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❙❡❝t✐♦♥❛❧ ❛❧❣❡❜r❛s ♦❢ s❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ❯▼P❆ ✲ ❊◆❙ ▲②♦♥ ▼❛② ✾✱ ✷✵✶✾ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✶ ✴ ✶✾

▼♦t✐✈❛t✐♦♥ ❚❤❡♦r❡♠ ▲❡t R ❛♥❞ S ❜❡ ✉♥✐t❛❧✱ ❞✐s❝r❡t❡✱ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ r✐♥❣s ❛♥❞ X ❛♥❞ Y ❜❡ ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧✱ ❧♦❝❛❧❧② ❝♦♠♣❛❝t✱ ❍❛✉s❞♦r✛ s♣❛❝❡s✳ ❆♥② r✐♥❣ ✐s♦♠♦r♣❤✐s♠ ϕ : C c ( X , R ) → C c ( Y , S ) ✐s ♦❢ t❤❡ ❢♦r♠ ϕ ( f )( y ) = χ φ ( y ) ( f ( φ ( y )) ❢♦r s♦♠❡ ❤♦♠❡♦♠♦r♣❤✐s♠ φ : Y → X ❛♥❞ ❛ ✜❡❧❞ χ = { χ x } x ∈ X ♦❢ r✐♥❣ ✐s♦♠♦r♣❤✐s♠s ξ x : R → S ✳ X × R ✏✐s ❣r❛❞❡❞✑ ♦✈❡r X ❀ ❋♦r♠❛❧✐③❡ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ❣r❛❞❡❞ ❛❧❣❡❜r❛❄ ●♦❞❡♠❡♥t✱ ❑❛♣❧❛♥s❦②✱ ❋❡❧❧✱ ❑✉♠❥✐❛♥✱✳ ✳ ✳ ✿ C ∗ ✲❛❧❣❡❜r❛✐❝ ❜✉♥❞❧❡s✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ ❛ s❡tt✐♥❣ ✇❤✐❝❤ ❛❧s♦ ❡♠❝♦♠♣❛ss❡s ♣♦ss✐❜❧② ❙t❡✐♥❜❡r❣ ❛❧❣❡❜r❛s❄ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✷ ✴ ✶✾

❉❡✜♥✐t✐♦♥ ❆ s❡♠✐❣r♦✉♣♦✐❞ ✐s ❛ ❣r❛♣❤ s , r : Λ → Λ ( ✵ ) ✇✐t❤ ❛ ♣r♦❞✉❝t Λ ( ✷ ) = { ( x , y ) ∈ Λ × Λ : s ( x ) = r ( y ) } ✱ ( x , y ) �→ xy ✱ s❛t✐s❢②✐♥❣ s ( xy ) = s ( y ) ✱ r ( xy ) = r ( x ) ❀ x ( yz ) = ( xy ) z ✇❤❡♥❡✈❡r s❡♥s✐❜❧❡✳ ❊①❛♠♣❧❡ ❈❛t❡❣♦r✐❡s✱ ❣r♦✉♣♦✐❞s✱ s❡♠✐❣r♦✉♣s✱ ❛r❡ s❡♠✐❣r♦✉♣♦✐❞s✳ ❆ s❡♠✐❣r♦✉♣♦✐❞ Λ ✐s ✐♥✈❡rs❡ ✐❢ ❢♦r ❡✈❡r② a ∈ Λ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ a ∗ ∈ Λ s✉❝❤ t❤❛t aa ∗ a = a ❛♥❞ a ∗ aa ∗ = a ∗ ✳ ❍♦♠♦♠♦r♣❤✐s♠s❂❢✉♥❝t♦rs✿ φ : Λ → Γ s✉❝❤ t❤❛t Λ ( ✷ ) ⊆ ( φ × φ ) − ✶ (Γ ( ✷ ) ) ❛♥❞ φ ( xy ) = φ ( x ) φ ( y ) ✇❤❡♥❡✈❡r s❡♥s✐❜❧❡✳ ❉❡✜♥✐t✐♦♥ ❍♦♠♦♠♦r♣❤✐s♠ φ : Λ → Γ ✐s r✐❣✐❞ ✐❢ ( φ × φ ) − ✶ (Γ ( ✷ ) ) = Λ ( ✷ ) ✳ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✸ ✴ ✶✾

R ✲❜✉♥❞❧❡s ▲❡t R ❜❡ ❛ ✉♥✐t❛❧ r✐♥❣✳ ❉❡✜♥✐t✐♦♥ ❆♥ R ✲❜✉♥❞❧❡ ✐s ❛ s✉r❥❡❝t✐✈❡ r✐❣✐❞ s❡♠✐❣r♦✉♣♦✐❞ ❤♦♠♦♠♦r♣❤✐s♠ π : Λ → Γ s✉❝❤ t❤❛t π − ✶ ( γ ) ✐s ❛♥ R ✲❜✐♠♦❞✉❧❡ ❢♦r ❛❧❧ γ ∈ Γ ❀ ❚❤❡ ♣r♦❞✉❝t ♠❛♣ ♦❢ Λ π − ✶ ( γ ✶ ) × π − ✶ ( γ ✷ ) → π − ✶ ( γ ✶ γ ✷ ) ✐s R ✲❜❛❧❛♥❝❡❞ ❢♦r ❛❧❧ ( γ ✶ , γ ✷ ) ∈ Γ ( ✷ ) ✭❡q✉✐✈❛❧❡♥t❧② ✐t ✐s ❛♥ R ✲❜✐♠♦❞✉❧❡ ❤♦♠♦♠♦r♣❤✐s♠ π − ✶ ( γ ✶ ) ⊗ π − ✶ ( γ ✷ ) → π − ✶ ( γ ✶ γ ✷ ) ✮✳ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✹ ✴ ✶✾

❙❡❝t✐♦♥❛❧ ❛❧❣❡❜r❛s ❈♦♥✈❡♥t✐♦♥s ❆❧❧ s❡♠✐❣r♦✉♣♦✐❞s ❛r❡ ét❛❧❡ ✭❛❧❧ r❡❧❡✈❛♥t ♠❛♣s ❛r❡ ❧♦❝❛❧ ❤♦♠❡♦♠♦r♣❤✐s♠s✱ ✐♥❝❧✉❞✐♥❣ ❜✉♥❞❧❡s✮✱ ❍❛✉s❞♦r✛✱ ❛♥❞ ❛♠♣❧❡ ✭③❡r♦✲❞✐♠❡♥s✐♦♥❛❧✮✳ ❆❧❧ r✐♥❣s ❛r❡ ❞✐s❝r❡t❡✳ ❚❤❡ s❡❝t✐♦♥❛❧ ❛❧❣❡❜r❛ A ( π ) ≡ A (Λ → Γ) ♦❢ π : Λ → Γ ✐s t❤❡ s❡t ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s α : Γ → Λ s✉❝❤ t❤❛t α ✐s ❝♦♥t✐♥✉♦✉s❀ supp( α ) := { γ ∈ Γ : α ( γ ) � = ✵ } ✐s ❝♦♠♣❛❝t✳ ❚❤❡ ❜✐♠♦❞✉❧❡ str✉❝t✉r❡ ♦❢ A ( π ) ✐s ♣♦✐♥t✇✐s❡✳ ❚❤❡ ♣r♦❞✉❝t ✐s � αβ ( γ ) = α ( γ ✶ ) β ( γ ✷ ) . γ ✶ γ ✷ = γ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✺ ✴ ✶✾

❚✇♦ ❡①❛♠♣❧❡s ❊①❛♠♣❧❡ ■❢ A = ⊕ s ∈ S A s ✐s ❛ s❡♠✐❣r♦✉♣ ❣r❛❞❡❞ ❛❧❣❡❜r❛✱ t❤❡♥ A = A ( ⊔ s ∈ S A s → S ) ✳ ❚❤✐s ❛❧s♦ ✇♦r❦s ❢♦r ❣r♦✉♣♦✐❞ ✭❛♥❞ s❡♠✐❣r♦✉♣♦✐❞✮ ❣r❛❞❡❞ ❛❧❣❡❜r❛s✳ ❊①❛♠♣❧❡ ■❢ A ✐s ❛♥② ❛❧❣❡❜r❛✱ t❤❡ s❡♠✐❣r♦✉♣♦✐❞ ❛❧❣❡❜r❛ ♦❢ Γ ✐s A Γ = A (Γ × A → Γ) ✳ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✻ ✴ ✶✾

✏◆❛ï✈❡✑ ❝r♦ss❡❞ ♣r♦❞✉❝ts ❆♥ ❛❝t✐♦♥ θ : S � Λ ♦❢ ❛♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣♦✐❞ S ♦♥ ❛ s❡♠✐❣r♦✉♣♦✐❞ Λ ✐s ❛ ❝♦❧❧❡❝t✐♦♥ { θ s : s ∈ S } ♦❢ ♠❛♣s s✉❝❤ t❤❛t ❢♦r ❛❧❧ s ∈ S ✱ θ s : dom( θ s ) → ran( θ s ) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ ✐❞❡❛❧s ♦❢ Λ ❀ ❢♦r ❛❧❧ ( s , t ) ∈ S ( ✷ ) ✱ θ st = θ s ◦ θ t : θ − ✶ t (dom( θ s )) → θ s (ran( θ t )) ✳ ❚❤❡ s❡♠✐❞✐r❡❝t ♣r♦❞✉❝t S ⋉ Λ = { ( s , λ ) : λ ∈ dom( θ s ) } ❤❛s ♣r♦❞✉❝t � � st , θ − ✶ ( s , λ )( t , µ ) = t ( λθ t ( µ )) ✇❤❡♥❡✈❡r s❡♥s✐❜❧❡✳ ❲❡ ♦♥❧② ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡♥ ✐t ✐s ❛ss♦❝✐❛t✐✈❡✳ ■❢ A ✐s ❛♥ ❛❧❣❡❜r❛ ❛♥❞ θ : S � A ❜② ❛❧❣❡❜r❛ ✐s♦♠♦r♣❤✐s♠s ♦❢ ❛❧❣❡❜r❛ ✐❞❡❛❧s✱ t❤❡ ✏♥❛ï✈❡✑ ❝r♦ss❡❞ ♣r♦❞✉❝t ✐s S ⋆ A = A ( S ⋉ A → S ) . ❆❧❧ ♦❢ t❤✐s ✇♦r❦s ❢♦r ♣❛rt✐❛❧ ❛❝t✐♦♥s ❛♥❞ ∧ ✲♣r❡❛❝t✐♦♥s ✭❊①❡❧❀ ▼❝❆❧❧✐st❡r✲❘❡✐❧❧②✮✳ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✼ ✴ ✶✾

◆❛ï✈❡ ❝r♦ss❡❞ ♣r♦❞✉❝ts ❛♥❞ s❡♠✐❞✐r❡❝t ❜✉♥❞❧❡s ❆♥ ❛❝t✐♦♥ θ ♦❢ S ♦♥ ❛ ❜✉♥❞❧❡ π : Λ → Γ ❝♦♥s✐sts ♦❢ t✇♦ ❛❝t✐♦♥s θ Λ : S � Λ ❛♥❞ θ Γ : S � Γ ✇❤✐❝❤ ❛r❡ ✐♥t❡rt✇✐♥❡❞ ❜② π ✿ θ Γ s ◦ π = π ◦ θ Λ ❢♦r ❛❧❧ s ∈ S . s ✭❛♥❞ θ Λ r❡s♣❡❝ts t❤❡ R ✲❜✐♠♦❞✉❧❡ str✉❝t✉r❡s ♦❢ Λ ✮✳ ❙✉♣♣♦s❡ t❤❛t S ✐s ❞✐s❝r❡t❡ ❛♥❞ S ⋉ Γ ✐s ♦♣❡♥ ✐♥ S × Γ ✳ ❲❡ ❞❡✜♥❡ ❛♥ ❛❝t✐♦♥ Θ: S � A ( π ) ❜② � � α ∈ A ( π ) : supp( α ) ⊆ dom( θ Γ dom(Θ s ) = s ) s ) − ✶ ◦ α ◦ θ Γ Θ s ( α ) = ( θ Λ s ∗ ♦♥ dom( θ s ∗ ) , ✵ ❡✈❡r②✇❤❡r❡ ❡❧s❡ . ❆t t❤❡ ❜✉♥❞❧❡ ❧❡✈❡❧✱ ✇❡ ❝r❡❛t❡ ❛ ♥❡✇ ❜✉♥❞❧❡ S ⋉ π : S ⋉ Λ → S ⋉ Γ ✱ ( s , λ ) �→ ( s , π ( λ )) ✳ ❚❤❡♦r❡♠ S ⋆ A ( π ) ∼ = A ( S ⋉ π ) ✳ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✽ ✴ ✶✾

◗✉♦t✐❡♥ts ♦❢ s❡♠✐❣r♦✉♣♦✐❞s ❆ r✐❣✐❞ ❝♦♥❣r✉❡♥❝❡ ∼ ♦♥ ❛ s❡♠✐❣r♦✉♣♦✐❞ Λ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ∼ s✉❝❤ t❤❛t x ∼ y ✐♠♣❧✐❡s s ( x ) = s ( y ) ❛♥❞ r ( x ) = r ( y ) ❀ x ∼ y ❛♥❞ z ∼ w ✐♠♣❧✐❡s xz ∼ yw ✭✇❤❡♥❡✈❡r s❡♥s✐❜❧❡✮✳ ❚❤❡ q✉♦t✐❡♥t Λ / ∼ ❤❛s ❛ ♥❛t✉r❛❧ s❡♠✐❣r♦✉♣♦✐❞ str✉❝t✉r❡ ♦✈❡r Λ ( ✵ ) ✳ ■❢ Λ ✐s t♦♣♦❧♦❣✐❝❛❧✴ét❛❧❡✴❛♠♣❧❡ ❛♥❞ ∼ ✐s ♦♣❡♥ t❤❡♥ Λ / ∼ ✐s t♦♣♦❧♦❣✐❝❛❧✴ét❛❧❡✴❛♠♣❧❡✳ ▲✉✐③ ●✳ ❈♦r❞❡✐r♦ ✭❊◆❙ ✲ ▲②♦♥✮ ❙❡♠✐❣r♦✉♣♦✐❞ ❜✉♥❞❧❡s ▼❛② ✾✱ ✷✵✶✾ ✾ ✴ ✶✾