υo.  † † ‡ §  id  G S A G G [Z 2 , Z 2 ] = {0, id} S [Z 2 , Z 2 ] = {0, id, e} e(0) = 1, e(1) = 1 A G (A i ) i∈I C C S C P Ob(C S ) = {(α i : A i −→ X) I |X ∈ (C)}, [(α i : A i −→ X) I , (β i : A i −→ Y ) I ] C S = {δ : X −→ Y ∈ [X, Y ] C | β i = δα i , ∀i ∈ I} C S C Ob(C P ) = {(α i : X −→ A i ) I |X ∈ (C)}, [(α i : X −→ A i ) I , (β i : Y −→ A i ) I ] C P = {γ : X −→ Y ∈ [X, Y ] C | β i = γα i , ∀i ∈ I} C P C ∗ [(α i : A i −→ X) I , (β i : A i −→ Y ) I ] C S ⊂ [X, Y ] C ∗ 1 (α i :A i −→X) I = 1 X ∗ C S C 1 (α i :A i −→X) I = 1 X A B C O B U A Ob(O B ) = {(X −→ B)|X ∈ (C)}, [X α −→ B, Y β −→ B] O B = {γ : X −→ Y ∈ [X, Y ] C |βγ = α} Ob(U A ) = {(A −→ Y )|Y ∈ (C)}, [A α −→ Y, A β −→ X] U A = {δ : X −→ Y ∈ [X, Y ] C |δβ = α} α β βα βα αβ α β α C [α] C f g βαf = βαg β αf = αg α f = g βα f g αf = αg βαf = βαg βα f = g α β N α −→ Z β −→ N n −→ n z −→ |z| βα = id N β |z 1 | = |z 2 | z 1 = z 2 N Mor(N) a b ∈ Mor(N) a ∼ b ⇐⇒ a b 2 2 Mor(N) [0], [1] N Ob(N) = {N}, Mor(N) = {[0], [1]} [a], [b] [a][b] = [ab] =    [1] a b [0] N [2] N Mon ) j : N −→ Z g 1 g 2 Z M n ∈ Z g 1 (n) = g 2 (n) g 1 (−n) = g 2 (−n) n −n /∈ N g 1 j = g 2 j j Div q : Q −→ Q/Z qf = qg f g : G −→ Q G qh = 0 h = f − g h(x) x ∈ G h(x) = 0 h  x 2008h(x)  = 1 2008 qh  x 2008h(x)  = 0 qh = 0 h(x) = 0 q A O −→ A A −→ O 1 A a ⇐⇒ b a ⇐⇒ d a =⇒ b) A f g : A −→ X f.0 OA = g.0 OA f = g [A, X] O −→ A (b =⇒ a) O −→ A O −→ A A 0 AO // O 0 OA // A ∗ 0 OA .0 AO = 0 AA ∈ [A, A] 1 AA ∈ [A, A] 0 AA 0 OA = 1 AA 0 OA = 0 OA 0 OA 0 AA = 1 AA ∗ 0 AO .0 OA = 0 OO = 1 OO a ⇐⇒ b a =⇒ d) (d =⇒ a) 1 A A 0 XA ∈ [X, A] f ∈ [X, A] 1 A f = 0 XA = 1 A 0 XA f = 0 XA 1 A a ⇐⇒ d C P p 1 // p 2  B 1 β 1  D 2 β 2 // B [...]... Categories, Types and StructuresD ws ressD ftp://ftp.di.ens.fr/pub/users/longo/CategTypesStructures/book.pdfF SĂch l thƯy cừa cĂc thƯy rƯu hát Ă ti liằu trản 1ãu õ tÔi 1 h http://www.vnmath.com/2009/11/pham-tru-ham-tu.html 28