ( F j)

gH (Fj )

∀M "∈ Ob( M " ) ∀g : ∆ → M " ( F j ) ∈ Fl( Ens )
∃H ∈ H j ∃g H : R " j , H → M "∈ Fl( M " )
gH (F j ) o f j,H = g

Figure 7

7

Ens

R " j , H (F j )

R " j , H (¬ j )

R " j, H ( j )

R " j , H (¬ ( E j , j ))

R " j, H ( E j )

Figure 8

8

∆
δ

f j, H

R " j, H ( F j )

Assert. 7

=

R " j, H ( j )

R " j , H (¬ j )

δ j, H

R " j , H (¬ ( E j , j ))

R " j,H (E j )

∆
f j, H

δ

R " j,H (F j )

Assert. 8

=
R " j, H ( j )
R " j , H (¬ j )

δ ¬ j, H

R " j , H (¬ ( E j , j ))

R " j,H (E j )

Figure 9

9

M"

M"

r "( j , H ), X

U

U'

P "( j , H ), X

M, M '
R( j , H )

r( j , H )

U (M " )
p( j , H ), X

U (r "( j , H ), X )

P( j , H ), X = U ' ( P "( j , H ), X )

∈ Fl(M )
∈ Fl( M " ) ∪ Fl( M ' )

∀M "∈ Ob( M " ) ∀( j , H ) ∈ I ∀r( j , H ) : R( j , H ) → U ( M " ) ∈ Fl( M )
∃ X ∈ X ( j , H ) ∃ r "( j , H ), X : P "( j , H ), X → M "∈ Fl( M " )
U (r "( j , H ), X ).p( j , H ), X = r( j , H )

Figure 10

10

M, M '

R( j , H )

r( j , H )
M

p( j , H ), X
r '( j , H ), X

P( j , H ), X

∈ Fl(M )
∈ Fl( M ' )

∀M ∈ Ob( M ' ) = Ob( M ) ∀( j , H ) ∈ I ∀r( j , H ) : R( j , H ) → M ∈ Fl( M )
∃ X ∈ X ( j , H ) ∃ r '( j , H ), X : P( j , H ), X → M ∈ Fl( M ' )
r (' j , H ), X .p( j , H ), X = r( j , H )

Figure 11

11

R( F j )
r(Fj )

M (Fj )

R " (¬ j )
M " (¬ j )

m( F j )

R( j )

M ( j)

N (F j )
r(E j )

R " (¬ ( E j , j ))
R( E j )

M (E j )

m( E j )

N (E j )

M " (¬ ( E j , j ))

N ( j)

N " (¬ j )

N " (¬ ( E j , j ))

Figure 12

12

R ( F j0 )
r ( F j0 )

M ( F j0 )

R " (¬ j0 )
M " (¬ j0 )

m( F j )
0

R( j0 )

M ( j0 )

N (F j )
0

r(E j )
0

R " (¬ ( E j0 , j0 ))

R( E j )
0

M ( E j0 )

M " (¬ ( E j0 , j0 ))

m( E j )
0

N (E j )
0

N ( j0 )

N " (¬ j0 )

N " (¬ ( E j0 , j0 ))

Figure 13

13

×

R ( F j0 )
r ( F j0 )

M ( F j0 )

R " (¬ j0 )
M " (¬ j0 )

m( F j )
0

R( j0 )

M ( j0 )

N (F j )
0

r(E j )
0

R " (¬ ( E j0 , j0 ))

R( E j )
0

M ( E j0 ) ϕ

m( E j )
0

N (E j )
0

M " (¬ ( E j0 , j0 ))

N ( j0 )

N " (¬ j0 )

N " (¬ ( E j0 , j0 ))

Figure 14

14

×

ε

R ( F j0 )
r ( F j0 )

M ( F j0 )

R " (¬ j0 )
M " (¬ j0 )

m( F j )
0

R( j0 )

M ( j0 )

N (F j )
0

r(E j )
0

R " (¬ ( E j0 , j0 ))

R( E j )
0

M ( E j0 ) ϕ

m( E j )
0

N (E j )
0

ψ

M " (¬ ( E j0 , j0 ))

N ( j0 )

N " (¬ j0 )

N " (¬ ( E j0 , j0 ))

Figure 15

15

×

ε

R ( F j0 )
r ( F j0 )

M ( F j0 )

R " (¬ j0 )
M " (¬ j0 )
m( F j )

R( j0 )

0

m.r ( F j0 )

M ( j0 )

N (F j )
0

r(E j )
0

R " (¬ ( E j0 , j0 ))

R( E j )
0

M ( E j0 ) ϕ

m( E j )
0

N (E j )
0

ψ

×

M " (¬ ( E j0 , j0 ))

N ( j0 )

+

N " (¬ j0 )

N " (¬ ( E j0 , j0 ))

Figure 16

16

ε

M (Fj )

M " (¬ j )

m( F j )
M ( j)

N (F j )

M (E j )

m( E j )

M " (¬ ( E j , j ))
N ( j)

N " (¬ j )

N " (¬ ( E j , j ))

N (E j )

Figure 17

17

∆
ε
f j0 , Hθ

M " (¬ j0 ) . s j0 ,θ
R " j0 , Hθ ( F j0 )
s j0 ,θ

r " j0 , Hθ ( F j0 )

M ( F j0 )

M " (¬ j0 )
M ( j0 )

m( F j )
0

N (F j )
0

M ( E j0 )

θ M " (¬ ( E j0 , j0 ))

m( E j )
0

N ( j0 )

N " (¬ j0 )

N " (¬ ( E j0 , j0 ))

N (E j )
0

Figure 18

18

∆

ε
f j0 , Hθ

R " j0 , Hθ ( F j0 )
r " j0 , Hθ ( F j0 )

M ( F j0 )
s j0 ,θ

R " j0 , Hθ (¬ j0 )
M ( j0 )

M " (¬ j0 )

m( F j )
0

N (F j )
0

δ ¬ j0 , Hθ
R " j0 , Hθ (¬ ( E j0 , j0 ))

M ( E j0 )

θ

m( E j )
0

N (E j )
0

N ( j0 )

M " (¬ ( E j0 , j0 ))

N " (¬ j0 )

N " (¬ ( E j0 , j0 ))

Figure 19

19

r " j0 , Hθ (¬ ( E j0 , j0 ))

R " j0 , Hθ ( F j0 )

r " j0 , H θ ( F j0 )

M ( F j0 )

R " j0 , Hθ (¬ j0 )

M ( j0 )

M " (¬ j0 )

m( F j )
0

N (F j )
0

δ ¬ j0 , Hθ

P

R " j0 , Hθ (¬ ( E j0 , j0 ))

M ( E j0 )

θ

m( E j )
0

N ( j0 )

M " (¬ ( E j0 , j0 ))

r " j0 , Hθ (¬ ( E j0 , j0 ))

m " (¬ ( E j0 , j0 ))

N " (¬ j0 )

m " (¬ ( E j0 , j0 )) . r " j0 , Hθ (¬ ( E j0 , j0 ))

N (E j )
0

N " (¬ ( E j0 , j0 ))

Figure 20

20

I

...

..... I .....

L X 'L

L X "L

X 'I

X "I

...

X I = X ' I U X "I
M

. . . . . RI . . . . .

pI , X '

pI , X "

L PI , X ' L

L PI , X " L

∈ Fl( M ' )
∉ Fl( M ' )

Figure 21

21

M

s

RI

=

m

pI , X "

1

N
tI , X "

1

∈ Fl( M ' )
∉ Fl( M ' )
∈ Fl( M )

Figure 22

22

PI , X "

1

s

M

RI

=
m

Assert. 9

pI , X "

1

=
N
tI , X "

1

∈ Fl( M ' )
∉ Fl( M ' )
∈ Fl( M )

Figure 23

23

PI , X "

1

s

M

sI , X

RI

=
pI , X
pI , X "

m

1

PI , X
N
tI , X "

1

∈ Fl( M ' )
∉ Fl( M ' )
∈ Fl( M )

Figure 24

24

PI , X "

1

s

M

RI

sI , X

Assert. 10
0
p

m

I,X

?

pI , X "

1

PI , X
PI , X "

N

1

tI , X "

1

s

M

RI

× Assert. 9

m Assert. 11

N
tI , X "

1

∈ Fl( M ' )
∉ Fl( M ' )
∈ Fl( M )

Figure 25

25

pI , X "

1

PI , X "

1

M

r 'I

m

N

∈ Fl( M ' )
∈ Fl( M )

Figure 26

26

RI

r 'I

M

RI

=

=
pI , X

m

N
tI , X
∈ Fl( M ' )
∈ Fl( M )

Figure 27

27

PI , X

r 'I

M

RI

=

=
?

m

N
tI , X
∈ Fl( M ' )
∉ Fl( M ' )
∈ Fl( M )

Figure 28

28

? pI , X
PI , X

M

r 'I
r ' I , X "2

RI

=
pI , X "2

?

m

? pI , X "1

PI , X "2
N

tI , X "

1

∈ Fl( M ' )
∉ Fl( M ' )
∈ Fl( M )

Figure 29

29

PI , X "

1

X = X "1

M

r 'I
r ' I , X "2

×

RI

=
pI , X "2

?

m

? pI , X "1

PI , X "2
N

tI , X "

1

∈ Fl( M ' )
∉ Fl( M ' )
∈ Fl( M )

Figure 30

30

PI , X "

1

X = X "1

Ext( E , q )
E

FQ

f qY

FQ

Y

Figure 31

31

Ext( E , q )

Q

E

FQ
M•

qY
f qY
QY
FQ

M

Y

M∩
Ens

Hom M (Q, M ) = M ∩ ( FQ )

Hom M (qY , M ) = M ∩ ( f qY )

Hom M (QY , M ) = M ∩ ( FQY )

Figure 32

32

M

SurExt( E , q )
Ext( E , q )

E

FQ
jq
Eq

f qY
gq

FQ

eq

Y

σ q,Y

Σq

Figure 33

33

SurExt( E , q )
Ext( E , q )
E

FQ

jq

Eq

f qY
gq

FQ

eq

Y

σ q ,Y

Σq
T∧

T

Ens

T ( FQ )
∧

injection canonique = T ( jq )
T ( f qY )

image = T ∧ ( Eq )

T ∧ ( g q ) = unique factorisat ion
surjection canonique = T ∧ (eq )

T ( FQ )
Y

T ∧ (σ q,Y ) = coprojection
T ∧ (Σ q ) = somme canonique = réunion disjointe

Figure 34

34

Compl(SurExt( E , q ), jq )

SurExt( E , q)
Ext( E , q )

¬ ( Eq , jq )
¬ jq

E

FQ
jq

Eq

f qY
gq

FQ

eq

Y

σ q,Y
Σq

V¬

V

Ens

complémentaire = V ¬ (¬ ( Eq , jq ))
V ( FQ )
injection = V ( jq )

image = V ( Eq )

V ( f qY )
V ( g q ) = unique factorisation

surjection = V (eq )

V ( FQ )
Y

V (σ q,Y ) = coprojection
V (Σ q ) = somme

Figure 35

35

V¬
V

complément aire = V ¬ (¬ ( Eq , jq ))
V ( FQ )
injection = V ( jq )

v

V ( f qY )

image = V ( Eq )

v

V ( g q ) = unique factorisation
surjection = V (eq )

?

V ( FQ )
Y

V (σ q ,Y ) = coprojection
V (Σ q ) = somme

W¬
W

v

complément aire = W ¬ (¬ ( Eq , jq ))

v

W ( FQ )
injection = W ( jq )
W ( f qY )

image = W ( Eq )

W ( g q ) = unique factorisation

surjection = W (eq )

W ( FQ )
Y

W (σ q ,Y ) = coprojection
W (Σ q ) = somme

Figure 36

36