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❖r❞❡r❡❞ ❋❛❝❡ ❙tr✉❝t✉r❡s ❛♥❞ ▼❛♥②✲t♦✲♦♥❡ ❈♦♠♣✉t❛❞s

▼❛r❡❦ ❩❛✇❛❞♦✇s❦✐ ❏✉♥❡ ✷✷✱ ✷✵✵✼✱ ❈❛r✈♦❡✐r♦✱ P♦rt✉❣❛❧

▼❛♥②✲t♦✲♦♥❡ ❝♦♠♣✉t❛❞s ❛r❡ ω✲❝❛t❡❣♦r✐❡s t❤❛t ❛r❡ ❢r❡❡ ✬❧❡✈❡❧✇✐s❡✬ ✐✳❡✳ ✇❡ ❛❞❥♦✐♥ n + 1 ❞✐♠❡♥✲ s✐♦♥❛❧ ❣❡♥❡r❛t♦rs ✭❂ ✐♥❞❡ts✮ ❛♥❞ ✇❡ ✜① t❤❡✐r ❞♦♠❛✐♥s ❛♥❞ ❝♦❞♦♠❛✐♥s ♦♥❧② ❛❢t❡r ❣❡♥❡r❛t✐♦♥ ♦♥ ❛❧❧ n ❞✐♠❡♥s✐♦♥❛❧ ❝❡❧❧✳ ▼♦r❡♦✈❡r ✇❡ ✐♥✲ s✐sts t❤❛t t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ✐♥❞❡ts ❛r❡ ✐♥❞❡ts ❛❣❛✐♥ ✭❂ ♠❛♥②✲t♦✲♦♥❡✮✳ ❲❤② ♠❛♥②✲t♦✲♦♥❡ ❝♦♠♣✉t❛❞s❄ ❚❤❡② s❡❡♠ t♦ ❜❡ ✐♥ t❤❡ ❝❡♥t❡r ♦❢ t✇♦ ❛♣♣r♦❛❝❤❡s t♦ ✇❡❛❦ ω✲❝❛t❡❣♦r✐❡s✿ ♠✉❧t✐t♦♣✐❝ ❛♥❞ ♦♣❡t♦♣✐❝✳ ❚❤❡ ❝❛t❡❣♦r② Compm/1 ♦❢ ♠❛♥②✲t♦✲♦♥❡ ❝♦♠✲ ♣✉t❛❞s ❛♥❞ ❝♦♠♣✉t❛❞s ♠❛♣ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝❛t❡❣♦r② ♦❢ ♠✉❧t✐t♦♣✐❝ s❡ts MltSets✳✳✳ ✳✳✳❛♥❞ ♣r♦❜❛❜❧② t♦ t❤❡ ❝❛t❡❣♦r② ♦❢ ♦♣❡t♦♣✐❝ s❡ts✱ ❛s ✇❡❧❧✳

✶

❖r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s ❛r❡ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡s ❞❡s❝r✐❜✐♥❣ t❤❡ ✬t②♣❡s✬ ♦❢ ✭❛❧❧✮ ❝❡❧❧s ✐♥ ♠❛♥②✲t♦✲♦♥❡ ❝♦♠♣✉t❛❞s✳ ❊①❛♠♣❧❡s✳

• ✲
• x6
• ✒

❅ ❅ ❅ ❅ ❘

⇓

• ✒
• x2

❅ ❅ ❅ ❅ ❘ ✲

x1

✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✶

x3 ⇓ ⇓a1 s0

✲

x0 ❉♦♠❛✐♥s✿ δ(a1) = {x2, x3} ❈♦❞♦♠❛✐♥s✿ γ(a1) = x1 ❚✇♦ ♦r❞❡rs✳ ▲♦✇❡r✿ x1 <− x0 ❛s γ(x1) ∈ δ(x0)✱ ✳✳✳ ❛♥❞ ❜② tr❛♥s✐t✐✈✐t②✿ x6 <− x0 ❯♣♣❡r✿ x3 <+ x1 ❛s x3 ∈ δ(a3) ❛♥❞ γ(a3) = x1✱ ✳✳✳ ❛♥❞ ❜② tr❛♥s✐t✐✈✐t②✿ x6 <− x1

✷

❍♦✇❡✈❡r ✇❡ ♠❛② ❤❛✈❡ ❡♠♣t②✲❞♦♠❛✐♥ ❢❛❝❡s✳✳✳ ❛♥❞ ❤❡♥❝❡ ❡♠♣t② ❢❛❝❡s ❛♥❞ ❧♦♦♣s s3 s1

✲

x5 s2 x7

• ✒

x6

❅ ❅ ❅ ❘

⇓a5 s4 x8

• ✒

s0 x2

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✲

x0 ⇓a1

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ a4 x4

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ a3 x3 ⇒

✔ ✕ ✟ ✟ ✟ ✟ ✙

a6 x9 ⇐

✗ ✖ ❍❍❍ ❍ ❥

a2 x1

δ(a2) = 1s0✱ γ(a2) = x1 ❛♥❞ ♠♦r❡ ❧♦♦♣s

✫ ✪

⇓b1

✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❑

s

y0

✫ ✪

⇓b0

✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❑

s

y0

= ⇒ β

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ b3 y2

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ b2 y1

❲❤❛t ❛❜♦✉t ❧♦✇❡r ♦r❞❡r ♦❢ x3 ❛♥❞ x4❄✳✳✳ ♦r y2 ❛♥❞ y3❄ ❲❡ ♥❡❡❞ t❤✐s ♦r❞❡r ❛s ❛♥ ❛❞❞✐t✐♦♥❛❧ ♣❛rt ♦❢ ❞❛t❛ <∼ ❝♦♥t❛✐♥❡❞ ✐♥ <−✦ ❲❡ ❤❛✈❡ x4 <∼ x3 ❜✉t ♥♦t x3 <∼ x4✦ ❙✐♠✐❧❛r❧② y2 <∼ y1✳

✸

❉❛t❛ ❢♦r ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s ❢❛❝❡s✿ {Sn}n∈ω❀ Sn ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ❢❛❝❡s ♦❢ ❞✐♠❡♥s✐♦♥ n❀ ❛❧♠♦st ❛❧❧ Sn✬s ❛r❡ ❡♠♣t②❀ ❞♦♠❛✐♥ r❡❧❛t✐♦♥ δ✿ δ(α) ✐s ❡✐t❤❡r ❛ ✜♥✐t❡ ♥♦♥✲ ❡♠♣t② s❡t ♦❢ ❢❛❝❡s ♦r ❛♥ ❡♠♣t② ❢❛❝❡s❀ ❇❡❧♦✇ ✇❡ ❤❛✈❡✿ δ(a2) = 1s0 ❛♥❞ δ(a1) = {x1, x2, x3, x4, x5, x8, x9} ❝♦❞♦♠❛✐♥ ❢✉♥❝t✐♦♥ γ✿ ❡✳❣✳ γ(a1) = x0 ❧♦✇❡r ♦r❞❡r <∼✿ ❡✳❣✳ x4 <∼ x3 s3 s1

✲

x5 s4 x8

• ✒

s0 x2

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✲

x0 ⇓a1

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ a4 x4

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ a3 x3 ⇒

✔ ✕ ✟ ✟ ✟ ✟ ✙

a6 x9 ⇐

✗ ✖ ❍❍❍ ❍ ❥

a2 x1

❚❤❡ ✉♣♣❡r ♦r❞❡r <+ ✐s ❞❡✜♥❛❜❧❡ ❢r♦♠ γ ❛♥❞ δ✳

✹

❆①✐♦♠s ❢♦r ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s ■♥ t❤❡ ♦r❞❡r❡❞ ❢❛❝❡s str✉❝t✉r❡ s3 s1

✲

x3 s2 x5

• ✒

x4

❅ ❅ ❅ ❘

⇓a3

s4 x6

• ✒

s0 x1

❅ ❅ ❅ ❘ ✲

x0

✏✏✏✏✏✏✏✏✏✏✏✏ ✏ ✶

x2

⇓a2 ⇓a1

= ⇒ α s3 s1 s2 x5

• ✒

x4

❅ ❅ ❅ ❘

⇓a0

s4 x6

• ✒

s0 x1

❅ ❅ ❅ ❘ ✲

x0 ✇❡ ❤❛✈❡ γγ(α) = x0✱ δδ(α) = {x1, x2, x3, x4, x5, x6} δγ(α) = {x1, x4, x5, x6}✱ γδ(α) = {x0, x2, x3}

• ❧♦❜✉❧❛r✐t② ❛①✐♦♠ ✭♣♦s✐t✐✈❡ ❝❛s❡✮

γγ(α) = γδ(α) − δδ(α)✱ δγ(α) = δδ(α) − γδ(α)

✺

❇✉t ✇❤❡♥ ✇❡ ❤❛✈❡ ❧♦♦♣s ✐♥ t❤❡ ❞♦♠❛✐♥ ❛s ✐♥ b1 ♦r ❡♠♣t②✲❞♦♠❛✐♥ ❧♦♦♣ ❛s ✐♥ b2

✫ ✪

⇓b1

✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❑

s

y0

✫ ✪

⇓b0

✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❑

s

y0

= ⇒ β

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ b3 y2

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ b2 y1

✇❡ ❤❛✈❡ γγ(bi) = δδ(bi) = δγ(bi) = γδ(bi) = s✱ ❛♥❞ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛s ❞♦❡s ♥♦t ✇♦r❦✳ ❲❡ ❤❛✈❡ t♦ ❞r♦♣ ❜♦t❤ ❧♦♦♣s ❛♥❞ ❡♠♣t② ❢❛❝❡s✳

• ❧♦❜✉❧❛r✐t② ❛①✐♦♠

γγ(α) = γδ(α) − δ ˙ δ−λ(α) δγ(α) ≡1 δδ(α) − γ ˙ δ−λ(α) ≡1 ✐s ✬❡q✉❛❧✐t②✬ t❤❛t ✐❣♥♦r❡s ❡♠♣t② ❢❛❝❡s✱ ✐✳❡✳ t❤❡ ❡♠♣t② ❢❛❝❡s t❤❛t ♠✐❣❤t ♦❝❝✉r ♦♥ t❤❡ r✐❣❤t s✐❞❡ ♦❢ t❤❡ s✐❣♥ ≡1 ♠✉st ❜❡ ❡♠♣t② ♦♥ ❡t❤❡r ❞♦♠❛✐♥ ♦r ❝♦❞♦♠❛✐♥ ♦❢ ❛ ❢❛❝❡ t❤❛t ❜❡❧♦♥❣s t♦ t❤❡ ❧❡❢t s✐❞❡✳

✻

❖t❤❡r ❛①✐♦♠s ♦❢ ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s t❛❧❦s ❛❜♦✉t t❤❡ ✉♣♣❡r <+ ❛♥❞ ❧♦✇❡r <∼ ♦r❞❡rs✳ ❚❤❡② ❛r❡ str✐❝t✱ ❞✐s❥♦✐♥t ❛♥❞ <∼ ✐s ♠❛①✐♠❛❧ s✉❝❤ ❝♦♥t❛✐♥❡❞ ✐♥ <−✳ ❚❤❡ ✉♣♣❡r ♦r❞❡r ♦♥ 0✲❝❡❧❧s ✐s ❧✐♥❡❛r✳ ◆♦ t✇♦ ❢❛❝❡s ✐♥ ❛ ❞♦♠❛✐♥ ♦❢ ❛ ❢❛❝❡ ♠✐❣❤t ❜❡ ❝♦♠♣❛r❛❜❧❡ ✐♥ t❤❡ ✉♣♣❡r ♦r❞❡r <+✳ ■♥❝✐❞❡♥t ❢❛❝❡s ♠✉st ❜❡ ❝♦♠♣❛r❛❜❧❡ ✐♥ ♦♥❡ ♦❢ t❤❡s❡ ♦r❞❡rs✳ ❊✈❡r② ❧♦♦♣ ♠✉st ❜❡ ✜❧❧❡❞ ✐♥✱ ✐✳❡✳ ♠✉st ❜❡ ❛ ❝♦❞♦♠❛✐♥ ♦❢ ❛ ❝❡❧❧ ✇❤✐❝❤ ✐s ♥♦t ❛ ❧♦♦♣✳

✼

❚❤❡r❡ ❛r❡ t✇♦ ❜❛s✐❝ ❦✐♥❞s ♦❢ ♠♦r♣❤✐s♠s ♦❢ ♦r✲ ❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s✳ ❆ ❧♦❝❛❧ ♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s f : S → T ✐s ❛ ❢❛♠✐❧② ♦❢ ❢✉♥❝t✐♦♥s fk : Sk → Tk✱ ❢♦r k ∈ ω✱ s✉❝❤ t❤❛t t❤❡ ❞✐❛❣r❛♠s Sk Tk

✲

fk Sk+1 Tk+1

✲

fk+1

❄

γ

❄

γ Sk ⊔ · 1Sk−1 Tk ⊔ · 1Tk−1

✲

fk + 1fk−1 Sk+1 Tk+1

✲

fk+1

❄

δ

❄δ

❝♦♠♠✉t❡✳ ❋♦r t❤❡ r✐❣❤t sq✉❛r❡ ✐t ♠❡❛♥s ♠♦r❡ t❤❡♥ ❝♦♠♠✉t❛t✐♦♥ ♦❢ r❡❧❛t✐♦♥s✱ ✇❡ ❞❡♠❛♥❞ t❤❛t ❢♦r ❛♥② a ∈ S≥1✱ fa : (˙ δ(a), <∼) − → (˙ δ(f(a)), <∼) ❜❡ ❛♥ ♦r❞❡r ✐s♦♠♦r♣❤✐s♠✱ ✇❤❡r❡ fa ✐s t❤❡ r❡✲ str✐❝t✐♦♥ ♦❢ f t♦ ˙ δ(a) ✭✐❢ δ(a) = 1u ✇❡ ♠❡❛♥ ❜② t❤❛t δ(f(a)) = 1f(u)✮✳ ❆ ❣❧♦❜❛❧ ✭♠♦♥♦t♦♥❡✮ ♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s f : S → T ✐s ❛ ❧♦❝❛❧ ♠♦r♣❤✐s♠ t❤❛t ♣r❡s❡r✈❡s ❧♦✇❡r ♦r❞❡r <∼ ✭❣❧♦❜❛❧❧②✮✳

✽

❊①❛♠♣❧❡s✳ f1 : T1 → S ✐s ♠♦♥♦t♦♥❡✿ S : s

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ b y

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ a x

T1 : s2 s1

✲

y

s0

✲

x

f2 : T2 → S ✐s ♥♦t ♠♦♥♦t♦♥❡ ❜✉t ✐t ✐s ❧♦❝❛❧✿ T2 : s s

✲

x s

✲

y s

✲

y s

✲

x

✖✕ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❖

⇓ b y

✖✕ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❖

⇓ a x

✖✕ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❖

⇓ b y

❚❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s ❛r❡ ♥♦t ✐s♦♠♦r♣❤✐❝ ✭❣❧♦❜❛❧❧②✮ ❜✉t t❤❡② ❛r❡ ✐s♦♠♦r✲ ♣❤✐❝ ❧♦❝❛❧❧②✿

✫ ✪

⇓b

✁ ✁ ✁ ✁ ✁ ✁ ✻

s

z

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ c y

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ a x

✫ ✪

⇓b

❆ ❆ ❆ ❆ ❆ ❆ ❑

s

z

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ c y

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ a x z<∼x x<∼z

✾

• Fs ✭oFsloc✮ ✲ ✐s t❤❡ ❝❛t❡❣♦r② ♦❢ ♦r❞❡r❡❞ ❢❛❝❡

str✉❝t✉r❡s ❛♥❞ ♠♦♥♦t♦♥❡ ✭❧♦❝❛❧✮ ♠❛♣s ■♥ oFs ✇❡ ❤❛✈❡ ♦♣❡r❛t✐♦♥s ♦❢ t❤❡ k✲t❤ ❞♦♠❛✐♥

d(k) ❛♥❞ k✲t❤ ❝♦❞♦♠❛✐♥ c(k)✳ ❋♦r ❛♥ ♦r❞❡r ❢❛❝❡

str✉❝t✉r❡ S ❛s ❢♦❧❧♦✇s S

s3 s1

✲

s2 x8

• ✒

x7

❅ ❅ ❅ ❅ ❅ ❅ ❘

x6 ⇓a7 s0

✲

x2

✖✕ ✣✢

∧ ⇓

α a6

✖✕ ✣✢

∧ ⇓

α′ a5

✫ ✪ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ ✄ ✄ ✄ ✄ ✄ ✄✄ ✗

⇓a1

✁ ✁ ✁ ✁ ✁ ✻

⇓

✫ ✪ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❑

⇓a2

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ a4 x5

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ a3 x4

✖✕ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❖

⇓ a0 x0 x1 x3

✐ts 1✲❞♦♠❛✐♥ ✐s

d(1)S

s3 s1 s2 x8

• ✒

x7

❅ ❅ ❅ ❅ ❅ ❅ ❘

s0

✲

x2

✶✵

t❤❡ ❝♦♥✈❡① s✉❜s❡t ♦❢ S ❞❡✜♥✐♥❣ 1✲❝♦❞♦♠❛✐♥ ✐s c(1)S

s3 s1

✲

x6 s0

✫ ✪ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ ✄ ✄ ✄ ✄ ✄ ✄✄ ✗ ✖✕ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❖

x0 x1

❛♥❞ ✜♥❛❧❧② t❤❡ 1✲❝♦❞♦♠❛✐♥ ♦❢ S ✐s t❤❡ str❡t❝❤✲ ✐♥❣ ♦❢ c(1)S

c(1)S

s3 s1

✲

x6 (s0,∅,{x0})

✲

x1 (s0,{x0},∅)

✲

x0

❲❡ ❤❛✈❡ ♠♦♥♦t♦♥❡ ♠❛♣s ❡♠❜❡❞❞✐♥❣ k✲t❤ ❞♦✲ ♠❛✐♥ ❛♥❞ k✲t❤ ❝♦❞♦♠❛✐♥ ✐♥t♦ ❛♥ ♦r❞❡r ❢❛❝❡ str✉❝t✉r❡ S✿

d(k)S

S

✲

d(k)

S

c(k)S

✛ c(k)

S

✶✶

❲❡ ❛❧s♦ ❤❛✈❡ ❛♥ ♦♣❡r❛t✐♦♥ ♦❢ k✲t❡♥s♦r ♦❢ t✇♦ ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s S ❛♥❞ T s✉❝❤ t❤❛t

c(k)S = d(k)T✳ c(k)S

T

✲

d(k)

T

S S ⊗k T

✲

κS

✻

c(k)

S

✻κT

❊①❛♠♣❧❡s S S ⊗0 T

✲

κS T

✛

κT

✖✕ ✁ ✁ ✁ ✁ ✁ ✻

⇓ a x

s

✖✕ ❆ ❆ ❆ ❆ ❆ ❑

⇓ b y

✖✕ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❖

⇓ a x

s

✖✕ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❖

⇓ b y

s

x<∼y

❚❤❡ t❡♥s♦r ✐s ❛ ♣✉s❤♦✉t ❧♦❝❛❧❧②✳ ❇✉t <∼ ✐s ♥♦t ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤✐s✳ ❚❤❡ ❛❞❞✐t✐♦♥❛❧ r✉❧❡ ✐s t❤❛t ✐♥ ❝❛s❡ ♦❢ ❞♦✉❜ts ❢❛❝❡s ❢r♦♠ S ❝♦♠❡s ❜❡❢♦r❡ ❢❛❝❡s ❢r♦♠ T✳ ❚❤✐s ✐s ✇❤② x <∼ y✱ ❛❜♦✈❡✳

✶✷

S

• ✲
• ✒

❅ ❅ ❅ ❘

x

⇓

• ✲

y0

✍✌ ✒ ✑

∧ ⇓ a

✍✌ ✁ ✁ ✁ ✻

⇓ a1 y2

✍✌ ❆ ❆ ❆ ❑

⇓ a0 y1

T

• ✲

x

• ✲
• ✲

✄ ✄ ✄✄ ✗ ❈ ❈ ❈❈ ❲ ✟✟✟✟✟✟✟✟ ✟ ✯

⇓b1 ⇓b2 y0 y1 y2

✍✌ ✒ ✑

∧ ⇓ b

✍✌ ✄ ✄✄ ❈ ❈ ❈ ❖

⇓

• ✲

x

• ✲

y2

• ✲

y1

• ✲

y0

c(1)S = d(1)T

❛♥❞ t❤❡ 1✲t❡♥s♦r S ⊗1 T ✐s S ⊗1 T

• ✲
• ✒

❅ ❅ ❅ ❅ ❘

⇓

• ✲

y0

✍✌ ✒ ✑

∧ ⇓ a

✍✌ ✒ ✑

∧ ⇓ b

✫ ✪ ✁ ✁ ✁ ✁ ✁ ✄ ✄ ✄ ✄ ✄ ✄ ✗

⇓b1

✁ ✁ ✁ ✻

⇓

✫ ✪ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❑

⇓b2

✍✌ ✁ ✁ ✁ ✻

⇓ a1 y2

✍✌ ❆ ❆ ❆ ❑

⇓ a0 y1

✍✌ ✄ ✄✄ ❈ ❈ ❈ ❖

⇓

a <∼ b

✶✸

❚❤❡♦r❡♠✳ ❚❤❡ ❝❛t❡❣♦r② oFs ✐s ❛ ♠♦♥♦✐❞❛❧ ❣❧♦❜✉❧❛r ❝❛t❡❣♦r② ✐♥ t❤❡ s❡♥s❡ ♦❢ ❇❛t❛♥✐♥✱ ✇✐t❤ k✲t❡♥s♦r sq✉❛r❡s ❜❡✐♥❣ ♣✉s❤♦✉t ❧♦❝❛❧❧② ✭✐✳❡✳ ✐♥

• Fsloc✮✳

❲❡ ❤❛✈❡ ❛ ❢✉❧❧ ❡♠❜❡❞❞✐♥❣ ❢✉♥❝t♦r (−)∗ : oFsloc − → Compm/1 ❋✐① S ✐♥ oFsloc✳ k✲❝❡❧❧s ✐♥ S∗✿ ✭♠♦♥♦t♦♥❡ ✐s♦ ❝❧❛ss❡s ♦❢✮ ❧♦❝❛❧ ♠❛♣s f : T − → S ✇✐t❤ dim(T) ≤ k❀ f0 ✐s ❡q✉✐✈❛❧❡♥t t♦ f1 ✐✛ t❤❡r❡ ✐s ❛ ♠♦♥♦t♦♥❡ ✐s♦ h s✉❝❤ t❤❛t t❤❡ tr✐❛♥❣❧❡ T0 T1

✲

h S f0

❅ ❅ ❅ ❘

f1

• ✠

❝♦♠♠✉t❡s✳

✶✹

❞♦♠❛✐♥s ❛♥❞ ❝♦❞♦♠❛✐♥s S∗✿ T S

✲

f

c(k)T d(k)T c(k)

T

d(k)

T

❍❍❍❍❍❍❍ ❍ ❥ ✟✟✟✟✟✟✟ ✟ ✯

d(k)(f) = f ◦ d(k)

T

❛♥❞ c(k)(f) = f ◦ c(k)

T

❝♦♠♣♦s✐t✐♦♥s ✐♥ S∗✿ ✐❢ c(k)(f0) = d(k)(f1) T0 ⊗k T1 S

✲

[f0, f1] T1 T0 ❍❍❍❍❍❍❍

❍ ❥ ✟✟✟✟✟✟✟ ✟ ✯

c(k)T0

✟✟✟✟✟✟✟ ✟ ✯ ❍❍❍❍❍❍❍ ❍ ❥

d(k)

T1

c(k)

T0

❄ ✻

f1 f0 t❤❡♥ f1 ◦k f0 = [f0, f1] (−)∗ ❛❝ts ♦♥ ♠♦r♣❤✐s♠ ❜② ❝♦♠♣♦s✐t✐♦♥s✳

✶✺

❚❤❡♦r❡♠✳ (−)∗ : oFsloc − → Compm/1 ✐♥❞✉❝❡s t❤❡ ❢✉♥❝t♦r

Compm/1

SetoFsop

loc

✲

C

Comp((−)∗, C)

✲

✇❤✐❝❤ ✐s ❢✉❧❧ ❛♥❞ ❢❛✐t❤❢✉❧✱ ❛♥❞ ✇❤♦s❡ ❡ss❡♥✲ t✐❛❧ ✐♠❛❣❡ ❝♦♥s✐sts ♦❢ ❢✉♥❝t♦rs s❡♥❞✐♥❣ t❡♥s♦r sq✉❛r❡s ✐♥ oFsop

loc t♦ ♣✉❧❧❜❛❝❦s ✐♥ Set✳

✶✻