exposé

Xe
X
x∈X π1(X, x)
x
π1(X, x)
π1
f: (Y, y)→(X, x)f∗π1(Y, y)
(X, x)
π1(X, x)H⊂π1(X, x)
e
X/H f :Y→X
f∗π1(Y, y)y
X
π1(X, x)
f:Y→X Yx=f−1(x)
π1(X, x)
g= [γ]γ: [0,1] →X
x y ∈f−1(x)eγ: [0,1] →Yeγ(0) = y
f◦eγ=γ g.y =eγ(1) f:Y→X
f−1(x)X
π1(X, x)π1(X, x)E
e
X×E π1(X, x)
π1(X, x)
xe X
e
X/H →X H π1(X, x)
π1(X, x)/H e
X/H e X/N N =∩g∈π1gHg−1
π1(X, x)H
(X, x)x π1

π1bπ1
X
X
X
X→XqY
XqX→X
Spec()→Spec(k)/k
k f :A1\ {0} → A1\ {0}f(x) = xnn k
k p > 0f:A1→A1f(x) = xp−x
n:A→A n k
X= Spec(Z)X=Pn
k
CD={Xα//Y
β,γ ////Z}
C C =Ens α
X x0∈X0β(x0) = γ(x0)
D(T) =
{X(T)//Y(T)////Z(T)}T X(T) = HomC(T, X)
α(β, γ)C
C
C,C0
F:C→C0F
F
F
F D

X Y
ooZ
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C
α:X0→XXX0
ooX0×XX0
oooo
CX∈CG⊂AutC(X)
Y∈CHomC(X, Y )G g.f =f◦g−1
F:C→Ens
F(Y) = HomC(X, Y )G.
X/G
X G
CX= (Xi)i∈I
Pro(C)
X= (Xi)i∈IY= (Yj)j∈J
HomPro(C)(X, Y )d´ef = lim
←−
j
lim
−→
i
HomC(Xi, Yj).
C C
Pro(C)C Ens
Pro(C)
E
C
C
C
u:Y→XCYY0→X
X
F:C→E
F
F
uCF(u)
F
F
X
Spec(Ω) →X
C(π)
π F :C(π)→E

CF
π F
F:C→C(π)
C
FP= (Pi)i∈I
πAut(P)◦
FC(π)F
G:C(π)→CE(E×P)/π
F
F
u:Y→X F (u)F
u∆u:Y→Y×XY
u:Y→X F (u)F
Eu F (u)
F(u)YY0→X Y →Y0
u0:Y0→X X F (u)
F(u0)F(u0)
u0u
C. . . →X2→X1→X0
. . . →F(X2)→F(X1)→F(X0)F(X0)
Xn
X=∅XC
X P ∈C
C
P P P
X u :X→P X P0→P
X→P0X P 0P0→P
P u
P=XqY X
X →P
Y
X=P1q···qPr=Q1qQsPi
QjF(X)F(Qj)j F (Pi)F(Qj)
F(Pi×XQj) = F(Pi)∩F(Qj)Pi×XQj
Pi→X Qj→X Pi'Qj
F X Y
X→YC

X X qX
(X, x)X∈Cx∈F(X)u:X→Y u(x)
F(u)(x)
(P, p) (X, x)P u : (P, p)→
(X, x)u, v i :Y→P
F(i) : F(Y)→F(P)F(u)F(v)
F(p)∈F(Y)F(Y)6=∅Y6=∅P i
u=v
(X1, x1),...,(Xr, xr) (P, p)P
(Xi, xi)X=X1×···×Xrx= (x1, . . . , xr)
(X, x)X=P1q ··· q Pr
F(X)F(Pi)j
x∈F(Pj)P=Pj
X(P, p)P X
evp: Hom(P, X)→F(X)u u(p)I
(P, p)P= (Pi)i∈I∈Pro(C)
Hom(P,·) = lim
−→Hom(P, ·)→F
F(X) = {x1, . . . , xr}(X, xi)
u:P→X P X X =YqZ
Y y ∈F(Y)p∈F(P)F(u)
(Q, q)Q(Y, y) (P, p)
Q→P→X Q →Y→X q y
Q→P Q →P→X
Q→Y→X F (X) = F(Y)Z
PHom(P, P ) = Aut(P)P
u:P→P P P0→P P 06=∅P0→P
u F (u)F(P)
u
P
evp: Aut(P)→F(P)Pevp
pAut(P)
F(P)
u=i◦f=j◦g ϕ :Z→Z0
Y×ZY
Y Z pr1,pr2
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fHom(Z, Z0)//Hom(Y, Z0)////Hom(Y×ZY, Z0)
j◦g◦pr1=u◦pr1=u◦pr2=j◦g◦pr2g◦pr1=g◦pr2
j ϕ :Z→Z0
g=ϕ◦f f g ψ :Z0→Z
f=ψ◦g f g ϕ ψ