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)
F
j j
¬
. . .
E) ,
(E j
Compl
) ,
(E j
¬ E
. . .
. . . .
Figure 1
RI
X
PI, X
pI,
. . . . . .
. . . . . .
I
X
. . . . . .
. . . . . .
I
XI
M
. . . . . .
) (M
∈Fl
Figure 2
RI
X
PI, X
pI,
'
,M
M
M
rI
X
r'I,
Résol. 1
I X I X I
X I X I I
I I
r p
r
M P
r X
M R
r I
M
=
∈
→
∃
∈
∃
∈
→
∀
∈
∀
=
∈
∀
, ,
, ,
'
) ' ( :
'
) ( :
) ( ) ' (
.
Fl
Fl Ob
Ob
M X
M I
M M
) (M
∈Fl
) '
(M
∈Fl
Figure 3
RI
'
,M
M
M
rI
Résol. 2
) ' ( )
' ( :
) ( :
) ( :
) ( )
(
M M
M I
M M
M
Fl . Fl
Fl
Fl Ob
Ob
∉
∉ ⇒
→
∈
→
∀
∈
∀
∈
→
∀
∈
∀
∈
∀
I I
I
I I
r m M
R r
M R
r I
N M m N
M
) (M
∈Fl
) '
(M
∉Fl N
m m.rI
Figure 4
RI
'
,M
M
M
r'I
Résol. 3
)]
' ( :
[
)]
' ( :
' )
' ( :
' [
) ( :
) ( )
(
M
M M
I
M M
M
Fl
Fl .
Fl
Fl Ob
Ob
∈
→ ⇒
∈
→
∈
→
∀
∈
∀ ∀ ∈ ∀ ∈ ∀ → ∈
N M m
N R r m M
R r I
N M m N
M
I I I
I
) '
(M
∈Fl N
m m.r'I
)
(M
∈Fl
Figure 5
Q
Y1
Q
Y1
q M
s
Y2
s
N tY1
Y2
Q m
Y2
q
Y1
. . . . . .
Y
M
Equil. 1
) )
( :
(
) )
( :
(
) ( :
)) ( :
(
) ( :
) ( )
(
2 2 2
2
1 1 1
1
2 1
Y Y Y
Y
Y Y Y
Y Y Y
q s s M
Q s Y
q t s m N
Q t Y
M Q s Q
Q q q
N M m N
M
. . .
=
∈
→
∃
∈
∃ ⇒
=
∈
→
∃
∈
∃
∈
→
∀
∈
∈
→
=
∀
∈
→
∀
∈
∀
∈
∀
M Y
M Y
M q
M
M M
M
Fl Fl
Fl Fl
Fl Ob
Ob
Figure 6
j) F M"(
H
fj,
)
( j
H F g
∆
g
Ens
) (
"j,H Fj R
"
M
. . .
H. . .
Hj
g f
F g
M R
g H
F M g
M
H j j H
H j H j
j
=
∈
→
∃
∈
∃
∈
→
∀
∈
∀
, ,
) (
)
"
(
"
"
:
) ( ) (
"
: )
"
(
"
o
M H
Ens M
Fl Fl
Ob ∆
"
M
H
R"j,
. . . . . .
gH
Figure 7
)
"j,H Fj
R (
) (
" , j R j H )
(
" , j
R j H ¬
Ens
)) , ( (
" , E j
R j H ¬ j R"j,H(Ej)
Figure 8
∆
δ fj,H
) (
"j,H Fj R
j,H
δ
) (
" , j R jH )
(
" , j
R jH ¬
)) , ( (
" , E j
R j H ¬ j R"j,H (Ej)
Assert. 7
=
∆
δ fj,H
) (
"j,H Fj R
) (
" , j
R jH ¬
) (
" , j R jH
)) , ( (
" , E j
R jH ¬ j R"j,H (Ej)
H j,
δ¬
Assert. 8
=
Figure 9
) , (j H
R
)
"
(
' ( , ),
), ,
(jH X P j H X
P =U
X H
p(j, ),
'
,M
M
)
"
(M U
) , (jH
r
)
"
(r (j,H),X U
) , ( ), , ( ), ,
), , ( ), , ( ) , (
) , ( ) , ( )
"
(
)
"
(
"
"
:
"
) ( )
"
( :
) , )
"
(
"
H j X H j X H j
X H j X H j H
j
H j H j
r p
r
M P
r X
M R
r H
j M
=
∈
→
∃
∈
∃
∈
→
∀
∈
∀
∈
∀
.
Fl
Fl (
Ob
U (
M X
M U
I M
(M)
∈Fl
) ' ( )
"
(M Fl M
Fl ∪
∈
X H
P"(j, ),
"
M
X H
r"(j, ),
"
M
U U'
Figure 10
) , (j H
R
X H
P(j, ), X H
p(j, ),
'
,M
M
M
) , (j H
r
X H
r'(j, ),
) , ( ), , ( ), ,
), , ( ), , ( ) , (
) , ( ) , (
'
) ' ( :
'
) ( :
) , ) ( ) ' (
H j X H j X H j
X H j X H j H
j
H j H j
r p
r
M P
r X
M R
r H
j M
=
∈
→
∃
∈
∃
∈
→
∀
∈
∀
=
∈
∀
.
Fl
Fl (
Ob Ob
(
M X
M I
M M
) (M
∈Fl
) '
(M
∈Fl
Figure 11
) (Ej R )
(Ej M
) (j M
) (Ej
m N(j) ) (Ej r
) (Fj M
) (Fj N
) (Fj R )
(Fj r
) (Fj
m R(j)
) (Ej
N N"(¬(Ej, j))
)) , ( (
" E j
R ¬ j ) (
" j R ¬ )
(
" j M ¬
) (
" j N ¬
)) , ( (
" E j
M ¬ j
Figure 12
) (Ej0
R )
(Ej0
M
) (j0 M
) (Ej0
m N(j0) ) (Ej0
r
) (Fj0
M
) (Fj0
N
) (Fj0
R )
(Fj0
r
) (Fj0
m R(j0)
) (Ej0
N
×
××
×
)) , ( (
" 0
0 j E M ¬ j
)) , ( (
" 0
0 j E N ¬ j
)) , ( (
" 0
0 j E R ¬ j
) (
" j0 R ¬ )
(
" j0 M ¬
) (
" j0 N ¬
Figure 13
) (Ej0
R )
(Ej0
M
) (j0 M
) (Ej0
m N(j0) ) (Ej0
r
) (Fj0
M
) (Fj0
N
) (Fj0
R )
(Fj0
r
) (Fj0
m R(j0)
) (Ej0
N
×
××
×
)) , ( (
" 0
0 j E M ¬ j
)) , ( (
" 0
0 j E N ¬ j
)) , ( (
" 0
0 j E R ¬ j
) (
" j0 R ¬ )
(
" j0 M ¬
) (
" j0 N ¬
ε ϕ
Figure 14
) (Ej0
R )
(Ej0
M
) (j0 M
) (Ej0
m N(j0) ) (Ej0
r
) (Fj0
M
) (Fj0
N
) (Fj0
R )
(Fj0
r
) (Fj0
m R(j0)
) (Ej0
N
×
××
×
)) , ( (
" 0
0 j E M ¬ j
)) , ( (
" 0
0 j E N ¬ j
)) , ( (
" 0
0 j E R ¬ j
) (
" j0 R ¬ )
(
" j0 M ¬
) (
" j0 N ¬
ε ϕ
ψ
Figure 15
) (Ej0
R )
(Ej0
M
) (j0 M
) (Ej0
m N(j0) ) (Ej0
r
) (Fj0
M
) (Fj0
N
) (Fj0
R )
(Fj0
r
) (Fj0
m R(j0)
) (Ej0
N
×
××
×
)) , ( (
" 0
0 j E M ¬ j
)) , ( (
" 0
0 j E N ¬ j
)) , ( (
" 0
0 j E R ¬ j
) (
" j0 R ¬ )
(
" j0 M ¬
) (
" j0 N ¬
ε ϕ
ψ
) (Fj0
r m
.
+
Figure 16
) (Ej M
) (j M
) (Ej
m N(j)
) (Fj M
) (Fj N
) (Fj m
) (Ej
N N"(¬(Ej, j))
) (
" j M ¬
) (
" j N ¬
)) , ( (
" E j
M ¬ j
Figure 17
) (Ej0
M
) (j0 M
) (Ej0
m N(j0) ) (Fj0
M
) (Fj0
N
) (Ej0
N "( ( , 0))
0 j E N ¬ j
) (
" j0 M ¬
) (
" j0 N ¬
)) , ( (
" 0
0 j E M ¬ j
∆
θ
0, sj
ε
Hθ
fj ,
0
) (Fj0
m
) (
"
0 0,H j
j F
R θ
) (
"
0 0,H j
j F
r θ θ
,
0) 0
(
" j sj
M ¬
.
θ
Figure 18
) (Ej0
M
) (j0 M
) (Ej0
m N(j0) ) (Fj0
M
) (Fj0 N
) (Ej0
N "( ( , 0))
0 j E N ¬ j
) (
" j0 M ¬
) (
" j0 N ¬
)) , ( (
" E 0 j0 M ¬ j
∆
,θ j0
s ε
Hθ
fj ,
0
) (Fj0 m
) (
"
0
0,H j
j F
R θ
) (
"
0
0,H j
j F
r θ
) (
" , 0
0 j
R j Hθ ¬
)) , ( (
" , 0
0
0 E j
R j Hθ ¬ j δ j ,Hθ
¬ 0
)) , ( (
" , 0
0
0 E j
r j Hθ ¬ j θ
Figure 19
) (Ej0
M
) (j0 M
) (Ej0
m N(j0) ) (Fj0
M
) (Fj0 N
) (Ej0
N "( ( , 0))
0 j E N ¬ j
) (
" j0 M ¬
) (
" j0 N ¬
)) , ( (
" 0
0 j E M ¬ j )
(Fj0 m
) (
"
0
0,H j
j F
R θ
) (
"
0
0,H j
j F
r θ
) (
" , 0
0 j
R j Hθ ¬
)) , ( (
" , 0
0
0 E j
R j Hθ ¬ j δ j ,Hθ
¬ 0
θ " , ( ( , 0))
0
0 E j
r j Hθ ¬ j
P
)) , ( (
" 0
0 j E m ¬ j
)) , ( (
"
)) , ( (
" 0 , 0
0 0
0 j r E j
E
m ¬ j
.
j Hθ ¬ jFigure 20
RI
' ,X
pI
. . . . . .
M
) ' (M
∉Fl
I
. . . . . .
I
'I
X
. . . . . .
"I
X L LX"
L LX'
U
I II X' X"
X =
L
LPI,X' LPI,X"L
"
,X
pI
) (M'
∈Fl
Figure 21
RI
) ' (M
∉Fl
"1 ,X
PI
"1 ,X
pI
) ' (M
∈Fl M
N
) (M
∈Fl s
m
"1 ,X
tI
=
Figure 22
Assert. 9
RI
) (M'
∉Fl
"1 ,X
PI
"1 ,X
pI
) ' (M
∈Fl M
N
) (M
∈Fl s
m
"1 ,X
tI
=
=
Figure 23
=
RI
) (M'
∉Fl
"1 ,X
PI
"1 ,X
pI
) ' (M
∈Fl M
N
) (M
∈Fl s
m
"1 ,X
tI
X
pI,
X
PI, X
sI,
Figure 24
Assert. 11
X
pI,
?
RI
) '
(M
∉Fl
) '
(M
∈Fl M
)
(M
∈Fl s
X
PI, X
sI,
RI
"1 ,X
PI
"1 ,X
pI
M
N
s
"1 ,X
tI
×× ×
×
"1 ,X
pI
N m
"1 ,X
tI I,X"1
P
m
Assert. 10 0
Assert. 9
Figure 25
RI
) ' (M
∈Fl M
N
) (M
∈Fl r'I
m
Figure 26
RI
X
PI, X
pI,
) ' (M
∈Fl M
N
) (M
∈Fl r'I
m
X
tI,
= =
Figure 27
RI
) (M'
∉Fl
X
PI, X
pI,
) ' (M
∈Fl M
N
) (M
∈Fl r'I
m
X
tI,
= =
? ?
Figure 28
=
RI
) ' (M
∉Fl
"1 ,X
PI
"1 ,X
pI
) ' (M
∈Fl M
N
) (M
∈Fl r'I
m
"1 ,X
tI
"2
,X
pI
"2
,X
PI
"2
'I,X
r
? ?
"1
X X =
Figure 29
=
RI
) ' (M
∉Fl
"1 ,X
PI
"1 ,X
pI
) ' (M
∈Fl M
N
) (M
∈Fl r'I
m
"1 ,X
tI
"2
,X
pI
"2
,X
PI
"2
'I,X
r
? ?
"1
X X =
×
××
×
Figure 30
E ) ,
(E q
Ext
FQ
QY
F
qY
f
Figure 31
) ( )
,
(Q M =M∩ FQ
HomM
E ) ,
(E q
Ext
FQ
QY
F
qY
f
Ens
∩ M M
) ( )
,
(qY M =M∩ fqY
HomM
) ( )
,
(QY M =M∩ FQY
HomM
Q
QY
qY
M •
M
Figure 32
jq
gq
E ) ,
(E q
Ext ) ,
(E q
SurExt
FQ
QY
F
qY
f
Σq
eq
Eq
Y
σq,
Figure 33
jq
gq
E ) , (E q Ext
) , (E q SurExt
FQ
QY F
qY
f
Σq eq
Eq
Ens
) (FQ T
) (fqY
T
disjointe réunion
canonique somme
)
( = =
∧ q
T Σ
∧ T T
Y
σq,
) (FQY T )
( canonique
injection =T∧ jq
) ( canonique
surjection =T∧ eq
ion factorisat unique
)
( =
∧ gq T )
( image=T∧ Eq
on coprojecti )
( , =
∧ qY
T σ
Figure 34
) (FQ V
somme ) ( q = V Σ
) ( injection=V jq
jq
gq
q,Y
σ
E ) , (E q Ext
) , (E q SurExt
) ), , ( (SurExt E q jq Compl
FQ
QY F
qY
f
Σq eq
Eq )
, (Eq jq
¬
jq
¬
on coprojecti )
( q,Y = V σ Ens
) (fqY V V
) (FQY ) V
( surjection=V eq
)) , ( ( aire
complément =V¬¬ Eq jq V¬
ion factorisat unique ) (gq = V )
( image=V Eq
Figure 35
v
?
) (FQ V
somme ) ( q = V Σ
) ( injection=V jq
on coprojecti )
( q,Y = Vσ
) (fqY V
) (FQY ) V
( surjection=Veq
ion factorisat unique ) (gq = V ))
, ( ( aire
complément =V¬¬ Eq jq
) (FQ W
somme ) ( q = W Σ
on coprojecti )
( q,Y = W σ
) (fqY
W
) (FQY W
ion factorisat unique ) (gq = W ) ( injection=W jq
) ( image=V Eq
)) , ( ( aire
complément =W¬¬ Eq jq
) ( image=W Eq
) ( surjection=W eq
v
v v
V
W V¬
W¬
Figure 36
