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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: Cc(X, R) → Cc(Y , S) ✐s ♦❢ t❤❡ ❢♦r♠ ϕ(f )(y) = χφ(y)(f (φ(y)) ❢♦r s♦♠❡ ❤♦♠❡♦♠♦r♣❤✐s♠ φ: Y → X ❛♥❞ ❛ ✜❡❧❞ χ = {χx}x∈X ♦❢ r✐♥❣ ✐s♦♠♦r♣❤✐s♠s ξx : 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) =

• γ✶γ✷=γ

α(γ✶)β(γ✷).

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

❚✇♦ ❡①❛♠♣❧❡s

❊①❛♠♣❧❡

■❢ A = ⊕s∈SAs ✐s ❛ s❡♠✐❣r♦✉♣ ❣r❛❞❡❞ ❛❧❣❡❜r❛✱ t❤❡♥ A = A(⊔s∈SAs → S)✳ ❚❤✐s ❛❧s♦ ✇♦r❦s ❢♦r ❣r♦✉♣♦✐❞ ✭❛♥❞ s❡♠✐❣r♦✉♣♦✐❞✮ ❣r❛❞❡❞ ❛❧❣❡❜r❛s✳

❊①❛♠♣❧❡

■❢ A ✐s ❛♥② ❛❧❣❡❜r❛✱ t❤❡ s❡♠✐❣r♦✉♣♦✐❞ ❛❧❣❡❜r❛ ♦❢ Γ ✐s AΓ = 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✐♦♥ {θs : s ∈ S} ♦❢ ♠❛♣s s✉❝❤ t❤❛t ❢♦r ❛❧❧ s ∈ S✱ θs : dom(θs) → ran(θs) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ ✐❞❡❛❧s ♦❢ Λ❀ ❢♦r ❛❧❧ (s, t) ∈ S(✷)✱ θst = θs ◦ θt : θ−✶

t (dom(θs)) → θs(ran(θt))✳

❚❤❡ s❡♠✐❞✐r❡❝t ♣r♦❞✉❝t S ⋉ Λ = {(s, λ) : λ ∈ dom(θs)} ❤❛s ♣r♦❞✉❝t (s, λ)(t, µ) =

• st, θ−✶

t (λθt(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 θΛ : S Λ ❛♥❞ θΓ : S Γ ✇❤✐❝❤ ❛r❡ ✐♥t❡rt✇✐♥❡❞ ❜② π✿ θΓ

s ◦ π = π ◦ θΛ s

❢♦r ❛❧❧ s ∈ S. ✭❛♥❞ θΛ 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 dom(Θs) =

• α ∈ A(π) : supp(α) ⊆ dom(θΓ

s )

• Θs(α) = (θΛ

s )−✶ ◦ α ◦ θΓ s∗ ♦♥ dom(θs∗),

✵ ❡✈❡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 ✾✱ ✷✵✶✾ ✾ ✴ ✶✾

◗✉♦t✐❡♥ts ♦❢ R✲❜✉♥❞❧❡s

❆ ❝♦♥❣r✉❡♥❝❡ ∼ ♦♥ ❛♥ R✲❜✉♥❞❧❡ π: Λ → Γ ❝♦♥s✐sts ♦❢ r✐❣✐❞ ♦♣❡♥ ❝♦♥❣r✉❡♥❝❡s ∼Λ ❛♥❞ ∼Γ s✉❝❤ t❤❛t π ✐s ❛ ♠♦r♣❤✐s♠ ❢r♦♠ ∼Λ t♦ ∼Γ❀ ∼Λ ✐s ❛ ❝♦♥❣r✉❡♥❝❡ ❢♦r t❤❡ R✲❜✐♠♦❞✉❧❡ str✉❝t✉r❡s ♦❢ π−✶(γ) ✭γ ∈ Γ✮❀ ■❢ x, y ∈ Λ ❛♥❞ π(x) ∼ π(y) ✐♥ Γ✱ t❤❡♥ t❤❡r❡ ❡①✐sts y′ ∈ Λ s✉❝❤ t❤❛t y ∼ y′ ✐♥ Λ ❛♥❞ π(x) = π(y′) ❚❤❡ ✜rst ♣r♦♣❡rt② ❣✐✈❡s ✉s ❛ q✉♦t✐❡♥t ♠❛♣ π/∼: Λ/∼ → Γ/∼✳ ❚❤❡ s❡❝♦♥❞ ❛♥❞ t❤✐r❞ ♣r♦♣❡rt✐❡s ❣✐✈❡ t❤❡ R✲❜✐♠♦❞✉❧❡ str✉❝t✉r❡ ♦❢ t❤❡ ✜❜❡rs ♦❢ π/∼✱ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s ❛♥ R✲❜✉♥❞❧❡✳

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

◗✉♦t✐❡♥ts ✈s✳ q✉♦t✐❡♥ts

❚❤❡ ❛♣♣r♦♣r✐❛t❡ ❝❧❛ss ♦❢ ❛♥❞ ❡❧❡♠❡♥t x ♦❢ Γ ♦r Λ ✐s ❞❡♥♦t❡❞ x✳

❚❤❡♦r❡♠

❚❤❡ ♠❛♣ T : A(π) → A(π/∼) T(α)( γ) =

• δ∼γ
• α(δ)

✐s ❛ s✉r❥❡❝t✐✈❡ ❛❧❣❡❜r❛ ❤♦♠♦♠♦r♣❤✐s♠✳

◗✉❡st✐♦♥

■s t❤❡r❡ ❛ ❜❡tt❡r ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❦❡r♥❡❧ ♦❢ T✱ ♦t❤❡r t❤❛♥ t❤❡ ♦❜✈✐♦✉s ♦♥❡❄

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

❚❤❡ ❦❡r♥❡❧ ♦❢ T

❚❤❡ ❢✉❧❧ s❡♠✐❣r♦✉♣ ✭♦r ♣s❡✉❞♦❣r♦✉♣✮ [[∼]] ♦❢ ∼ ✭✐♥ ❡✐t❤❡r Γ ♦r Λ✮ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❤♦♠❡♦♠♦r♣❤✐s♠s ϕ: V → W ✱ ✇❤❡r❡ V ❛♥❞ W ❛r❡ ♦♣❡♥✱ s✉❝❤ t❤❛t ϕ(v) ∼ v ❢♦r ❛❧❧ v ∈ V ✳ ■❢ α ∈ A(π)✱ ✇❡ ♠❛② ✏❝♦♥❥✉❣❛t❡✑ α ❜② ❛♣♣r♦♣r✐❛t❡ ❡❧❡♠❡♥ts ♦❢ [[∼Γ]] ❛♥❞ [[∼Λ]]✿ ❙✉♣♣♦s❡ ψ: A → B ✐♥ [[∼Λ]] ❛♥❞ ϕ: π(B) → π(A) ✐♥ [[∼Γ]] s✉❝❤ t❤❛t ϕ ◦ π = π ◦ ψ ♦♥ A✳ ■❢ α ∈ A(π) ✐s s✉❝❤ t❤❛t supp(α) ⊆ π(B)✱ t❤❡♥ ✇❡ ♠❛② ❞❡✜♥❡ ψαϕ = ψ ◦ α ◦ ϕ ♦♥ π(A) ❛♥❞ ✵ ❡✈❡r②✇❤❡r❡ ❡❧s❡. ❚❤❡♥ ψαϕ ✐s ❝♦♥❥✉❣❛t❡ t♦ α✱ ❛♥❞ α − ψαϕ ∈ ker T✳

❚❤❡♦r❡♠

■❢ ✵(Γ) ✐s ∼Λ✲s❛t✉r❛t❡❞✱ t❤❡♥ t❤❡ ❦❡r♥❡❧ ♦❢ T ✐s ❣❡♥❡r❛t❡❞ ❛s ❛♥ ❛❞❞✐t✐✈❡ ❣r♦✉♣ ❜② ❛❧❧ α − ψαϕ ❛s ❛❜♦✈❡✳

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

❆♥ ❛♣♣❧✐❝❛t✐♦♥

❚❤❡ ♥❛t✉r❛❧ ♦r❞❡r ♦❢ ❛♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣♦✐❞ ✐s s ≤ t ✐✛ s = ts∗s✳ ▲❡t θ ❜❡ ❛♥ ♦♣❡♥ ❛❝t✐♦♥ ♦❢ ❛ ❞✐s❝r❡t❡ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣♦✐❞ S ♦♥ ❛ ❣r♦✉♣♦✐❞ G✳ ❚❤❡ ❣r♦✉♣♦✐❞ ♦❢ ❣❡r♠s ♦❢ ●❡r♠(θ) ✐s t❤❡ q✉♦t✐❡♥t ♦❢ t❤❡ s❡♠✐❞✐r❡❝t ♣r♦❞✉❝t S ⋉ G ❜② t❤❡ r❡❧❛t✐♦♥ (s, g) ∼ (t, h) ⇐ ⇒ ∃h = g ❛♥❞ u ≤ s, t s✉❝❤ t❤❛tu ∈ dom(θu). ✭✐t ✐s ✉♥✐✈❡rs❛❧ ❢♦r ♠♦r♣❤✐s♠s ❢r♦♠ S ⋉ G t♦ ❣r♦✉♣♦✐❞s✮✳ ▲❡t Θ: S RG ❜② dom(Θs) = {f ∈ RG : supp(f ) ⊆ dom(θs)} Θs(f ) = f ◦ θ−✶

s

♦♥ dom(θs∗), ✵ ❡✈❡r②✇❤❡r❡ ❡❧s❡.

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

❆♥ ❛♣♣❧✐❝❛t✐♦♥

❚❤❡♦r❡♠

R●❡r♠(θ) ∼ = S ⋆ RG f δs − f δt : s ≤ t, f ∈ dom(Θs) ✭t❤❡ r✐❣❤t s✐❞❡ ✐s t❤❡ ✏♥♦♥✲♥❛ï✈❡✑ ❝r♦ss❡❞ ♣r♦❞✉❝t✮✱ ✇❤❡r❡ f δs ✐s t❤❡ ❢✉♥❝t✐♦♥ t❛❦✐♥❣ s → f ✐♥ S ⋆ RG✱ ❛♥❞ ❛❧❧ t = s t♦ ✵✳

❘❡♠❛r❦

❚❤✐s ✐s ✈❛❧✐❞ ♠♦r❡ ❣❡♥❡r❛❧❧② ❢♦r ❛♠♣❧❡✱ ♥♦♥✲❍❛✉s❞♦r✛ s❡♠✐❣r♦✉♣♦✐❞s✳ ❚❤✐s ✐s ❛ ❢❛r✲r❡❛❝❤✐♥❣ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❦♥♦✇♥ r❡s✉❧ts ♦❢ t❤❡ t❤❡♦r② ♦❢ ❙t❡✐♥❜❡r❣ ❛❧❣❡❜r❛s ✭❇❡✉t❡r✲❈✳✱ ❍❛③r❛t✲▲✐✮✳

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

❚❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ♣r♦❞✉❝t ❜✉♥❞❧❡s

▲❡t π: Λ → Γ ❜❡ ❛♥ R✲❜✉♥❞❧❡✱ ❛♥❞ E ❛♥♦t❤❡r ✭❛♠♣❧❡✱ ❍❛✉s❞♦r✛✮ s❡♠✐❣r♦✉♣♦✐❞✳ ❈♦♥str✉❝t t❤❡ ♥❡✇ ❜✉♥❞❧❡ π × E = π × idE : Λ × E → Γ × E✳

❚❤❡♦r❡♠

❚❤❡ ♠❛♣ T : A(π) ⊗

R R RE → A(π × E)

T(α ⊗ f )(γ, e) = (α(γ)f (e), e) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳

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

❚❡♥s♦r ♣r♦❞✉❝ts

❈♦r♦❧❧❛r②

R(Γ✶ × Γ✷) ∼ = RΓ✶ ⊗ RΓ✷✱ ❢♦r ❛♥② r✐♥❣ R ❛♥❞ ❛♠♣❧❡ ❍❛✉s❞♦r✛ ❣r♦✉♣♦✐❞ Γ✶✱ Γ✷✳

❈♦r♦❧❧❛r②

■❢ A ✐s ❛♥ R✲❛❧❣❡❜r❛✱ t❤❡♥ AΓ ∼ = A ⊗

R R RΓ✳

✭●❡♥❡r❛❧✐③❡s r❡s✉❧ts ♦❢ ❘✐❣❜② ✬✶✽✮

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

❙♠❛s❤ ♣r♦❞✉❝ts ♦❢ ❛❧❣❡❜r❛s

▲❡t G ❜❡ ❛ ❣r♦✉♣ ❛♥❞ A = ⊕g∈GAg ❛ G✲❣r❛❞❡❞ ❛❧❣❡❜r❛✳ ▲❡t pg : A → Ag ❜❡ t❤❡ ♣r♦❥❡❝t✐♦♥✳ ❚❤❡ s♠❛s❤ ♣r♦❞✉❝t A#G ✐s t❤❡ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② s②♠❜♦❧s a#g✱ a ∈ A✱ g ∈ G✱ ✇✐t❤ ❡♥tr②✇✐s❡ ♠♦❞✉❧❡ str✉❝t✉r❡ ❛♥❞ ♣r♦❞✉❝t (a#g)(b#h) = apgh−✶(b)#h ❚❤❡♥ G ❛❝ts ♦♥ A#G ❜② g(a#h) = a#(hg−✶)✳

❚❤❡♦r❡♠ ✭❉✉❛❧✐t② t❤❡♦r❡♠s✱ ❈♦❤❡♥✲▼♦♥t❣♦♠❡r② ✬✽✹✮

MG,✵(A) ∼ = G ⋉ (A#G) ❛♥❞ A ∼ = (A#G)G ✭✜①❡❞ ❛❧❣❡❜r❛✮✱ ✇❤❡r❡ MG,✵(A) ✐s t❤❡ ❛❧❣❡❜r❛ ♦❢ ✜♥✐t❡❧② s✉♣♣♦rt❡❞ ♠❛tr✐❝❡s ✐♥❞❡①❡❞ ❜② G × G✱ ❛♥❞ ❝♦❡✣❝✐❡♥ts ✐♥ A✳

❘❡♠❛r❦ ✭❈✳ ✬❚✉❡s❞❛②✮

❚❤❡r❡ ✐s ❛ ✈❡rs✐♦♥ ❢♦r ❣r♦✉♣♦✐❞ ❣r❛❞✐♥❣s ❛s ✇❡❧❧✳

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

✏❙❦❡✇ ♣r♦❞✉❝ts✑

▲❡t G ❜❡ ❛ ❣r♦✉♣✱ Γ ❛ s❡♠✐❣r♦✉♣♦✐❞ ❛♥❞ c : Γ → G ❛ ❤♦♠♦♠♦r♣❤✐s♠✳ ❚❤❡♥ Γ ✏❛❝ts ♦♥ G ❜② ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥✑✿ γ · g = c(γ)g❀ ❛♥❞ ✇❡ ❝❛♥ ❝r❡❛t❡ t❤❡ s❦❡✇ ♣r♦❞✉❝t Γ#G = Γ × G ✇✐t❤ ♣r♦❞✉❝t (γ✶, g)(γ✷, h) = (γ✶γ✷, h) ✇❤❡♥❡✈❡r g = c(γ✷)h ■❢ ✇❡ ❤❛✈❡ ❛♥ R✲❜✉♥❞❧❡ π: Λ → Γ t❤❡♥ c ◦ π: Λ → G✳ ❈♦♥s✐❞❡r t❤❡ ✏s❦❡✇ ❜✉♥❞❧❡✑ π#G = π × idG : Λ#G → Γ#G✳

❚❤❡♦r❡♠

A(π#G) ∼ = A(π)#G✳

• ❡♥❡r❛❧✐③❡s ❛ r❡s✉❧t ♦❢ ❍❛③r❛t✲▲✐ ✬✶✽ ❢♦r ❙t❡✐♥❜❡r❣ ❛❧❣❡❜r❛s✳

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

◆❛t✉r❛❧ q✉❡st✐♦♥s

✶ ■s♦♠♦r♣❤✐s♠ t❤❡♦r❡♠ ❢♦r A(π) ✭❛♥❛❧♦❣♦✉s t♦ t❤❡ ♦♥❡ ✐♥ t❤❡

♠♦t✐✈❛t✐♦♥✮❄

✷ ■❢ S ✐s ✐♥✈❡rs❡✱ ✐s RS ∼

= RG ❢♦r s♦♠❡ ❣r♦✉♣♦✐❞ G ✭tr✉❡ ✐♥ t❤❡ ✏tr✐✈✐❛❧✑ ❝❛s❡ ♦❢ ❛ ♣r♦❞✉❝t ♦❢ ❛♥ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ❛♥❞ ❛ ❣r♦✉♣♦✐❞✮✳

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