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❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

P✐❡rr❡✲❆❧❛✐♥ ❏❛❝q♠✐♥ ❈❛t❡❣♦r② ❚❤❡♦r② ✷✵✶✼ ❏✉❧② ✷✵✱ ✷✵✶✼

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❋❛♠♦✉s ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠s

❨♦♥❡❞❛ ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢✉❧❧② ❢❛✐t❤❢✉❧ ❡♠❜❡❞❞✐♥❣ C ֒ → SetCop ✇❤✐❝❤ ♣r❡s❡r✈❡s s♠❛❧❧ ❧✐♠✐ts✳ ❇❛rr✬s ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ r❡❣✉❧❛r ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢✉❧❧② ❢❛✐t❤❢✉❧ ❡♠❜❡❞❞✐♥❣ C ֒ → SetD ✇❤✐❝❤ ♣r❡s❡r✈❡s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✳ ▲✉❜❦✐♥✬s ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛♥ ❡①❛❝t ❝♦♥s❡r✈❛t✐✈❡ ❡♠❜❡❞❞✐♥❣ C ֒ → Ab .

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❋❛♠♦✉s ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠s

❨♦♥❡❞❛ ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢✉❧❧② ❢❛✐t❤❢✉❧ ❡♠❜❡❞❞✐♥❣ C ֒ → SetCop ✇❤✐❝❤ ♣r❡s❡r✈❡s s♠❛❧❧ ❧✐♠✐ts✳ ❇❛rr✬s ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ r❡❣✉❧❛r ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢✉❧❧② ❢❛✐t❤❢✉❧ ❡♠❜❡❞❞✐♥❣ C ֒ → SetD ✇❤✐❝❤ ♣r❡s❡r✈❡s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✳ ▲✉❜❦✐♥✬s ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛♥ ❡①❛❝t ❝♦♥s❡r✈❛t✐✈❡ ❡♠❜❡❞❞✐♥❣ C ֒ → Ab .

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❋❛♠♦✉s ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠s

❨♦♥❡❞❛ ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢✉❧❧② ❢❛✐t❤❢✉❧ ❡♠❜❡❞❞✐♥❣ C ֒ → SetCop ✇❤✐❝❤ ♣r❡s❡r✈❡s s♠❛❧❧ ❧✐♠✐ts✳ ❇❛rr✬s ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ r❡❣✉❧❛r ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢✉❧❧② ❢❛✐t❤❢✉❧ ❡♠❜❡❞❞✐♥❣ C ֒ → SetD ✇❤✐❝❤ ♣r❡s❡r✈❡s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✳ ▲✉❜❦✐♥✬s ❡♠❜❡❞❞✐♥❣ ❊✈❡r② s♠❛❧❧ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛♥ ❡①❛❝t ❝♦♥s❡r✈❛t✐✈❡ ❡♠❜❡❞❞✐♥❣ C ֒ → Ab .

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡♦r❡♠ ✭❈❛r❜♦♥✐ ✲ P❡❞✐❝❝❤✐♦ ✲ P✐r♦✈❛♥♦✱ ✶✾✾✷✮ ❚❤❡ ❢♦❧❧♦✇✐♥❣s ❝♦♥❞✐t✐♦♥s ♦♥ ❛ ✜♥✐t❡❧② ❝♦♠♣❧❡t❡ ❝❛t❡❣♦r② C ❛r❡ ❡q✉✐✈❛❧❡♥t✿

✶ ❛♥② r❡✢❡①✐✈❡ r❡❧❛t✐♦♥ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✱ ✷ ❛♥② r❡✢❡①✐✈❡ r❡❧❛t✐♦♥ ✐s s②♠♠❡tr✐❝✱ ✸ ❛♥② r❡✢❡①✐✈❡ r❡❧❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✱ ✹ ❡✈❡r② r❡❧❛t✐♦♥ ✐s ❞✐❢✉♥❝t✐♦♥❛❧✳

■♥ t❤✐s ❝❛s❡✱ ✇❡ s❛② t❤❛t C ✐s ❛ ▼❛❧✬ts❡✈ ❝❛t❡❣♦r②✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❘❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡♦r❡♠ ✭❈❛r❜♦♥✐ ✲ ▲❛♠❜❡❦ ✲ P❡❞✐❝❝❤✐♦✱ ✶✾✾✵✮ ❚❤❡ ❢♦❧❧♦✇✐♥❣s ❝♦♥❞✐t✐♦♥s ♦♥ ❛ r❡❣✉❧❛r ❝❛t❡❣♦r② C ❛r❡ ❡q✉✐✈❛❧❡♥t✿

✶ C ✐s ❛ ▼❛❧✬ts❡✈ ❝❛t❡❣♦r②✱ ✷ ❢♦r ❛♥② ♣❛✐r (R, S) ♦❢ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s ♦♥ ❛ s❛♠❡ ♦❜❥❡❝t✱ t❤❡✐r

❝♦♠♣♦s✐t✐♦♥ RS ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✱

✸ ❢♦r ❛♥② ♣❛✐r (R, S) ♦❢ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s ♦♥ ❛ s❛♠❡ ♦❜❥❡❝t✱ RS = SR✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❆✐♠ ❋✐♥❞ ❛ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② M s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② s♠❛❧❧ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② C✱ t❤❡r❡ ❡①✐sts ❛ ❢❛✐t❤❢✉❧ ❝♦♥s❡r✈❛t✐✈❡ ❡♠❜❡❞❞✐♥❣ C ֒ → MD ✇❤✐❝❤ ♣r❡s❡r✈❡s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❆✐♠ ❋✐♥❞ ❛ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② M s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② s♠❛❧❧ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② C✱ t❤❡r❡ ❡①✐sts ❛ ❢❛✐t❤❢✉❧ ❝♦♥s❡r✈❛t✐✈❡ ❡♠❜❡❞❞✐♥❣ C ֒ → MD ✇❤✐❝❤ ♣r❡s❡r✈❡s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥ ❆ ✜♥✐t❛r② ❡ss❡♥t✐❛❧❧② ❛❧❣❡❜r❛✐❝ t❤❡♦r② ✐s ❛ q✉✐♥t✉♣❧❡ Γ = (S, Σ, E, Σt, Def) ✇❤❡r❡ S ✐s ❛ s❡t ♦❢ s♦rts✱ Σ ✐s ❛ s❡t ♦❢ S✲s♦rt❡❞ ✜♥✐t❛r② ♦♣❡r❛t✐♦♥ s②♠❜♦❧s✱ E ✐s ❛ s❡t ♦❢ Σ✲❡q✉❛t✐♦♥s✱ Σt ⊆ Σ ✐s t❤❡ s✉❜s❡t ♦❢ ❵t♦t❛❧ ♦♣❡r❛t✐♦♥ s②♠❜♦❧s✬✱ ❢♦r ❡❛❝❤ σ ∈ Σ \ Σt✱ Def(σ) ✐s ❛ ✜♥✐t❡ s❡t ♦❢ Σt✲❡q✉❛t✐♦♥s ✭✐♥ t❤❡ ✈❛r✐❛❜❧❡s ♦❢ σ✮✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥ ❆ Γ✲♠♦❞❡❧ A ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛♥ S✲s♦rt❡❞ s❡t (As)s∈S ∈ SetS✱ ❢♦r ❡❛❝❤ σ : s1 × · · · × sn → s ✐♥ Σ✱ ❛ ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥ σA : As1 × · · · × Asn → As s❛t✐s❢②✐♥❣ ❢♦r ❡❛❝❤ σ ∈ Σt✱ σA ✐s t♦t❛❧❧② ❞❡✜♥❡❞✱ ❢♦r ❡❛❝❤ σ ∈ Σ \ Σt✱ σ(a1, . . . , an) ✐s ❞❡✜♥❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ (a1, . . . , an) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ Def(σ) ✐♥ A✱ A s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ E ✇❤❡♥❡✈❡r t❤❡② ❛r❡ ❞❡✜♥❡❞✳ ❚❤✐s ❣✐✈❡s r✐s❡ t♦ t❤❡ ❝❛t❡❣♦r② Mod(Γ)✳ ❚❤❡♦r❡♠ ✭●❛❜r✐❡❧ ✲ ❯❧♠❡r✱ ✶✾✼✶✮ ❯♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✱ t❤❡ ❝❛t❡❣♦r✐❡s ♦❢ t❤❡ ❢♦r♠ Mod(Γ) ❛r❡ ❡①❛❝t❧② t❤❡ ❧♦❝❛❧❧② ✜♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡ ❝❛t❡❣♦r✐❡s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥ ❆ Γ✲♠♦❞❡❧ A ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛♥ S✲s♦rt❡❞ s❡t (As)s∈S ∈ SetS✱ ❢♦r ❡❛❝❤ σ : s1 × · · · × sn → s ✐♥ Σ✱ ❛ ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥ σA : As1 × · · · × Asn → As s❛t✐s❢②✐♥❣ ❢♦r ❡❛❝❤ σ ∈ Σt✱ σA ✐s t♦t❛❧❧② ❞❡✜♥❡❞✱ ❢♦r ❡❛❝❤ σ ∈ Σ \ Σt✱ σ(a1, . . . , an) ✐s ❞❡✜♥❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ (a1, . . . , an) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ Def(σ) ✐♥ A✱ A s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ E ✇❤❡♥❡✈❡r t❤❡② ❛r❡ ❞❡✜♥❡❞✳ ❚❤✐s ❣✐✈❡s r✐s❡ t♦ t❤❡ ❝❛t❡❣♦r② Mod(Γ)✳ ❚❤❡♦r❡♠ ✭●❛❜r✐❡❧ ✲ ❯❧♠❡r✱ ✶✾✼✶✮ ❯♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✱ t❤❡ ❝❛t❡❣♦r✐❡s ♦❢ t❤❡ ❢♦r♠ Mod(Γ) ❛r❡ ❡①❛❝t❧② t❤❡ ❧♦❝❛❧❧② ✜♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡ ❝❛t❡❣♦r✐❡s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥ ❆ Γ✲♠♦❞❡❧ A ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛♥ S✲s♦rt❡❞ s❡t (As)s∈S ∈ SetS✱ ❢♦r ❡❛❝❤ σ : s1 × · · · × sn → s ✐♥ Σ✱ ❛ ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥ σA : As1 × · · · × Asn → As s❛t✐s❢②✐♥❣ ❢♦r ❡❛❝❤ σ ∈ Σt✱ σA ✐s t♦t❛❧❧② ❞❡✜♥❡❞✱ ❢♦r ❡❛❝❤ σ ∈ Σ \ Σt✱ σ(a1, . . . , an) ✐s ❞❡✜♥❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ (a1, . . . , an) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ Def(σ) ✐♥ A✱ A s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ E ✇❤❡♥❡✈❡r t❤❡② ❛r❡ ❞❡✜♥❡❞✳ ❚❤✐s ❣✐✈❡s r✐s❡ t♦ t❤❡ ❝❛t❡❣♦r② Mod(Γ)✳ ❚❤❡♦r❡♠ ✭●❛❜r✐❡❧ ✲ ❯❧♠❡r✱ ✶✾✼✶✮ ❯♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✱ t❤❡ ❝❛t❡❣♦r✐❡s ♦❢ t❤❡ ❢♦r♠ Mod(Γ) ❛r❡ ❡①❛❝t❧② t❤❡ ❧♦❝❛❧❧② ✜♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡ ❝❛t❡❣♦r✐❡s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥ ❆ Γ✲♠♦❞❡❧ A ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛♥ S✲s♦rt❡❞ s❡t (As)s∈S ∈ SetS✱ ❢♦r ❡❛❝❤ σ : s1 × · · · × sn → s ✐♥ Σ✱ ❛ ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥ σA : As1 × · · · × Asn → As s❛t✐s❢②✐♥❣ ❢♦r ❡❛❝❤ σ ∈ Σt✱ σA ✐s t♦t❛❧❧② ❞❡✜♥❡❞✱ ❢♦r ❡❛❝❤ σ ∈ Σ \ Σt✱ σ(a1, . . . , an) ✐s ❞❡✜♥❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ (a1, . . . , an) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ Def(σ) ✐♥ A✱ A s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ E ✇❤❡♥❡✈❡r t❤❡② ❛r❡ ❞❡✜♥❡❞✳ ❚❤✐s ❣✐✈❡s r✐s❡ t♦ t❤❡ ❝❛t❡❣♦r② Mod(Γ)✳ ❚❤❡♦r❡♠ ✭●❛❜r✐❡❧ ✲ ❯❧♠❡r✱ ✶✾✼✶✮ ❯♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✱ t❤❡ ❝❛t❡❣♦r✐❡s ♦❢ t❤❡ ❢♦r♠ Mod(Γ) ❛r❡ ❡①❛❝t❧② t❤❡ ❧♦❝❛❧❧② ✜♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡ ❝❛t❡❣♦r✐❡s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥ ❆ Γ✲♠♦❞❡❧ A ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛♥ S✲s♦rt❡❞ s❡t (As)s∈S ∈ SetS✱ ❢♦r ❡❛❝❤ σ : s1 × · · · × sn → s ✐♥ Σ✱ ❛ ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥ σA : As1 × · · · × Asn → As s❛t✐s❢②✐♥❣ ❢♦r ❡❛❝❤ σ ∈ Σt✱ σA ✐s t♦t❛❧❧② ❞❡✜♥❡❞✱ ❢♦r ❡❛❝❤ σ ∈ Σ \ Σt✱ σ(a1, . . . , an) ✐s ❞❡✜♥❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ (a1, . . . , an) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ Def(σ) ✐♥ A✱ A s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ E ✇❤❡♥❡✈❡r t❤❡② ❛r❡ ❞❡✜♥❡❞✳ ❚❤✐s ❣✐✈❡s r✐s❡ t♦ t❤❡ ❝❛t❡❣♦r② Mod(Γ)✳ ❚❤❡♦r❡♠ ✭●❛❜r✐❡❧ ✲ ❯❧♠❡r✱ ✶✾✼✶✮ ❯♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✱ t❤❡ ❝❛t❡❣♦r✐❡s ♦❢ t❤❡ ❢♦r♠ Mod(Γ) ❛r❡ ❡①❛❝t❧② t❤❡ ❧♦❝❛❧❧② ✜♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡ ❝❛t❡❣♦r✐❡s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❉❡✜♥✐t✐♦♥ ❆ Γ✲♠♦❞❡❧ A ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛♥ S✲s♦rt❡❞ s❡t (As)s∈S ∈ SetS✱ ❢♦r ❡❛❝❤ σ : s1 × · · · × sn → s ✐♥ Σ✱ ❛ ♣❛rt✐❛❧ ❢✉♥❝t✐♦♥ σA : As1 × · · · × Asn → As s❛t✐s❢②✐♥❣ ❢♦r ❡❛❝❤ σ ∈ Σt✱ σA ✐s t♦t❛❧❧② ❞❡✜♥❡❞✱ ❢♦r ❡❛❝❤ σ ∈ Σ \ Σt✱ σ(a1, . . . , an) ✐s ❞❡✜♥❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ (a1, . . . , an) s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ Def(σ) ✐♥ A✱ A s❛t✐s✜❡s t❤❡ ❡q✉❛t✐♦♥s ♦❢ E ✇❤❡♥❡✈❡r t❤❡② ❛r❡ ❞❡✜♥❡❞✳ ❚❤✐s ❣✐✈❡s r✐s❡ t♦ t❤❡ ❝❛t❡❣♦r② Mod(Γ)✳ ❚❤❡♦r❡♠ ✭●❛❜r✐❡❧ ✲ ❯❧♠❡r✱ ✶✾✼✶✮ ❯♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✱ t❤❡ ❝❛t❡❣♦r✐❡s ♦❢ t❤❡ ❢♦r♠ Mod(Γ) ❛r❡ ❡①❛❝t❧② t❤❡ ❧♦❝❛❧❧② ✜♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡ ❝❛t❡❣♦r✐❡s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❊①❛♠♣❧❡

❚❤❡ ❝❛t❡❣♦r② Cat ✐s ♦❢ t❤❡ ❢♦r♠ Mod(Γ)✿ t✇♦ s♦rts✿ O ❛♥❞ A ♦♣❡r❛t✐♦♥s✿ A2

m

A

d

• c

O

e

• Def(m) = {(f, g) ∈ A2 | c(f) = d(g)}

✰ ❛①✐♦♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❊①❛♠♣❧❡

❚❤❡ ❝❛t❡❣♦r② Cat ✐s ♦❢ t❤❡ ❢♦r♠ Mod(Γ)✿ t✇♦ s♦rts✿ O ❛♥❞ A ♦♣❡r❛t✐♦♥s✿ A2

m

A

d

• c

O

e

• Def(m) = {(f, g) ∈ A2 | c(f) = d(g)}

✰ ❛①✐♦♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❊①❛♠♣❧❡

❚❤❡ ❝❛t❡❣♦r② Cat ✐s ♦❢ t❤❡ ❢♦r♠ Mod(Γ)✿ t✇♦ s♦rts✿ O ❛♥❞ A ♦♣❡r❛t✐♦♥s✿ A2

m

A

d

• c

O

e

• Def(m) = {(f, g) ∈ A2 | c(f) = d(g)}

✰ ❛①✐♦♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❊①❛♠♣❧❡

❚❤❡ ❝❛t❡❣♦r② Cat ✐s ♦❢ t❤❡ ❢♦r♠ Mod(Γ)✿ t✇♦ s♦rts✿ O ❛♥❞ A ♦♣❡r❛t✐♦♥s✿ A2

m

A

d

• c

O

e

• Def(m) = {(f, g) ∈ A2 | c(f) = d(g)}

✰ ❛①✐♦♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❊①❛♠♣❧❡

❚❤❡ ❝❛t❡❣♦r② Cat ✐s ♦❢ t❤❡ ❢♦r♠ Mod(Γ)✿ t✇♦ s♦rts✿ O ❛♥❞ A ♦♣❡r❛t✐♦♥s✿ A2

m

A

d

• c

O

e

• Def(m) = {(f, g) ∈ A2 | c(f) = d(g)}

✰ ❛①✐♦♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡♦r❡♠ ✭▼❛❧✬ts❡✈✱ ✶✾✺✹✮ ❆ ✈❛r✐❡t② ♦❢ ✉♥✐✈❡rs❛❧ ❛❧❣❡❜r❛s V ✐s ❛ ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts t❤❡♦r② ❝♦♥t❛✐♥s ❛ t❡r♥❛r② ♦♣❡r❛t✐♦♥ p(x, y, z) s❛t✐s❢②✐♥❣ t❤❡ ✐❞❡♥t✐t✐❡s

• p(x, y, y) = x

p(x, x, y) = y. ❚❤❡♦r❡♠ ▲❡t Γ ❜❡ ❛♥ ❡ss❡♥t✐❛❧❧② ❛❧❣❡❜r❛✐❝ t❤❡♦r②✳ ❚❤❡♥ Mod(Γ) ✐s ❛ ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② ✐❢ ❛♥❞ ♦♥❧② ✐❢✱ ❢♦r ❡❛❝❤ s♦rt s ∈ S✱ t❤❡r❡ ❡①✐sts ✐♥ Γ ❛ t❡r♠ ps : s3 → s s✉❝❤ t❤❛t ps(x, x, y) ❛♥❞ ps(x, y, y) ❛r❡ ❡✈❡r②✇❤❡r❡✲❞❡✜♥❡❞ ❛♥❞ ps(x, x, y) = y ❛♥❞ ps(x, y, y) = x ❛r❡ t❤❡♦r❡♠s ♦❢ Γ✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡♦r❡♠ ✭▼❛❧✬ts❡✈✱ ✶✾✺✹✮ ❆ ✈❛r✐❡t② ♦❢ ✉♥✐✈❡rs❛❧ ❛❧❣❡❜r❛s V ✐s ❛ ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts t❤❡♦r② ❝♦♥t❛✐♥s ❛ t❡r♥❛r② ♦♣❡r❛t✐♦♥ p(x, y, z) s❛t✐s❢②✐♥❣ t❤❡ ✐❞❡♥t✐t✐❡s

• p(x, y, y) = x

p(x, x, y) = y. ❚❤❡♦r❡♠ ▲❡t Γ ❜❡ ❛♥ ❡ss❡♥t✐❛❧❧② ❛❧❣❡❜r❛✐❝ t❤❡♦r②✳ ❚❤❡♥ Mod(Γ) ✐s ❛ ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② ✐❢ ❛♥❞ ♦♥❧② ✐❢✱ ❢♦r ❡❛❝❤ s♦rt s ∈ S✱ t❤❡r❡ ❡①✐sts ✐♥ Γ ❛ t❡r♠ ps : s3 → s s✉❝❤ t❤❛t ps(x, x, y) ❛♥❞ ps(x, y, y) ❛r❡ ❡✈❡r②✇❤❡r❡✲❞❡✜♥❡❞ ❛♥❞ ps(x, x, y) = y ❛♥❞ ps(x, y, y) = x ❛r❡ t❤❡♦r❡♠s ♦❢ Γ✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡ ❝❛t❡❣♦r② Mod(ΓMal)

❲❡ ❝♦♥str✉❝t ❛ ✜♥✐t❛r② ❡ss❡♥t✐❛❧❧② ❛❧❣❡❜r❛✐❝ t❤❡♦r② ΓMal s✉❝❤ t❤❛t✿ ❢♦r ❡❛❝❤ s♦rt s ∈ SMal✱ t❤❡r❡ ❡①✐sts ❛ s♦rt s ❛♥❞ ♦♣❡r❛t✐♦♥ s②♠❜♦❧s s3

ρs

s

s

αs

• s❛t✐s❢②✐♥❣ t❤❡ ❛①✐♦♠s

     ρs(x, y, y) = αs(x) ρs(x, x, y) = αs(y) πs(αs(x)) = x ❛♥❞ s✉❝❤ t❤❛t πs(αs(x)) ✐s ❡✈❡r②✇❤❡r❡✲❞❡✜♥❡❞✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡ ❝❛t❡❣♦r② Mod(ΓMal)

❲❡ ❝♦♥str✉❝t ❛ ✜♥✐t❛r② ❡ss❡♥t✐❛❧❧② ❛❧❣❡❜r❛✐❝ t❤❡♦r② ΓMal s✉❝❤ t❤❛t✿ ❢♦r ❡❛❝❤ s♦rt s ∈ SMal✱ t❤❡r❡ ❡①✐sts ❛ s♦rt s ❛♥❞ ♦♣❡r❛t✐♦♥ s②♠❜♦❧s s3

ρs

s

s

αs

• s❛t✐s❢②✐♥❣ t❤❡ ❛①✐♦♠s

     ρs(x, y, y) = αs(x) ρs(x, x, y) = αs(y) πs(αs(x)) = x ❛♥❞ s✉❝❤ t❤❛t πs(αs(x)) ✐s ❡✈❡r②✇❤❡r❡✲❞❡✜♥❡❞✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡ ❝❛t❡❣♦r② Mod(ΓMal)

❲❡ ❝♦♥str✉❝t ❛ ✜♥✐t❛r② ❡ss❡♥t✐❛❧❧② ❛❧❣❡❜r❛✐❝ t❤❡♦r② ΓMal s✉❝❤ t❤❛t✿ ❢♦r ❡❛❝❤ s♦rt s ∈ SMal✱ t❤❡r❡ ❡①✐sts ❛ s♦rt s ❛♥❞ ♦♣❡r❛t✐♦♥ s②♠❜♦❧s s3

ρs

s

πs

• s

αs

• s❛t✐s❢②✐♥❣ t❤❡ ❛①✐♦♠s

     ρs(x, y, y) = αs(x) ρs(x, x, y) = αs(y) πs(αs(x)) = x ❛♥❞ s✉❝❤ t❤❛t πs(αs(x)) ✐s ❡✈❡r②✇❤❡r❡✲❞❡✜♥❡❞✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡ ❝❛t❡❣♦r② Mod(ΓMal)

Mod(ΓMal) ✐s ❛ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r②✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❤♦❧❞s✿ ❚❤❡♦r❡♠ ❊✈❡r② s♠❛❧❧ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢❛✐t❤❢✉❧ ❝♦♥s❡r✈❛t✐✈❡ ❡♠❜❡❞❞✐♥❣ C ֒ → Mod(ΓMal)Sub(1) ✇❤✐❝❤ ♣r❡s❡r✈❡s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡ ❝❛t❡❣♦r② Mod(ΓMal)

Mod(ΓMal) ✐s ❛ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r②✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❤♦❧❞s✿ ❚❤❡♦r❡♠ ❊✈❡r② s♠❛❧❧ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢❛✐t❤❢✉❧ ❝♦♥s❡r✈❛t✐✈❡ ❡♠❜❡❞❞✐♥❣ C ֒ → Mod(ΓMal)Sub(1) ✇❤✐❝❤ ♣r❡s❡r✈❡s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡ ❝❛t❡❣♦r② Mod(ΓMal)

Mod(ΓMal) ✐s ❛ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r②✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❤♦❧❞s✿ ❚❤❡♦r❡♠ ❊✈❡r② s♠❛❧❧ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r② C ❛❞♠✐ts ❛ ❢❛✐t❤❢✉❧ ❝♦♥s❡r✈❛t✐✈❡ ❡♠❜❡❞❞✐♥❣ C ֒ → Mod(ΓMal)Sub(1) ✇❤✐❝❤ ♣r❡s❡r✈❡s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✳ ▼♦r❡♦✈❡r✱ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s ❛r❡ s❡♥t t♦ ❝♦♠♣♦♥❡♥t✇✐s❡ s✉r❥❡❝t✐✈❡ ❤♦♠♦♠♦r♣❤✐s♠s✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❆♣♣❧✐❝❛t✐♦♥

Pr♦♣♦s✐t✐♦♥ ✭❇♦✉r♥✱ ✷✵✵✸✮ ▲❡t C ❜❡ ❛ r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r②✳ ❋♦r ❛♥② ❝♦♠♠✉t❛t✐✈❡ ❞✐❛❣r❛♠ X ×Y Z

• λ

U ×V W

• Z

g

• δ

W

k

• X
• f
• γ

U

• h
• Y

t

• β

V

v

• ✐❢ γ ❛♥❞ δ ❛r❡ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠s✱ t❤❡♥ t❤❡ ❝♦♠♣❛r✐s♦♥ ♠♦r♣❤✐s♠ λ ✐s

❛❧s♦ ❛ r❡❣✉❧❛r ❡♣✐♠♦r♣❤✐s♠✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❡ ✈❛r✐❡t❛❧ ♣r♦♦❢

▲❡t (u, w) ∈ U ×V W✳ ❙♦ u ∈ U ❛♥❞ w ∈ W ❛r❡ s✉❝❤ t❤❛t h(u) = k(w)✳ ❙✐♥❝❡ γ ❛♥❞ δ ❛r❡ s✉r❥❡❝t✐✈❡✱ t❤❡r❡ ❡①✐st x ∈ X ❛♥❞ z ∈ Z s✉❝❤ t❤❛t γ(x) = u ❛♥❞ δ(z) = w✳ ▲❡t z′ = p(z, tg(z), tf(x)) ∈ Z✳ (x, z′) ∈ X ×Y Z s✐♥❝❡ g(z′) = p(g(z), gtg(z), gtf(x)) = p(g(z), g(z), f(x)) = f(x). λ(x, z′) = (u, w) s✐♥❝❡ γ(x) = u ❛♥❞ δ(z′) = p(δ(z), δtg(z), δtf(x)) = p(δ(z), vkδ(z), vhγ(x)) = p(w, vk(w), vh(u)) = p(w, vk(w), vk(w)) = w.

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❯s✐♥❣ t❤❡ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠

▲❡t s ❜❡ ❛ s♦rt ✐♥ ΓMal ▲❡t (u, w) ∈ U ×V W✳ ❙♦ u ∈ U ❛♥❞ w ∈ W ❛r❡ s✉❝❤ t❤❛t h(u) = k(w)✳ ❙✐♥❝❡ γ ❛♥❞ δ ❛r❡ s✉r❥❡❝t✐✈❡✱ t❤❡r❡ ❡①✐st x ∈ X ❛♥❞ z ∈ Z s✉❝❤ t❤❛t γ(x) = u ❛♥❞ δ(z) = w✳ ▲❡t z′ = p(z, tg(z), tf(x)) ∈ Z✳ (x, z′) ∈ X ×Y Z s✐♥❝❡ g(z′) = p(g(z), gtg(z), gtf(x)) = p(g(z), g(z), f(x)) = f(x). λ(x, z′) = (u, w) s✐♥❝❡ γ(x) = u ❛♥❞ δ(z′) = p(δ(z), δtg(z), δtf(x)) = p(δ(z), vkδ(z), vhγ(x)) = p(w, vk(w), vh(u)) = p(w, vk(w), vk(w)) = w.

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❯s✐♥❣ t❤❡ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠

▲❡t s ❜❡ ❛ s♦rt ✐♥ ΓMal ▲❡t (u, w) ∈ (U ×V W)s✳ ❙♦ u ∈ Us ❛♥❞ w ∈ Ws ❛r❡ s✉❝❤ t❤❛t h(u) = k(w)✳ ❙✐♥❝❡ γ ❛♥❞ δ ❛r❡ s✉r❥❡❝t✐✈❡✱ t❤❡r❡ ❡①✐st x ∈ X ❛♥❞ z ∈ Z s✉❝❤ t❤❛t γ(x) = u ❛♥❞ δ(z) = w✳ ▲❡t z′ = p(z, tg(z), tf(x)) ∈ Z✳ (x, z′) ∈ X ×Y Z s✐♥❝❡ g(z′) = p(g(z), gtg(z), gtf(x)) = p(g(z), g(z), f(x)) = f(x). λ(x, z′) = (u, w) s✐♥❝❡ γ(x) = u ❛♥❞ δ(z′) = p(δ(z), δtg(z), δtf(x)) = p(δ(z), vkδ(z), vhγ(x)) = p(w, vk(w), vh(u)) = p(w, vk(w), vk(w)) = w.

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❯s✐♥❣ t❤❡ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠

▲❡t s ❜❡ ❛ s♦rt ✐♥ ΓMal ▲❡t (u, w) ∈ (U ×V W)s✳ ❙♦ u ∈ Us ❛♥❞ w ∈ Ws ❛r❡ s✉❝❤ t❤❛t h(u) = k(w)✳ ❙✐♥❝❡ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t γ ❛♥❞ δ ❛r❡ s✉r❥❡❝t✐✈❡✱ t❤❡r❡ ❡①✐st x ∈ Xs ❛♥❞ z ∈ Zs s✉❝❤ t❤❛t γ(x) = u ❛♥❞ δ(z) = w✳ ▲❡t z′ = p(z, tg(z), tf(x)) ∈ Z✳ (x, z′) ∈ X ×Y Z s✐♥❝❡ g(z′) = p(g(z), gtg(z), gtf(x)) = p(g(z), g(z), f(x)) = f(x). λ(x, z′) = (u, w) s✐♥❝❡ γ(x) = u ❛♥❞ δ(z′) = p(δ(z), δtg(z), δtf(x)) = p(δ(z), vkδ(z), vhγ(x)) = p(w, vk(w), vh(u)) = p(w, vk(w), vk(w)) = w.

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❯s✐♥❣ t❤❡ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠

▲❡t s ❜❡ ❛ s♦rt ✐♥ ΓMal ▲❡t (u, w) ∈ (U ×V W)s✳ ❙♦ u ∈ Us ❛♥❞ w ∈ Ws ❛r❡ s✉❝❤ t❤❛t h(u) = k(w)✳ ❙✐♥❝❡ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t γ ❛♥❞ δ ❛r❡ s✉r❥❡❝t✐✈❡✱ t❤❡r❡ ❡①✐st x ∈ Xs ❛♥❞ z ∈ Zs s✉❝❤ t❤❛t γ(x) = u ❛♥❞ δ(z) = w✳ ▲❡t z′ = ρs(z, tg(z), tf(x)) ∈ Zs✳ (x, z′) ∈ X ×Y Z s✐♥❝❡ g(z′) = p(g(z), gtg(z), gtf(x)) = p(g(z), g(z), f(x)) = f(x). λ(x, z′) = (u, w) s✐♥❝❡ γ(x) = u ❛♥❞ δ(z′) = p(δ(z), δtg(z), δtf(x)) = p(δ(z), vkδ(z), vhγ(x)) = p(w, vk(w), vh(u)) = p(w, vk(w), vk(w)) = w.

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❯s✐♥❣ t❤❡ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠

▲❡t s ❜❡ ❛ s♦rt ✐♥ ΓMal ▲❡t (u, w) ∈ (U ×V W)s✳ ❙♦ u ∈ Us ❛♥❞ w ∈ Ws ❛r❡ s✉❝❤ t❤❛t h(u) = k(w)✳ ❙✐♥❝❡ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t γ ❛♥❞ δ ❛r❡ s✉r❥❡❝t✐✈❡✱ t❤❡r❡ ❡①✐st x ∈ Xs ❛♥❞ z ∈ Zs s✉❝❤ t❤❛t γ(x) = u ❛♥❞ δ(z) = w✳ ▲❡t z′ = ρs(z, tg(z), tf(x)) ∈ Zs✳ (αs(x), z′) ∈ (X ×Y Z)s s✐♥❝❡ g(z′) = ρs(g(z), gtg(z), gtf(x)) = ρs(g(z), g(z), f(x)) = f(αs(x)). λ(x, z′) = (u, w) s✐♥❝❡ γ(x) = u ❛♥❞ δ(z′) = p(δ(z), δtg(z), δtf(x)) = p(δ(z), vkδ(z), vhγ(x)) = p(w, vk(w), vh(u)) = p(w, vk(w), vk(w)) = w.

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❯s✐♥❣ t❤❡ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠

▲❡t s ❜❡ ❛ s♦rt ✐♥ ΓMal ▲❡t (u, w) ∈ (U ×V W)s✳ ❙♦ u ∈ Us ❛♥❞ w ∈ Ws ❛r❡ s✉❝❤ t❤❛t h(u) = k(w)✳ ❙✐♥❝❡ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t γ ❛♥❞ δ ❛r❡ s✉r❥❡❝t✐✈❡✱ t❤❡r❡ ❡①✐st x ∈ Xs ❛♥❞ z ∈ Zs s✉❝❤ t❤❛t γ(x) = u ❛♥❞ δ(z) = w✳ ▲❡t z′ = ρs(z, tg(z), tf(x)) ∈ Zs✳ (αs(x), z′) ∈ (X ×Y Z)s s✐♥❝❡ g(z′) = ρs(g(z), gtg(z), gtf(x)) = ρs(g(z), g(z), f(x)) = f(αs(x)). λ(αs(x), z′) = αs(u, w) s✐♥❝❡ γ(αs(x)) = αs(u) ❛♥❞ δ(z′) = ρs(δ(z), δtg(z), δtf(x)) = ρs(δ(z), vkδ(z), vhγ(x)) = ρs(w, vk(w), vh(u)) = ρs(w, vk(w), vk(w)) = αs(w).

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❯s✐♥❣ t❤❡ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠

▲❡t s ❜❡ ❛ s♦rt ✐♥ ΓMal ▲❡t (u, w) ∈ (U ×V W)s✳ ❙♦ u ∈ Us ❛♥❞ w ∈ Ws ❛r❡ s✉❝❤ t❤❛t h(u) = k(w)✳ ❙✐♥❝❡ ✇❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t γ ❛♥❞ δ ❛r❡ s✉r❥❡❝t✐✈❡✱ t❤❡r❡ ❡①✐st x ∈ Xs ❛♥❞ z ∈ Zs s✉❝❤ t❤❛t γ(x) = u ❛♥❞ δ(z) = w✳ ▲❡t z′ = ρs(z, tg(z), tf(x)) ∈ Zs✳ (αs(x), z′) ∈ (X ×Y Z)s s✐♥❝❡ g(z′) = ρs(g(z), gtg(z), gtf(x)) = ρs(g(z), g(z), f(x)) = f(αs(x)). λ(αs(x), z′) = αs(u, w) s✐♥❝❡ γ(αs(x)) = αs(u) ❛♥❞ δ(z′) = ρs(δ(z), δtg(z), δtf(x)) = ρs(δ(z), vkδ(z), vhγ(x)) = ρs(w, vk(w), vh(u)) = ρs(w, vk(w), vk(w)) = αs(w). (u, w) = πs(αs(u, w)) = πs(λ(αs(x), z′)) ∈ Im(λ)✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

■♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♣r♦♦❢

❚❤❡ ♣r♦♦❢ r❡❧✐❡s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❦❡② ✐♥❣r❡❞✐❡♥ts✿ ❆♣♣r♦①✐♠❛t❡ ▼❛❧✬ts❡✈ ♦♣❡r❛t✐♦♥s ✭❇♦✉r♥ ✲ ❏❛♥❡❧✐❞③❡✱ ✷✵✵✽✮✳ ❆ C✲♣r♦❥❡❝t✐✈❡ ❝♦✈❡r✐♥❣ ♦❢ Lex(C, Set)op ✭●r♦t❤❡♥❞✐❡❝❦ ✶✾✺✼✱ ❇❛rr ✶✾✽✻✮✳ ❚❤❡ t❤❡♦r② ♦❢ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❡①❛❝t♥❡ss ♣r♦♣❡rt✐❡s ✭❏✳ ✲ ❏❛♥❡❧✐❞③❡✮✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

■♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♣r♦♦❢

❚❤❡ ♣r♦♦❢ r❡❧✐❡s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❦❡② ✐♥❣r❡❞✐❡♥ts✿ ❆♣♣r♦①✐♠❛t❡ ▼❛❧✬ts❡✈ ♦♣❡r❛t✐♦♥s ✭❇♦✉r♥ ✲ ❏❛♥❡❧✐❞③❡✱ ✷✵✵✽✮✳ ❆ C✲♣r♦❥❡❝t✐✈❡ ❝♦✈❡r✐♥❣ ♦❢ Lex(C, Set)op ✭●r♦t❤❡♥❞✐❡❝❦ ✶✾✺✼✱ ❇❛rr ✶✾✽✻✮✳ ❚❤❡ t❤❡♦r② ♦❢ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❡①❛❝t♥❡ss ♣r♦♣❡rt✐❡s ✭❏✳ ✲ ❏❛♥❡❧✐❞③❡✮✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

■♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♣r♦♦❢

❚❤❡ ♣r♦♦❢ r❡❧✐❡s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❦❡② ✐♥❣r❡❞✐❡♥ts✿ ❆♣♣r♦①✐♠❛t❡ ▼❛❧✬ts❡✈ ♦♣❡r❛t✐♦♥s ✭❇♦✉r♥ ✲ ❏❛♥❡❧✐❞③❡✱ ✷✵✵✽✮✳ ❆ C✲♣r♦❥❡❝t✐✈❡ ❝♦✈❡r✐♥❣ ♦❢ Lex(C, Set)op ✭●r♦t❤❡♥❞✐❡❝❦ ✶✾✺✼✱ ❇❛rr ✶✾✽✻✮✳ ❚❤❡ t❤❡♦r② ♦❢ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❡①❛❝t♥❡ss ♣r♦♣❡rt✐❡s ✭❏✳ ✲ ❏❛♥❡❧✐❞③❡✮✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

• ❡♥❡r❛❧✐s❛t✐♦♥

❲❡ ❤❛✈❡ s✐♠✐❧❛r ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠s ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛ss❡s ♦❢ ❝❛t❡❣♦r✐❡s✿ n✲♣❡r♠✉t❛❜❧❡ ❝❛t❡❣♦r✐❡s✱ r❡❣✉❧❛r ✉♥✐t❛❧ ❝❛t❡❣♦r✐❡s✱ r❡❣✉❧❛r str♦♥❣❧② ✉♥✐t❛❧ ❝❛t❡❣♦r✐❡s✱ r❡❣✉❧❛r s✉❜tr❛❝t✐✈❡ ❝❛t❡❣♦r✐❡s✱ ✳ ✳ ✳

❆♥ ❡♠❜❡❞❞✐♥❣ t❤❡♦r❡♠ ❢♦r r❡❣✉❧❛r ▼❛❧✬ts❡✈ ❝❛t❡❣♦r✐❡s

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦