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2 1. Convenient category WHK C2 WHZ C2 WHZ C2 = WHK C2 = WH (C 2 ) Theorem 1 X X BBT- C2 : X X C2 X, x (x, x)
3 2. D C 2 WH A C 2 D WH. B f : D W D D W WH W D C λ Λ D λ D D = λ Λ D λ D D WHK C2, WHZ C2 = WH (C 2 ) WHZ(C 2 ), WH (C 2 ), WHZ(C 2 ) memo: classes
4 3. J. F. K. Proposition 2 A D WH D B C D B C D = WHK D D Corollary 3 WHK C2 A WH A WHK C2 WHK D = D WHK C2 WHZ C2 WH (C 2 ) memo: Kennison 1965, Vogt
5 4. A = {a : A X } X O X A- a A ( a 1 (O) A ) A S A a solution set a A s S b ( a = s b, b ) Proposition 4 S A O A- O S- ( O O S ) s, b ( a 1 O = b 1 s 1 O ) O a A O O O A- O O S O S a A
6 5. 2 (Prop. 2) 1 D X = { f MWH Sf D, Tf = X } X - D- D A, B, C D D X SD X SD X = A P(X ) { 1(A,O) X : (A, O) X (A, O) D } f : D X (f D X ) 1 f (D) X f f (D) : D f (D) X f (D) D (f (D), O) (f (D),O) 1 X f (f (D),O) : D (f (D), O) X B f (f (D),O) f (f (D),O) SD X [Prop. 2 ] Prop. 4 f : D X f = ε X f kd X : D k D X X 1) k D X = (X, O SDX ) D (A,O) (A, O) k 1 SD D X X X B C WHK D = D 2) ε X = 1 k DX X : k D X X (ι X ) ε X k D : WH D ι : D WH WH(ιD, X ) = D(D, k D X )( = WH(ιD, ιk D X ) ), ε X ιg g
7 C 2X X -D = C 2 X WH SC 2X = {1 L X : L X 1L X C 2X } C 2X φ = 1 φ(k) X φ φ(k) : K φ(k) X 3 P(X, Y ) { 1 dx X 1 Y : dx Y X Y } { 1 X φ : X K X Y φ C 2Y } Y WH SP(X, Y ) = { 1 dx X 1 Y : dx Y X Y } { 1 X 1 L Y 1L Y SC 2Y } BBT- X C2 Y (X Y, O SP(X,Y ) ) dx = (X, P(X )) BBT-
8 7. 4 ( ) 4 F(Y, Z) Top (, Z) C cpt(, Z) { φ : C cpt(y, Z) C cpt(k, Z) φ C 2Y } Y WH SF(Y, Z) = { (1 L Y ) : C cpt(y, Z) C cpt(l, Z) 1 L Y SC 2Y } C 2M (Y, Z) SF(Y, Z) O SF(Y,Z) C 2M (Y, Z) = (Top (Y, Z), O SF(Y,Z) ) BBT- Proposition 5 Y WH, X, Z Top e : Top (X C2 Y, Z) Top (X, C 2M (Y, Z)), f e(f ) : x f (x, ) meno: Y, Z WH C 2M (Y, Z) WH Escardo-Lawson-Simpson
9 8. e : Top (dx Y, Z) Top (dx, (Top (Y, Z), O)), g [x g(x, )] O well defined
10 9. the cetralizer of C 2 (WH, C2 ) C 2 the centralizer of C 2 ) WHZ(C 2 ) WHZ(C 2 ) = { X K C 2 ( K C2 X = X C2 K ) } WHZ(C 2 ) P(X, K) {1 X K } X C2 K = X K WHZ(C 2 ) = { X K C 2 ( K C2 X = K X ) } BBT- WHZ(C 2 ) WHZ(C 2 ) WH BBT- WH (C 2 ), WHZ C2 WH (C 2 ) = { X WH C2 : X X C2 X, x (x, x) } WHZ C2 = WHZ(C 2 ) WH (C 2 )
11 10. WHZ C2 is a convenient category WHZ C2 WH A, B, C ι : WHZ C2 = WHK WHZC2 WH k WHZC2 z C2 z C2 (X Y ) = z C2 X C2 z C2 Y = z C2 (X C2 Y ) WHZ C2 WH e : Map(X C2 Y, Z) = Map(X, Map(Y, Z)) MacLane Proposition 2 p.186 in chap. VII Haus, CGHaus, K WH, WHZ C2, z C2 Theorem 6 WHZ C2
12 11. WH (C 2 ) WHK C2!!! Lemma 7 Y WH (C 2 ) F Y C 2Y - C2 (F ) Y C2 Y (1 dy Y 1 Y ) 1 C2 (F ) = y F {y} {y} dy Y Y T 1 (1 Y φ) 1 C2 (F ) = G(φ φ 1 (F ) ) = G(φ) (Y φ 1 (F )) Theorem 1 WHK C2 = WHZ C2 = WH (C 2 ) F Y WH (C 2 ) C 2Y - F = 1 C 2 ( C2 (F )) ( WHK C2 WHZ C2 ) WH (C 2 ) WHK C2 memo: Y WH (C 2 ) Y WH C2 : Y Y C2 Y, y (y, y)
13 12. WHK C2 = WHZ C2 = WH (C 2 ) WHK C2 WHZ C2 WH (C 2 ) WH WH
14 13. some colimits WH Top WHZ C2 (= WHK C2 ) McCord (1968 Transaction) WHK C2 Proposition 8 f : (X, A) (Y, B) Top X, B WHZ C2 A, Y WHZ C2 Proposition 9 X X 0 X 1 X n X n X n WHZ C2 X WHZ C2
15 14. a Lemma Lemma 10 f : (X, A) (Y, B) 8 f L X f L : L X Y 1) f 1 (f (L)) = f 1 (f (L)) L = (f 1 (f (L A)) f 1 (f (L A))) L = f 1 (f (L A)) L L A L C 2 L A C 2 fb L A C 2B B WHZ C2 f (L A) = fb L A (L A) B Y f (L) Y 4) L = 1-point f Y T 1 5) F L F C 2 f (F ) f L Y T 1 memo: f (A) f (X A) B (Y B) = def. : A f A = f 1 f (A) X A f
16 15. Prop. 8 Proposition 8 f : (X, A) (Y, B) Top X, B WHZ C2 A, Y WHZ C2 1) 8 1 B Y, f : (B X, B A) (Y, B) f 2) A WHZ C2 B Y WH Y WHZ C2 3) Y WH φ : K Y G(φ) = { (k, φ(k)) k K } K Y [ ] (φ, 1 K ) : K Y K = G(φ), [ ] p Y : Y K Y p Y (G(φ)) = φ(k) 4) K 1 K f : K X K Y (1 K f ) 1 G(φ) K X
17 16. 5) K X = K C2 X (1 dk K 1 X ) 1 (1 K f ) 1 G(φ) = k K {k} f 1 (φ(k)) dk X Lemma 8 Y T 1 6) (1 K 1 L Y ) 1 (1 K f ) 1 G(φ) = (1 K f L ) 1 G(φ) K L 7) f L : L X Y (1 K f L ) G(φ) (1 K f L ) 1 G(φ) K L K L
18 17. Proposition 11 (1) X, Y WH k C2 (X Y ) X C2 Y X Y (2) X, Y WHK C2 k C2 (X Y ) = X C2 Y (1) O X Y P(X, Y )- C 2X Y - (2) SP(X, Y ) B, C X C2 Y WHK C2 (1) WHK C2 WHZ C2 WH
19 18. K C 2, Y WH SP(K, Y ) (dk Y ) (K k C2 Y ) K C2 Y
20 19. 1 X A = {a : A X } A a A p(a) = p(a, x 1,, x N ) A = { a p(a) } O X A- q(o) { O q(o) } O A = { O P(X ) q(o) } O A X, Y X C2 Y = (X Y, O P(X,Y ) ) X C2 Y C2
21 20. 2 A = C 2 φ, K, O 1 X, O φ, K, O 1 C 2 X,O φ, K, O 1, X, O MTop K, O 1 C 2 X, O O P(X ) C 2 X,O - q(o) φ K O 1 ( φ, K, O 1 C 2 X,O φ 1 O O 1 )
1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1.2 R A 1.3 X : (1)X (2)X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f
1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1. R A 1.3 X : (1)X ()X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f 1 (A) f X X f 1 (A) = X f 1 (A) = A a A f f(x) = a x
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