r s str - - PowerPoint PPT Presentation

• 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

✶

■♥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 ❘❡❧❛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 ▲❛❜❡❧✐♥❣s ❈❧❛ss✐✜❝❛t✐♦♥

• r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s

❑❡② r❡s✉❧ts ▼❛✐♥ ❚❤❡♦r❡♠ ❘❡❢❡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 ❣r❛♣❤ η : Γ ֒ → R2 ✭✇✐t❤ ♦✉t❡r✲❡❞❣❡s✮ ✐s s❛✐❞ t♦ ❜❡ ♣r♦❣r❡ss✐✈❡ ✐❢ ❢♦r ❡❛❝❤ ❡❞❣❡ e ♦❢ Γ✱ t❤❡ ❝♦♠♣♦s✐t✐♦♥ e

η

− → R2

proj2

− − − → 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 b1 · · · bn f a1 · · · am

• ↔

f : a1 ⊗ · · · ⊗ am → b1 ⊗ · · · ⊗ bn ∈ C ❚❤❡♥✱ 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 (c1 . . . cm) ∈ V ❢♦r ❡❛❝❤ ❝②❝❧❧✐❝ s❡q✉❡♥❝❡ ✐♥ C❀ ♦♣❡r❛t✐♦♥s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❧❛❜❡❧❡❞ ♣✐❝t✉r❡s s✉❝❤ ❛s

• c2

c3

• c4
• c1

c5 c6 < c : V (c∗

1c∗ 2c3c4c∗ 4)⊗V (c∗ 3c2c5c∗ 6) → V (c∗ 1c5c∗ 6) ,

✇❤✐❝❤ ✐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))✳

✶✵ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

❈♦❜♦r❞✐s♠s ♦❢ ❛rr❛♥❣❡♠❡♥ts

❉❡✜♥✐t✐♦♥ ✺ ❋♦r ❛ ♥♦♥✲✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ dn ≥ · · · ≥ d0 ♦❢ ✐♥t❡❣❡rs✱ ❞❡✜♥❡ ❛ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r② ArrCob(dn,...,d0) ❛s ❢♦❧❧♦✇s✿ ♦❜❥❡❝t ❛rr❛♥❣❡♠❡♥ts Y ♦❢ ❝❧♦s❡❞ ♦r✐❡♥t❡❞ ♠❛♥✐❢♦❧❞s ♦❢ s❤❛♣❡ [n] ♦❢ ❞✐♠❡♥s✐♦♥ (dn − 1, . . . , d0 − 1)✳ ♠♦r♣❤✐s♠ ❞✐✛❡♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ ❛rr❛♥❣❡♠❡♥ts W ♦❢ ❝♦♠♣❛❝t ♦r✐❡♥t❡❞ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❜♦✉♥❞❛r✐❡s ♦❢ s❤❛♣❡ [n] ♦❢ ❞✐♠❡♥s✐♦♥ (dn, . . . , d0) t♦❣❡t❤❡r ✇✐t❤ ❛ ❞✐✛❡♦♠♦r♣❤✐s♠ ∂W ∼ = −Y0 ∐ Y1 . ❝♦♠♣♦s✐t✐♦♥ ❣❧✉✐♥❣ ✭P❖❙❙■❇▲❊✦✦✦✮✳ ⊗✲str✉❝t✉r❡ ❞✐s❥♦✐♥t ✉♥✐♦♥ ∐✳ ❋♦r ♦✉r ♣✉r♣♦s❡✱ ArrCob(2,1) ✦

✶✶ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

❖♣❡r❛❞s ❢♦r s✉r❢❛❝❡s ✇✐t❤ str✐♥❣s

❉❡✜♥✐t✐♦♥ ✻ ❉❡✜♥❡ ✇✐❞❡ s✉❜❝❛t❡❣♦r✐❡s PlTang ⊂ SrfTang ⊂ ArrCob(2,1) t♦ ❝♦♥s✐st ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦r♣❤✐s♠s✿ ♠♦r♣❤✐s♠s ✐♥ SrfTang ❛r❡ t❤♦s❡ W : Y0 → Y1 s✉❝❤ t❤❛t π0(|Y1|) → π0(|W|) ✐s ❜✐❥❡❝t✐✈❡❀ ♠♦r♣❤✐s♠s ✐♥ PlTang s❛t✐s❢② ✐♥ ❛❞❞✐t✐♦♥ t❤❛t t❤❡ s✉r❢❛❝❡ |W| ✐s ♦❢ ❣❡♥✉s 0✳ Pr♦♣♦s✐t✐♦♥ ✼ ❚❤❡ s✉❜❝❛t❡❣♦r✐❡s SrfTang ❛♥❞ PlTang ❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ♠♦♥♦✐❞❛❧ ♣r♦❞✉❝ts✳ ▼♦r❡♦✈❡r✱ t❤❡s❡ ❝❛t❡❣♦r✐❡s ❛r❡ ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② ❝♦❧♦r❡❞ ♦♣❡r❛❞s ❛s s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r✐❡s✳

✶✷ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

❆❧❣❡❜r❛✐❝ ❞❡s❝r✐♣t✐♦♥

❞❡♣t❤ = 1 ❞❡♣t❤ = 0 ✐♥❞❡① = 0 ✐♥❞❡① = 1 ✐♥❞❡① = 1 ✐♥❞❡① = 2 ✐♥❞❡① = 1 ✐♥❞❡① = 0 ❘❡♠❛r❦ ❚❤❡ ♥✉♠❜❡r ♦❢ str✐♥❣s ♠❛② ✈❛r✐❡❞ ❡①❝❡♣t ❢♦r ❝✉♣ ❛♥❞ ❝❛♣✳

✶✸ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

❇♦♥✉s✿ ▲✐❢ts t♦ ∞✲❝♦♥t❡①ts

❇♦♥✉s✦ ❆ ❝♦♥str✉❝t✐♦♥ s✐♠✐❧❛r t♦ ∞✲❝❛t❡❣♦r② Cobd ♦❢ ❝♦❜♦r❞✐s♠s ✐♥ ❬▲✉r✐❡✱ ✷✵✵✾❪ ✇♦r❦s ❢♦r ❛rr❛♥❣❡❞ ❝♦❜♦r❞✐s♠s✳ ❖♥❡ ♦❜t❛✐♥s ❛ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ∞✲❝❛t❡❣♦r② ArrCob(2,1) t♦❣❡t❤❡r ✇✐t❤ ❛ 1✲tr✉♥❝❛t✐♦♥ ArrCob(2,1) → ArrCob(2,1) . ❉❡✜♥✐t✐♦♥ ✽ PlTang

• ·
• SrfTang
• ·
• ArrCob(2,1)
• PlTang

SrfTang ArrCob(2,1) Pr♦♣♦s✐t✐♦♥ ✾ PlTang ❛♥❞ SrfTang ❛r❡ ∞✲♦♣❡r❛❞s ✐♥ t❤❡ s❡♥s❡ ✐♥ ❬▲✉r✐❡✱ ✷✵✶✹❪✳

✶✹ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s

✶

■♥tr♦❞✉❝t✐♦♥

✷

❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s

✸

▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ▲❛❜❡❧✐♥❣s ❈❧❛ss✐✜❝❛t✐♦♥

• r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s

❑❡② r❡s✉❧ts ▼❛✐♥ ❚❤❡♦r❡♠ ❘❡❢❡r❡♥❝❡

✶✺ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

▲❛❜❡❧✐♥❣s

C✿ ❛ ✜①❡❞ ✭s♠❛❧❧✮ s❡t ✇✐t❤ ✐♥✈♦❧✉t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✶✵ ❆ C✲❧❛❜❡❧✐♥❣ ♦♥ ❛♥ ❛rr❛♥❣❡♠❡♥t X ♦❢ s❤❛♣❡ [1] ✐s ❥✉st ❛ ♠❛♣ X(0) → C . ArrCob(2,1)/C✿ t❤❡ ❝❛t❡❣♦r② ♦❢ C✲❧❛❜❡❧❡❞ ❛rr❛♥❣❡❞ ❝♦❜♦r❞✐s♠s✳ ❉❡✜♥✐t✐♦♥ ✶✶ SrfTang/C

• ·
• ArrCob(2,1)/C

❢♦r❣❡t

• SrfTang

ArrCob(2,1) PlTang/C

• ·
• ArrCob(2,1)/C

❢♦r❣❡t

• PlTang

ArrCob(2,1) ❘❡♠❛r❦ ❙✐♠✐❧❛r❧②✱ ♦♥❡ ❝❛♥ ❞❡✜♥❡ PlTang/C ⊂ SrfTang/C ⊂ ArrCob(2,1)/C✳

✶✻ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

❈❧❛ss✐✜❝❛t✐♦♥

❚❤❡♦r❡♠ ✶✷ ❆♥ ❛❧❣❡❜r❛ ♦✈❡r PlTang/C ✐s ❡①❛❝t❧② ❛ C✲❝♦❧♦r❡❞ ♣❧❛♥❛r ❛❧❣❡❜r❛✳ ❆ ♣❧❛♥❛r ❛❧❣❡❜r❛ V ✐♥ V ❞❡t❡r♠✐♥❡s ❛ ♠♦♥♦✐❞❛❧ V✲❝❛t❡❣♦r② CV ✇✐t❤ ♦❜❥❡❝ts s❡q✉❡♥❝❡s ♦❢ ❡❧❡♠❡♥ts ✐♥ C❀ ❤♦♠✲♦❜❥ CV (c1 . . . cm, d1 . . . dn) = V

• c1
• cm

✳ ✳ ✳

• d1
• dn

✳ ✳ ✳

• ❝♦♠♣♦s✐t✐♦♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❘❡♠❛r❦ ❚❤❡ ✐♥✈♦❧✉t✐♦♥ ♦♥ C ❡①t❡♥❞s t♦ ❛ ❢✉♥❝t♦r ( · )∗ : CV → Cop

V

s♦ t❤❛t c∗ ⊣ c , c∗∗ ∼ = c .

✶✼ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

• r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s

❆s♣❡❝t

• r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✐ ✐♥ C

= ♠♦♥♦✐❞❛❧ ❢✉♥❝t♦r CV → C ❢♦r s♦♠❡ C0✲❝♦❧♦r❡❞ ♣❧❛♥❛r ❛❧❣❡❜r❛ V ✳ ❚❖❉❆❨✿ ❋♦❝✉s ♦♥ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ✭∞✕✮❝❛t❡❣♦r✐❡s ✇✐t❤ ❞✉❛❧s✿ t❤❡ ❦❡② ✐♥❣r❡❞✐❡♥t ✐s ❚❤❡♦r❡♠ ✶✸ ✭❈♦❜♦r❞✐s♠ ❍②♣♦t❤❡s✐s ✐♥ ❞✐♠ 1✱ ❢♦❧❦❧♦r❡✱ ❬❇❛❡③ ❛♥❞ ❉♦❧❛♥✱ ✶✾✾✺❪✱ ❬▲✉r✐❡✱ ✷✵✵✾❪✮ ❋♦r ❛ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ∞✲❝❛t❡❣♦r② C✱ t❤❡ ❢✉♥❝t♦r Fun⊗(Cob1, C) → Core C ; Z → Z(+) ✐s ❛ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡✱ ✇❤❡r❡ t❤❡ ❞♦♠❛✐♥ ✐s t❤❡ ∞✲❝❛t❡❣♦r② ♦❢ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r✐❡s ❢r♦♠ t❤❡ ❝❛t❡❣♦r② ♦❢ 1✲❞✐♠ ❝♦❜♦r❞✐s♠s✱ ❛♥❞ Core C ✐s t❤❡ ♠❛①✐♠❛❧ ❣r♦✉♣♦✐❞ ♦❢ C✳

✶✽ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

❑❡② r❡s✉❧ts

▲❡♠♠❛ ✶✹ ❋♦r ❡❛❝❤ ❡❧❡♠❡♥t c ∈ C✱ t❤❡r❡ ❡①✐sts ❛ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❢✉♥❝t♦r ArrCob(2,1)/C → Cob1 ❜❡t✇❡❡♥ ∞✲❝❛t❡❣♦r✐❡s ✇❤✐❝❤ ❞♦❡s

✶ ❢♦r❣❡ts ❛❧❧ str✐♥❣s ❜✉t ♦♥❡s ❧❛❜❡❧❡❞ ❜② c ∈ C❀ ❛♥❞ ✷ ❢♦r❣❡ts t❤❡ ❛♠❜✐❡♥t ❝♦❜♦r❞✐s♠s✳ ✶✾ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

❑❡② r❡s✉❧ts

■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❤❛✈❡ ❛ ❢✉♥❝t♦r ψ : ArrCob(2,1)/C → Fun0(C, Cob1) , ✇❤❡r❡ Fun0(C, Cob1) ✐s t❤❡ ∞✲❝❛t❡❣♦r② ♦❢ ❢✉♥❝t♦rs ✇❤✐❝❤ ✈❛❧✉❡s t❤❡ ❡♠♣t② ❡①❝❡♣t ❢♦r ✜♥✐t❡❧② ♠❛♥② ♣♦✐♥ts ✐♥ C✳ ■♥ ❛❞❞✐t✐♦♥✱ ❢♦r ❡✈❡r② ♠❛♣ λ : C → Fun⊗(Cob1, C)✱ ✇❡ ❤❛✈❡ Ψ : ArrCob(2,1)/C

(λ,ϕ)

− − − → Fun⊗(Cob1, C) × Fun0(C, Cob1)

eval

− − → Fun0(C, C)

⊗

− → C . ❊①❛♠♣❧❡ ✶✺ ❇② ❈♦❜♦r❞✐s♠ ❍②♣♦t❤❡s✐s✱ ✇❡ ♠❛② ❝❤♦♦s❡ ❛ ♠❛♣ C0 ֒ → Core C → Fun⊗(Cob1, C) . ❍❡♥❝❡✱ ✇❡ ♦❜t❛✐♥ Ψ : ArrCob(2,1)/C0 → C .

✷✵ ✴ ✷✸

■♥tr♦❞✉❝t✐♦♥ ❖♣❡r❛❞s ♦❢ s✉r❢❛❝❡s ✇✐t❤ str✐♣s ▲❛❜❡❧✐♥❣s ❛♥❞ ❣r❛♣❤✐❝❛❧ ❝❛❧❝✉❧✉s ❘❡❢❡r❡♥❝❡

▼❛✐♥ ❚❤❡♦r❡♠

❚❤❡♦r❡♠ ✶✻ ❊✈❡r② s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ∞✲❝❛t❡❣♦r② C ✇✐t❤ ❞✉❛❧s ❣✐✈❡s r✐s❡ t♦ ❛ C0✲❝♦❧♦r❡❞ ♣❧❛♥❛r ❛❧❣❡❜r❛ ZC : SrfTang → ∞Grpd✳ ❙❦❡t❝❤ ❋♦r Y =

• c1
• cm

✳ ✳ ✳

• d1
• dn

✳ ✳ ✳✱ ♣✉t ZC(Y) := MapC(✶, Ψ(Y)) ∼ = MapC(✶, cm ⊗ · · · ⊗ c1 ⊗ d1 ⊗ · · · ⊗ dn) ; ❋♦r W : n

i=1 Yi → Y ✇✐t❤ ❡❛❝❤ |Yi| ❛♥❞ |Y| ❝♦♥♥❡❝t❡❞✱ ♣✉t

ZC(W) : ZC n

• i=1

Yi

• =

n

• i=1

MapC(✶, Ψ(Yi)) → MapC(✶,

• i

Ψ(Yi)) ∼ = MapC(✶, Ψ(

• i

Yi))

Ψ(W)

− − − − → MapC(✶, Y) = ZC(Y) .

✷✶ ✴ ✷✸

❘❡❢❡r❡♥❝❡ ■

❬❇❛❡③ ❛♥❞ ❉♦❧❛♥✱ ✶✾✾✺❪ ❇❛❡③✱ ❏✳ ❈✳ ❛♥❞ ❉♦❧❛♥✱ ❏✳ ✭✶✾✾✺✮✳ ❍✐❣❤❡r✲❞✐♠❡♥s✐♦♥❛❧ ❛❧❣❡❜r❛ ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ q✉❛♥t✉♠ ✜❡❧❞ t❤❡♦r②✳ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ✸✻✭✶✶✮✿✻✵✼✸✕✻✶✵✺✳ ❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴q✲❛❧❣✴✾✺✵✸✵✵✷✳ ❬❏♦♥❡s✱ ✶✾✾✾❪ ❏♦♥❡s✱ ❱✳ ❋✳ ❘✳ ✭✶✾✾✾✮✳ P❧❛♥❛r ❛❧❣❡❜r❛s✱ ■✳ ❛r❳✐✈✿♠❛t❤✴✾✾✵✾✵✷✼✳ ❬❏♦②❛❧ ❛♥❞ ❙tr❡❡t✱ ✶✾✾✶❪ ❏♦②❛❧✱ ❆✳ ❛♥❞ ❙tr❡❡t✱ ❘✳ ✭✶✾✾✶✮✳ ❚❤❡ ❣❡♦♠❡tr② ♦❢ t❡♥s♦r ❝❛❧❝✉❧✉s ■✳ ❆❞✈❛♥❝❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✽✽✭✶✮✿✺✺✕✶✶✷✳ ❬▲✉r✐❡✱ ✷✵✵✾❪ ▲✉r✐❡✱ ❏✳ ✭✷✵✵✾✮✳ ❖♥ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ t♦♣♦❧♦❣✐❝❛❧ ✜❡❧❞ t❤❡♦r✐❡s✳ ❛r❳✐✈✿✵✾✵✺✳✵✹✻✺✳

❘❡❢❡r❡♥❝❡ ■■

❬▲✉r✐❡✱ ✷✵✶✹❪ ▲✉r✐❡✱ ❏✳ ✭✷✵✶✹✮✳ ❍✐❣❤❡r ❛❧❣❡❜r❛✳ s❡❡ ▲✉r✐❡✬s ✇❡❜s✐t❡ ❤tt♣✿✴✴✇✇✇✳♠❛t❤✳❤❛r✈❛r❞✳❡❞✉✴✄❧✉r✐❡✴✳