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●r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ✐♥ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ✭ ∞ ✕✮❝❛t❡❣♦r✐❡s ✇✐t❤ ❞✉❛❧s ❏✉♥ ❨♦s❤✐❞❛ ●r❛❞✉❛t❡ ❙❝❤♦♦❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s✱ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❚♦❦②♦ ❏✉❧② ✷✶✱ ✷✵✶✼✱ ❈❚✷✵✶✼✱ ❯❇❈✱ ❱❛♥❝♦✉✈❡r

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ❈♦♥t❡♥ts ❖♣❡r❛❞s ❢♦r s✉r❢❛❝❡s ✇✐t❤ ■♥tr♦❞✉❝t✐♦♥ ✶ str✐♥❣s ❘❡✈✐❡✇ ♦♥ str✐♥❣ ❝❛❧❝✉❧✉s ❆❧❣❡❜r❛✐❝ ❞❡s❝r✐♣t✐♦♥ ❆♥ ❡①t❡♥s✐♦♥ ❇♦♥✉s✿ ▲✐❢ts t♦ ∞ ✲❝♦♥t❡①ts ❆♥s✇❡r ❢r♦♠ q✉❛♥t✉♠ ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ✸ t♦♣♦❧♦❣②✿ ♣❧❛♥❛r ❛❧❣❡❜r❛s ▲❛❜❡❧✐♥❣s ●♦❛❧ ♦❢ t❤❡ t❛❧❦ ❈❧❛ss✐✜❝❛t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ●r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ✷ ❘❡❧❛t✐✈❡ ♦❜❥❡❝ts ✐♥ s♠♦♦t❤ ❑❡② r❡s✉❧ts ❝❛t❡❣♦r② ▼❛✐♥ ❚❤❡♦r❡♠ ❈♦❜♦r❞✐s♠s ♦❢ ❛rr❛♥❣❡♠❡♥ts ❘❡❢❡r❡♥❝❡ ✷ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ■♥tr♦❞✉❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✶ ❘❡✈✐❡✇ ♦♥ str✐♥❣ ❝❛❧❝✉❧✉s ❆♥ ❡①t❡♥s✐♦♥ ❆♥s✇❡r ❢r♦♠ q✉❛♥t✉♠ t♦♣♦❧♦❣②✿ ♣❧❛♥❛r ❛❧❣❡❜r❛s ●♦❛❧ ♦❢ t❤❡ t❛❧❦ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ✷ ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ✸ ❘❡❢❡r❡♥❝❡ ✸ ✴ ✷✸

� � � � � � � � � � ■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ❘❡✈✐❡✇ ♦♥ str✐♥❣ ❝❛❧❝✉❧✉s ❉❡✜♥✐t✐♦♥ ✶ → R 2 ✭✇✐t❤ ♦✉t❡r✲❡❞❣❡s✮ ✐s s❛✐❞ t♦ ❜❡ ♣r♦❣r❡ss✐✈❡ ✐❢ ❆ ♣❧❛♥❛r ❣r❛♣❤ η : Γ ֒ ❢♦r ❡❛❝❤ ❡❞❣❡ e ♦❢ Γ ✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ proj 2 η → R 2 e − − − − → R = y ✲❛①✐s ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❛❧♦♥❣ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ❡❞❣❡ e ✳ � � ✹ ✴ ✷✸

� � � � ■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ❘❡✈✐❡✇ ♦♥ str✐♥❣ ❝❛❧❝✉❧✉s C ✿ ❛ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r②✳ Pr♦♣♦s✐t✐♦♥ ✷ ✭❬❏♦②❛❧ ❛♥❞ ❙tr❡❡t✱ ✶✾✾✶❪✮ ❋♦r ❛ ♣❧❛♥❛r ♣r♦❣r❡ss✐✈❡ ❣r❛♣❤ Γ ✱ ❝♦♥s✐❞❡r ❛ ❧❛❜❡❧✐♥❣ ✐♥ C s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r✉❧❡s✿ ❡❛❝❤ ❡❞❣❡ ♦❢ Γ ✐s ❧❛❜❡❧❡❞ ❜② ❛♥ ♦❜❥❡❝t ♦❢ C ❀ ❡❛❝❤ ✈❡rt❡① ♦❢ Γ ✐s ❧❛❜❡❧❡❞ ❜② ❛ ♠♦r♣❤✐s♠ ♦❢ C s♦ t❤❛t b 1 b n · · · ↔ f : a 1 ⊗ · · · ⊗ a m → b 1 ⊗ · · · ⊗ b n ∈ C f · · · a 1 a m ❚❤❡♥✱ Γ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❧❛❜❡❧✐♥❣ ❞❡t❡r♠✐♥❡s ❛ ♠♦r♣❤✐s♠ ✐♥ C ✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡s✉❧t✐♥❣ ♠♦r♣❤✐s♠ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r ✐s♦t♦♣✐❡s ♦❢ ♣❧❛♥❛r ♣r♦❣r❡ss✐✈❡ ❣r❛♣❤s✳ ✺ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ❆♥ ❡①t❡♥s✐♦♥ ❙❧♦❣❛♥ ❙tr✐♥❣ ❝❛❧❝✉❧✉s ✐♥ C = ❆ ❝❧❛ss ♦❢ ❣r❛♣❤s + ❧❛❜❡❧✐♥❣ r✉❧❡s ◗✉❡st✐♦♥ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❝♦♥s✐❞❡r ✇❤♦❧❡ ♣❧❛♥❛r ❣r❛♣❤s t♦ ♦❜t❛✐♥ ♥❡✇ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s❄ ❆♥s❡r ❨❊❙✦✦ ▲❡t✬s ❡♥❥♦② ♠♦r❡ ❣❡♦♠❡tr② ❛♥❞ ♠♦r❡ ❞✉❛❧✐t②✳ ✻ ✴ ✷✸

� � � � ■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ❆♥s✇❡r ❢r♦♠ q✉❛♥t✉♠ t♦♣♦❧♦❣②✿ ♣❧❛♥❛r ❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ ✸ ✭♠♦❞✐✜❡❞ ❢r♦♠ ❬❏♦♥❡s✱ ✶✾✾✾❪✮ ▲❡t C ❜❡ ❛ s❡t ✇✐t❤ ❛♥ ✐♥✈♦❧✉t✐♦♥ ( · ) ∗ : C → C ✳ ❚❤❡♥✱ ❛ C ✲❝♦❧♦r❡❞ ♣❧❛♥❛r ❛❧❣❡❜r❛ V ✐♥ ❛ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r② V ❝♦♥s✐sts ♦❢ ❛♥ ♦❜❥❡❝t V ( c 1 . . . c m ) ∈ V ❢♦r ❡❛❝❤ ❝②❝❧❧✐❝ s❡q✉❡♥❝❡ ✐♥ C ❀ ♦♣❡r❛t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❧❛❜❡❧❡❞ ♣✐❝t✉r❡s s✉❝❤ ❛s c 2 � c 5 c 1 : V ( c ∗ 1 c ∗ 2 c 3 c 4 c ∗ 4 ) ⊗ V ( c ∗ 3 c 2 c 5 c ∗ 6 ) → V ( c ∗ 1 c 5 c ∗ c 3 6 ) , c 4 � c 6 c < ✇❤✐❝❤ ✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ s✉❜st✐t✉t✐♦♥s ♦❢ ❞✐s❦s✳ ✼ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ●♦❛❧ ♦❢ t❤❡ t❛❧❦ ❇❛❞ ♥❡✇s✦ ❚❤❡ ♣r❡s❡♥t ❞❡✜♥✐t✐♦♥ ♦❢ ♣❧❛♥❛r ❛❧❣❡❜r❛s ✐s ♠♦r❡ ♦r ❧❡ss ❛❧❣❡❜r❛✐❝❀ ✐✳❡✳ ❜② ❣❡♥❡r❛t♦rs ❛♥❞ r❡❧❛t✐♦♥s✳ ❇✉t✳✳✳ ❲❊ ◆❊❊❉ ▼❖❘❊ ●❊❖▼❊❚❘❨✦✦✦✦ ❲❊ ◆❊❊❉ ▼❖❘❊ ●❊❖▼❊❚❘❨✦✦✦✦ ❲❊ ◆❊❊❉ ▼❖❘❊ ●❊❖▼❊❚❘❨✦✦✦✦ ●♦❛❧ ❚♦ ❞❡✜♥❡ ❛♥ ♦♣❡r❛❞ ♦❢ ♣❧❛♥❛r ❛❧❣❡❜r❛s ✐♥ ❛ ♣✉r❡❧② ❣❡♦♠❡tr✐❝ ✇❛②✳ ❇♦♥✉s✿ ❛ ♣r✐♦r✐ ❤✐❣❤❡r ❝♦❤❡r❡♥❝❡ ♣r♦❜❧❡♠s✳ ●r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ✐♥ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ∞ ✲❝❛t❡❣♦r✐❡s ✇✐t❤ ❞✉❛❧s✳ ✽ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ■♥tr♦❞✉❝t✐♦♥ ✶ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ✷ ❘❡❧❛t✐✈❡ ♦❜❥❡❝ts ✐♥ s♠♦♦t❤ ❝❛t❡❣♦r② ❈♦❜♦r❞✐s♠s ♦❢ ❛rr❛♥❣❡♠❡♥ts ❖♣❡r❛❞s ❢♦r s✉r❢❛❝❡s ✇✐t❤ str✐♥❣s ❆❧❣❡❜r❛✐❝ ❞❡s❝r✐♣t✐♦♥ ❇♦♥✉s✿ ▲✐❢ts t♦ ∞ ✲❝♦♥t❡①ts ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ✸ ❘❡❢❡r❡♥❝❡ ✾ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡ ❘❡❧❛t✐✈❡ ♦❜❥❡❝ts ✐♥ s♠♦♦t❤ ❝❛t❡❣♦r② ❲r✐t❡ [ n ] := { 0 < 1 < · · · < n } t❤❡ t♦t❛❧❧② ♦r❞❡r❡❞ s❡t ✇✐t❤ ( n + 1) ✲❡❧❡♠❡♥ts✳ ❉❡✜♥✐t✐♦♥ ✹ ❆♥ ❛rr❛♥❣❡♠❡♥t ♦❢ ♠❛♥✐❢♦❧❞s ♦❢ s❤❛♣❡ [ n ] ✐s ❛ ❢✉♥❝t♦r X : [ n ] → Emb ✐♥t♦ t❤❡ ❝❛t❡❣♦r② ♦❢ s♠♦♦t❤ ♠❛♥✐❢♦❧❞s ✭♣♦ss✐❜❧② ✇✐t❤ ❝♦r♥❡rs✮ ❛♥❞ s♠♦♦t❤ ❡♠❜❡❞❞✐♥❣s✳ � i < j ⇒ X ( i ) ✏✐s✑ ❛ s✉❜♠❛♥✐❢♦❧❞ ♦❢ X ( j ) ✳ ◆♦t❛t✐♦♥ ❚❤❡ ❛♠❜✐❡♥t ♠❛♥✐❢♦❧❞ |X| := X (max[ n ]) = X ( n ) ✳ ❚❤❡ ❞✐♠❡♥s✐♦♥ dim X := (dim X ( n ) , . . . , dim X (0)) ✳ ✶✵ ✴ ✷✸