Hệ nhân tử trong nhóm phạm trù phân bậc

Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (1.08 MB, 116 trang )

(ϕ, f)
(ϕ, f)
Γ
(ϕ, f)
(ϕ, f)
Γ
Γ
Γ
ObG
MorG
(0, g, d)
(1, l, r)
Π = π
0
G
A = π
1
G
S
G
Hom
(ϕ,f)
[S, S

]
(Π, A), (Π, A, k)


Γ
(Π, A, h)
(F,

F )
(F,
˘
F ,

F )
(H,

H), (G,

G)
(H
Γ
,

H
Γ
), (G
Γ
,

G
Γ
)
(R, M), (R, M, h)
H

i
(Π, A)
H
i
Γ
(Π, A)
H
i
MacL
(R, M)
H
i
Shu
(R, M)
M
A
Ext(Π, A, ψ)
M, (B, D, d, θ), B
d
→ D
Ext
B→D
(Q, B, ψ)
Ext
Γ
B→D
(Q, B, ψ)
G
G
G

I
G
(ϕ, f)
S S

(Π, A)
Γ (Π, A)
Γ
Γ
(R, M)
A
Γ
Γ
❅
❅
❅
❅
❅





Γ
Γ
❅
❅
❅

❅
❅
❅
❅
❅
❅
❅
❅
❅
❅
❅















✛
✛
✛
✲

✲
✲
✛
✛
✛
⊃
⊃
⊃
✛
✛
✛
⊃
⊃
⊃
C ⊗ : C × C → C
I ⊗
R R
X Y X ⊗Y Y ⊗X
I
X ⊗ Y = I = Y ⊗ X
G
G Π = π
0
G
G Π A = π
1
G G
Π Π A
(Π, A)
(Π, A)

Γ
Γ = 1
H
3
ab
(M, N)
Γ
Γ Γ
(Π, A)
Γ
R R M
H
3
MacL
(R, M)
(R, M) (R, M)
c
X,X
= id
X
H
3
Shu
(R, M)
G
G
k k
k k M B
H
3

Hoch
(B, M) k
K K
K = Z
Γ Γ Γ
Γ
Γ
(Π, A)
(ϕ, f)
(Π, A) (ϕ, f)
Γ
Γ
Γ
Γ = 1
Γ
Γ
Γ
Γ = 1
Γ
Γ = 1
XY X.Y
X ⊗ Y
(G, ⊗, I, a, l, r) G
⊗ : G × G → G I
a
X,Y,Z
: X ⊗ (Y ⊗ Z) → (X ⊗ Y ) ⊗ Z, l
X
: I ⊗ X → X , r
X

: X ⊗ I → X,
(a
X,Y,Z
⊗ id
T
) a
X,Y ⊗Z,T
(id
X
⊗ a
Y,Z,T
) = a
X⊗Y,Z,T
a
X,Y,Z⊗T
,
id
X
⊗ l
Y
= (r
X
⊗ id
Y
)a
X,I,Y
.
a
l, r
G

G π
0
G G
⊗
I π
1
G = Aut(I)
I
π
1
G π
0
G
su = γ
−1
X
δ
X
(u), X ∈ s, s ∈ π
0
G, u ∈ π
1
G,
γ
X
, δ
X
X X X X
I ⊗ X I ⊗ X X ⊗ I X ⊗ I.
✲

γ
X
(u)
✲
δ
X
(u)
✻
l
X
✲
u⊗id
✻
l
X
✻
r
X
✲
id⊗u
✻
r
X
G k ∈ Z
3
(π
0
G, π
1
G).

S
G
π
0
G (s, u) : s → s, s ∈ π
0
G, u ∈ π
1
G
π
1
G
(s, u) ◦ (s, v) = (s, u + v).
S
G
G
s = [X] ∈ π
0
G X
s
∈ G X
1
= I
X ∈ s i
X
: X
s
→ X i
X
s

= id (X
s
, i
X
)
G
i
I⊗X
s
= l
X
s
, i
X
s
⊗I
= r
X
s
.
(X
s
, i
X
)










G : G → S
G
G(X) = [X] = s
G(X
f
→ Y ) = (s, γ
−1
X
s
(i
−1
Y
fi
X
))









H : S
G

→ G
H(s) = X
s
H(s, u) = γ
X
s
(u).
G H
α = (i
X
) : HG
∼
=
id
G
, β = id : GH
∼
=
id
S
G
.
(G, H, α, β) S
G
s ⊗ t = s.t, s, t ∈ π
0
G,
(s, u) ⊗ (t, v) = (st, u + sv), u, v ∈ π
1
G.

S
G
a
s,r,t
=
(srt, k(s, r, t)) k ∈ Z
3
(π
0
G, π
1
G).
G H

G
X,Y
= G(i
X
⊗ i
Y
) ,

H
s,t
= i
−1
X
s
⊗X
t

: X
s
X
t
→ X
st
.
S
G
G
S
G
(Π, A, k) (Π, A) π
0
G, π
1
G
Π Π A
Γ Γ Π
Π Γ Π
Γ Γ A Π
σ(xa) = (σx)(σa) σ ∈ Γ x ∈ Π a ∈ A Γ f : Π → Π

Γ f(σx) = σf (x) σ ∈ Γ, x ∈ Π
Γ ∗
Γ Γ
G Γ gr : G → Γ
X ∈ Ob G σ ∈ Γ f
G X gr(f) = σ
Γ G = (G, gr, ⊗, I, a, l, r)

Γ (G, gr) Γ ⊗ : G ×
Γ
G → G
I : Γ → G
a
X,Y,Z
: (X ⊗ Y ) ⊗ Z
∼
→ X ⊗ (Y ⊗ Z), l
X
: I ⊗ X
∼
→
X, r
X
: X ⊗ I
∼
→ X
G
G G
G
G c
c = (c
X,Y
) : X ⊗ Y → Y ⊗ X a, l, r
(id
Y
⊗ c
X,Z
)a

Y,X,Z
(c
X,Y
⊗ id
Z
) = a
Y,Z,X
c
X,Y ⊗Z
a
X,Y,Z
,
(c
X,Z
⊗ id
Y
)a
−1
X,Z,Y
(id
X
⊗ c
Y,Z
) = a
−1
Z,X,Y
c
X⊗Y,Z
a
−1

X,Y,Z
.
c c
X,Y
◦ c
Y,X
= id
Y ⊗X
Γ Γ
(G, gr) Γ ⊗ : G ×
Γ
G → G I : Γ → G
a
X,Y,Z
: (X ⊗ Y ) ⊗ Z
∼
→ X ⊗ (Y ⊗ Z), l
X
: I ⊗ X
∼
→ X, r
X
: X ⊗ I
∼
→ X
c
X,Y
: X ⊗ Y
∼
→ Y ⊗ X I = I(∗)

G G Γ
G = (G, ⊗, I, a, l, r) G

= (G

, ⊗, I

, a

, l

, r

)
G G

(F,

F , F
∗
) F : G → G

F
∗
I

F I

F
X,Y

: F X ⊗ F Y → F (X ⊗ Y )
F (a
X,Y,Z
) ◦

F
X,Y Z
◦ (id
F X
⊗

F
Y,Z
) =

F
X⊗Y,Z
◦ (

F
X,Y
⊗ id
F Z
) ◦ a

F X,F Y,F Z
,
r

F X

= F (r
X
) ◦

F
X,I
◦ (id
F X
⊗ F
∗
), l

F X
= F (l
X
) ◦

F
I,X
◦ (F
∗
⊗ id
F X
).
α : (F,

F , F
∗
) → (F


,

F

, F

∗
)
G G

α : F → F

F

∗
= α
I
◦ F
∗
α
X⊗Y
◦

F
X,Y
=

F

X,Y

◦ (α
X
⊗ α
Y
).
F :
G → G

F

: G

→ G α : F

◦ F →
id
G
β : F ◦ F

→ id
G

(F,

F , F
∗
) F
G, G

Γ Γ

(F,

F , F
∗
) : G → G

Γ F : G → G


F
X,Y
: F X ⊗ FY → F (X ⊗ Y ), F
∗
: I

→ F I
(F,

F , F
∗
), (F

,

F

, F

∗
) Γ

Γ α : F
∼
→ F

α : F
∼
→ F

α
X
: F X → F

X
(G, c), (G

, c

) (F,

F ) :
G → G

c, c


F
Y,X
c

F X,F Y

= F (c
X,Y
)

F
X,Y
.
(G, gr) (G

, gr

) Γ
Γ (G, gr) (G

, gr

) (F,

F , F
∗
) F :
(G, gr) → (G

, gr

) Γ

F
X,Y
: F X ⊗ F Y → F (X ⊗ Y )

F
∗
: I

→ F I
(F,

F , F
∗
), (F

,

F

, F

∗
) Γ
Γ α : F
∼
→ F

A ⊕, ⊗ : A × A → A
0 ∈ Ob A a
+
, c, g, d (A, ⊕, a
+
, c, (0, g, d))
1 ∈ Ob A a, l, r (A, ⊗, a, (1, l, r))

L, R
L
A,X,Y
: A ⊗ (X ⊕ Y ) −→ (A ⊗ X) ⊕ (A ⊗ Y )
R
X,Y,A
: (X ⊕ Y ) ⊗ A −→ (X ⊗ A) ⊕ (Y ⊗ A)
A ∈ Ob A (L
A
,
˘
L
A
), (R
A
,
˘
R
A
)
L
A
= A ⊗ − R
A
= − ⊗ A
˘
L
A
X,Y
= L

A,X,Y
˘
R
A
X,Y
= R
X,Y,A
⊕ a
+
c
A, B, X, Y ∈ Ob A
(AB)(X ⊕ Y ) A(B(X ⊕ Y )) A(BX ⊕ BY )
(AB)X ⊕ (AB)Y A(BX) ⊕ A(BY )
❄
˘
L
AB
✛
a
A,B,X ⊕Y
✲
id
A
⊗
˘
L
B
❄
˘
L

A
✛
a
A,B,X
⊕a
A,B,Y
(X ⊕ Y )(BA) ((X ⊕ Y )B)A (XB ⊕ Y B)A
X(BA) ⊕ Y (BA) (XB)A ⊕ (Y B)A
❄
˘
R
BA
✲
a
X⊕Y,B,A
✲
˘
R
B
⊗id
A
❄
˘
R
A
✲
a
X,B,A
⊕a
Y,B,A

(A(X ⊕ Y )B A((X ⊕ Y )B) A(XB ⊕ Y B)
(AX ⊕ AY )B (AX)B ⊕ (AY )B A(XB) ⊕ A(Y B)
❄
˘
L
A
⊗id
B
✛
a
A,X⊕Y,B
✲
id
A
⊗
˘
R
B
❄
˘
L
A
✲
˘
R
B
✛
a⊕a
(A ⊕ B)X ⊕ (A ⊕ B)Y (A ⊕ B)(X ⊕ Y ) A(X ⊕ Y ) ⊕ B(X ⊕ Y )
(AX ⊕ BX) ⊕ (AY ⊕ BY ) (AX ⊕ AY ) ⊕ (BX ⊕ BY )

❄
˘
R
X
⊕
˘
R
Y
✛
˘
L
A⊕B
✲
˘
R
X⊕Y
❄
˘
L
A
⊕
˘
L
B
✲
v
v = v
U,V,Z,T
: (U ⊕ V ) ⊕ (Z ⊕ T ) −→ (U ⊕ Z) ⊕ (V ⊕ T )
⊕, a

+
, c, id (A, ⊕)
1 ∈ Ob A ⊗
1(X ⊕ Y )
1X ⊕ 1Y
(X ⊕ Y )1
X1 ⊕ Y 1
X ⊕ Y X ⊕ Y.
✲
˘
L
1
◗
◗
◗
◗s
l
X⊕Y
✑
✑
✑
✑✰
l
X
⊕l
Y
✲
˘
R
1

◗
◗
◗
◗s
r
X⊕Y
✑
✑
✑
✑✰
r
X
⊕r
Y
A
c
X,X
= id
A A

A A

(F,
˘
F ,

F , F
∗
) F : A → A


˘
F
X,Y
: F (X ⊕ Y ) → F (X) ⊕ F (Y );

F
X,Y
: F (X ⊗ Y ) → F (X) ⊗ F (Y )
F
∗
: F(1) → 1

(F,
˘
F )
⊕ (F,

F , F
∗
)
⊗
F (X(Y ⊕ Z)) F X.F (Y ⊕ Z) FX(F Y ⊕ F Z)
F (XY ⊕ XZ) F (XY ) ⊕ F(XZ)
F X.F Y ⊕ F X.F Z,
❄
F (L)
✲

F
✲

id⊕

F
❄
L

✲
˘
F
✲

F ⊕

F
F ((X ⊕ Y )Z) F (X ⊕ Y ).F Z (F X ⊕ F Y )F Z
F (XZ ⊕ Y Z) F (XZ) ⊕ F(Y Z)
F X.F Z ⊕ F Y.F Z.
❄
F (R)
✲

F
✲
˘
F ⊗id
❄
R

✲
˘

F
✲

F ⊕

F
F
α : (F,
˘
F ,

F , F
∗
) → (F

,
˘
F

,

F

, F

∗
)
⊕ ⊗
(F


,
˘
F

,

F

, F

∗
) : A

→ A
F

F
∼
→ id
A
, F F

∼
→ id
A

(F,
˘
F ,


F , F
∗
) A
A

A R = π
0
A A
+, × ⊕, ⊗ A M = π
1
A = Aut(0)
0
+ M R
S
A
A
π
0
A (s, u) : s →
s, s ∈ π
0
A, u ∈ π
1
A
(s, u) ◦ (s, v) = (s, u + v).
s ∈ π
0
A X
s
∈ Ob A X

0
= 0, X
1
= 1
i
X
: X → X
s
i
X
s
= id
X
s
A (X
s
, i
X
)
i
0⊕X
s
= g
X
s
, i
X
s
⊕0
= d

X
s
,
i
1⊗X
s
= l
X
s
, i
X
s
⊗1
= r
X
s
, i
0⊗X
s
=

R
X
s
, i
X
s
⊗0
=


L
X
s
.
A S
A









G : A → S
A
G(X) = [X] = s
G(X
f
→ Y ) = (s, γ
−1
X
s
(i
Y
fi
−1
X
))










H : S
A
→ A
H(s) = X
s
H(s, u) = γ
X
s
(u)
X, Y ∈ s f : X → Y γ
X
: Aut(0) → Aut(X)
γ
X
(u) = g
X
◦ (u ⊕ id) ◦ g
−1
X
.
S

A
s ⊕ t = G(H(s) ⊕ H(t)) = s + t,
s ⊗ t = G(H(s) ⊗ H(t)) = st,
(s, u) ⊕ (t, v) = G(H(s, u) ⊕ H(t, v)) = (s + t, u + v),
(s, u) ⊗ (t, v) = G(H(su) ⊗ H(t, v)) = (st, sv + ut),
s, t ∈ π
0
A u, v ∈ π
1
A A (0, id, id)
(1, id, id) h = (ξ, η, α, λ, ρ)
a
+
, c, a, L, R A H
˘
H = i
−1
X
s
⊕X
t
,

H = i
−1
X
s
⊗X
t
.