# Categorias monoidais e o Teorema de Mac Lane para a condição estrita

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### ▲❛♥❡ ♣❛r❛ ❛ ❝♦♥❞✐çã♦ ❡str✐t❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐✲ ❝❛❞❛✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❋ís✐❝❛s ❡ ▼❛t❡♠át✐❝❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❝♦♠ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ➪❧❣❡❜r❛✳

●❛❜r✐❡❧ ❙❛♠✉❡❧ ❞❡ ❆♥❞r❛❞❡ ❋❧♦r✐❛♥ó♣♦❧✐s

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### ▲❛♥❡ ♣❛r❛ ❛ ❝♦♥❞✐çã♦ ❡str✐t❛

♣♦r

●❛❜r✐❡❧ ❙❛♠✉❡❧ ❞❡ ❆♥❞r❛❞❡✶

❊st❛ ❉✐ss❡rt❛çã♦ ❢♦✐ ❥✉❧❣❛❞❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ✏▼❡str❡✑✱ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ➪❧❣❡❜r❛✱ ❡ ❛♣r♦✈❛❞❛ ❡♠ s✉❛ ❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✳

Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s ❈♦♦r❞❡♥❛❞♦r ❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛

Pr♦❢✳➟ ❉r❛✳ ❱✐r❣í♥✐❛ ❙✐❧✈❛ ❘♦❞r✐❣✉❡s ✭❖r✐❡♥t❛❞♦r❛ ✲ ❯❋❙❈✮

❆❜❞❡❧♠♦✉❜✐♥❡ ❆♠❛r ❍❡♥♥✐

✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮

▲✉③ ❆❞r✐❛♥❛ ▼❡❥í❛ ❈❛st❛ñ♦

✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮

❘❡❣✐♥❛ ▼❛r✐❛ ❞❡ ❆q✉✐♥♦

✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦ ✲ ❯❋❊❙✮

❙ér❣✐♦ ❚❛❞❛♦ ▼❛rt✐♥s

✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮

❋❧♦r✐❛♥ó♣♦❧✐s✱ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻✳

❇♦❧s✐st❛ ❞♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✲ ❈◆Pq

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✏■t ❤❛s ❧♦♥❣ ❜❡❡♥ ❛♥ ❛①✐♦♠ ♦❢ ♠✐♥❡ t❤❛t t❤❡ ❧✐tt❧❡ t❤✐♥❣s ❛r❡ ✐♥✜♥✐t❡❧② t❤❡ ♠♦st ✐♠♣♦rt❛♥t✳✑ ❙❤❡r❧♦❝❦ ❍♦❧♠❡s

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## ❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❞❡♠♦♥str❛r ♦ ❚❡♦r❡♠❛ ❞❡ ▼❛❝ ▲❛♥❡ ♣❛r❛ ❛ ❝♦♥❞✐çã♦ ❡str✐t❛✳ ❚❛❧ t❡♦r❡♠❛ ❛✜r♠❛ q✉❡ t♦❞❛ ❝❛t❡✲ ❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ❡ ❞❡♠♦♥str❛♠♦s q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ t❛♠❜é♠ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡sq✉❡❧ét✐❝❛✳

❯t✐❧✐③❛♠♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♣r✐♥❝✐♣❛❧ ❛s ♥♦t❛s ❞❡ ❛✉❧❛ ❯♥❛ ✐♥tr♦❞✉✲ ❝✐ó♥ ❛ ❧❛s ❝❛t❡❣♦rí❛s t❡♥s♦r✐❛❧❡s ② s✉s r❡♣r❡s❡♥t❛❝✐♦♥❡s ❞♦ Pr♦❢✳ ❉r✳ ▼❛rtí♥ ▼♦♠❜❡❧❧✐✳

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## ❆❜str❛❝t

❚❤❡ ♣r❡s❡♥t ✇♦r❦ ❛✐♠s t♦ ❞❡♠♦♥str❛t❡ ▼❛❝ ▲❛♥❡✬s ❙tr✐❝t♥❡ss ❚❤❡✲ ♦r❡♠✳ ❚❤✐s t❤❡♦r❡♠ st❛t❡s t❤❛t ❛♥② ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r② ✐s ♠♦♥♦✐❞❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❛ str✐❝t ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r②✳ ▼♦r❡♦✈❡r✱ ✇❡ ♣r❡s❡♥t ❛❜❡❧✐❛♥ ❝❛t❡❣♦r✐❡s ❛♥❞ ❞❡♠♦♥str❛t❡ t❤❛t ❛♥② ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r② ✐s ♠♦♥♦✐❞❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❛ s❦❡❧❡t❛❧ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r②✳

❲❡ ✉s❡❞ ❛s t❤❡ ♠❛✐♥ r❡❢❡r❡♥❝❡ t❤❡ ❝❧❛ss ♥♦t❡s ❯♥❛ ✐♥tr♦❞✉❝✐ó♥ ❛ ❧❛s ❝❛t❡❣♦rí❛s t❡♥s♦r✐❛❧❡s ② s✉s r❡♣r❡s❡♥t❛❝✐♦♥❡s ♦❢ t❤❡ Pr♦❢✳ ❉r✳ ▼❛rtí♥ ▼♦♠❜❡❧❧✐✳

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## ❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ Pré✲r❡q✉✐s✐t♦s ✹

✶✳✶ ❈❛t❡❣♦r✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ▼♦♥♦♠♦r✜s♠♦s✱ ❡♣✐♠♦r✜s♠♦s ❡ ✐s♦♠♦r✜s♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸ ❋✉♥t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✹ ❚r❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷ ❈❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ✸✻

✷✳✶ ◆ú❝❧❡♦s ❡ ❝♦♥ú❝❧❡♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✷ ❈❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✸ ❈❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✸ ❈❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ✻✻

✸✳✶ ❈❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✹ ▼❛❝ ▲❛♥❡✬s ❙tr✐❝t♥❡ss ❚❤❡♦r❡♠ ✾✽

✹✳✶ ❈♦♥str✉çã♦ ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✹✳✷ ❚❡♦r❡♠❛ ❞❡ ▼❛❝ ▲❛♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹

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## ■♥tr♦❞✉çã♦

❆ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s é ❛♣r❡s❡♥t❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ ✶✾✹✺✱ ♥♦ tr❛❜❛❧❤♦ ❞❡ ❙❛♠✉❡❧ ❊✐❧❡♥❜❡r❣ ❡ ❙❛✉♥❞❡rs ▼❛❝ ▲❛♥❡ ✐♥t✐t✉❧❛❞♦ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ◆❛t✉r❛❧ ❊q✉✐✈❛❧❡♥❝❡s✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡♥t❡♥❞❡r tr❛♥s❢♦r♠❛✲ çõ❡s ♥❛t✉r❛✐s✳ P♦r s❡r ✉♠❛ t❡♦r✐❛ tã♦ ❛❜str❛t❛ q✉❡ ❛♣❛r❡♥t❡♠❡♥t❡ ♥ã♦ t❡♠ ❝♦♥t❡ú❞♦✱ ❢♦✐ ❝❤❛♠❛❞❛ ❞❡ ✏❛❜str❛çã♦ s❡♠ s❡♥t✐❞♦✑✳ ❆t✉❛❧♠❡♥t❡✱ t♦r♥♦✉✲s❡ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♣♦❞❡r♦s❛✱ ✐♥❞✐s♣❡♥sá✈❡❧ ❡♠ ♠✉✐t❛s ár❡s ❞❛ ♠❛t❡♠át✐❝❛✱ ❝♦♠♦ ❣❡♦♠❡tr✐❛ ❛❧❣é❜r✐❝❛✱ t♦♣♦❧♦❣✐❛ ❡ t❡♦r✐❛ ❞❡ r❡♣r❡s❡♥✲ t❛çõ❡s✳

❉❡s❡♥✈♦❧✈✐♠❡♥t♦s ✐♠♣♦rt❛♥t❡s ❛❝♦♥t❡❝❡r❛♠ q✉❛♥❞♦ ❝❛t❡❣♦r✐❛s ❝♦✲ ♠❡ç❛r❛♠ ❛ s❡r❡♠ ✉s❛❞❛s ❡♠ t❡♦r✐❛ ❞❡ ❤♦♠♦❧♦❣✐❛ ❡ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛✳ ▼❛❝ ▲❛♥❡✱ ❇✉❝❤s❜❛✉♠✱ ●r♦t❤❡♥❞✐❡❝❦ ❡ ❍❡❧❧❡r ❝♦♥s✐❞❡r❛r❛♠ ❝❛t❡❣♦r✐❛s ❡♠ q✉❡ ❛s ❝♦❧❡çõ❡s ❞❡ ♠♦r✜s♠♦s ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s ✜①❛❞♦s tê♠ ✉♠❛ ❡str✉t✉r❛ ❛❞✐❝✐♦♥❛❧✳ P♦r ❡①❡♠♣❧♦✱ ❞❛❞♦s ♦❜❥❡t♦s X ❡Y ❞❡ ✉♠❛ ❝❛t❡✲

❣♦r✐❛ C✱ ♦ ❝♦♥❥✉♥t♦ HomC(X, Y) ❞❡ ♠♦r✜s♠♦s ❞❡X ❡♠Y ❢♦r♠❛ ✉♠

❣r✉♣♦ ❛❜❡❧✐❛♥♦✳

❉❡s❞❡ ❡♥tã♦✱ s✉r❣✐r❛♠ ♦✉tr♦s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s ❝♦♠ ❡str✉t✉✲ r❛s s❡♠❡❧❤❛♥t❡s às ❝♦♥❤❡❝✐❞❛s ❞❛ á❧❣❡❜r❛ ♦r❞✐♥ár✐❛✳ ❈♦♠♦ ♦❜s❡r✈❛❞♦ ❡♠ ❬✺❪✱ ✉♠❛ ❜♦❛ ♠❛♥❡✐r❛ ❞❡ ♣❡♥s❛r ❡♠ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s é ❝♦♠♦ ✉♠ r❡✜♥❛♠❡♥t♦ ✭♦✉ ✏❝❛t❡❣♦r✐✜❝❛çã♦✑✮ ❞❛ á❧❣❡❜r❛ ♦r❞✐♥ár✐❛✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❡①✐st❡ ✉♠ ❞✐❝✐♦♥ár✐♦ ❡♥tr❡ ❡st❛s ❞✉❛s ár❡❛s✱ t❛❧ q✉❡ ❡str✉t✉r❛s ❛❧❣é❜r✐❝❛s ❝♦♠✉♥s sã♦ ♦❜t✐❞❛s ❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡str✉t✉r❛s ❝❛t❡❣ó✲ r✐❝❛s ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❞❡ ✐s♦♠♦r✜s♠♦ ❞❡ ♦❜❥❡t♦s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ♥♦çã♦ ❞❡ ❝❛t❡❣♦r✐❛ ♣❡q✉❡♥❛ é ✉♠❛ ❝❛t❡❣♦r✐✜❝❛çã♦ ❞❛ ♥♦çã♦ ❞❡ ❝♦♥❥✉♥t♦✳ ❙✐♠✐❧❛r♠❡♥t❡✱ ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s sã♦ ✉♠❛ ❝❛t❡✲ ❣♦r✐✜❝❛çã♦ ❞❡ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ✭♦ q✉❡ ❥✉st✐✜❝❛ ❛ t❡r♠✐♥♦❧♦❣✐❛✮✳ ▼❛✐s ❣❡r❛❧♠❡♥t❡✱ ❛ ❝❛t❡❣♦r✐✜❝❛çã♦ ❞♦s ♠♦♥ó✐❞❡s✱ ✉♠❛ ❞❛s ❡str✉t✉r❛s ♠❛✐s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ á❧❣❡❜r❛ ♦r❞✐♥ár✐❛✱ ♦r✐❣✐♥❛ ❛s ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✳

❯♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é✱ ❜❛s✐❝❛♠❡♥t❡✱ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠✉♥✐❞❛ ❞❡ ✉♠ ❢✉♥t♦r⊗❡ ✉♠ ♦❜❥❡t♦1t❛✐s q✉❡ ♦s ♦❜❥❡t♦s(X⊗Y)⊗Z✱X⊗(Y⊗Z)

❡1⊗X✱X✱X⊗1❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ♣♦r ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛✐s✳ ◗✉❛♥❞♦

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t❛✐s ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s sã♦ ❛s r❡s♣❡❝t✐✈❛s ✐❞❡♥t✐❞❛❞❡s✱ ❞✐③❡♠♦s q✉❡ ❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ❡str✐t❛✳ ❖ ❚❡♦r❡♠❛ ❞❡ ▼❛❝ ♣❛r❛ ❛ ❝♦♥❞✐çã♦ ❡str✐t❛ ❛✜r♠❛ q✉❡ ♣♦❞❡♠♦s✱ ❡♠ ✉♠ ❝❡rt♦ s❡♥t✐❞♦✱ ❝♦♥s✐❞❡r❛r q✉❛✐sq✉❡r ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❝♦♠♦ ❡str✐t❛s✳

❊st❡ tr❛❜❛❧❤♦ é r❡s✉❧t❛❞♦ ❞♦s ❡st✉❞♦s ❞♦ ❛✉t♦r ❡ ❞❛ s✉❛ ♦r✐❡♥t❛❞♦r❛ s♦❜r❡ ❛s ♥♦t❛s ❞❡ ❛✉❧❛ ❯♥❛ ✐♥tr♦❞✉❝✐ó♥ ❛ ❧❛s ❝❛t❡❣♦rí❛s t❡♥s♦r✐❛❧❡s ② s✉s r❡♣r❡s❡♥t❛❝✐♦♥❡s ❞♦ Pr♦❢✳ ❉r✳ ▼❛rtí♥ ▼♦♠❜❡❧❧✐✳ ❉❡s❞❡ ♦ s❡❣✉♥❞♦ s❡✲ ♠❡str❡ ❞❡ ✷✵✶✹✱ ❛❜♦r❞❛♠♦s ♠✉✐t♦s ❛ss✉♥t♦s ❛tr❛✈és ❞❡ s❡♠✐♥ár✐♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❝❛t❡❣♦r✐❛s✱ ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s✱ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✱ ❝❛t❡✲ ❣♦r✐❛s t❡♥s♦r✐❛✐s ❡ ❝❛t❡❣♦r✐❛s ♠ó❞✉❧♦ s♦❜r❡ ❝❛t❡❣♦r✐❛s t❡♥s♦r✐❛s✳ ■♥✐❝✐✲ ❛❧♠❡♥t❡✱ ❡st❛s ú❧t✐♠❛s ❝❛t❡❣♦r✐❛s s❡r✐❛♠ ♦ ❛ss✉♥t♦ ❞❛ ❞✐ss❡rt❛çã♦✱ ♠❛s ♣♦r ❢❛❧t❛ ❞❡ t❡♠♣♦✱ ♥♦s ❧✐♠✐t❛♠♦s ❛ ❡s❝r❡✈❡r s♦❜r❡ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐✲ ❞❛✐s✳ ❆❧❡rt❛♠♦s ♦ ❧❡✐t♦r q✉❡ ♦ ❈❛♣ít✉❧♦ ✷ s♦❜r❡ ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ♣♦❞❡ s❡r ✐❣♥♦r❛❞♦✱ ❡♠ t❡r♠♦s ❞❡ ♣ré✲r❡q✉✐s✐t♦s ♣❛r❛ ❡ss❡ tr❛❜❛❧❤♦✱ ♣♦✐s ❛♣❡s❛r ❞❡ ❛❧❣✉♠❛s ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❝❛t❡❣♦r✐❛ ❞❡ ♠ó❞✉❧♦s✱ s❡r❡♠ ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✱ t❛❧ ❝❛♣ít✉❧♦ ♥ã♦ ❝♦♥tr✐❜✉✐ ♣❛r❛ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞♦ tr❛❜❛❧❤♦✳ ❊s❝♦❧❤❡♠♦s ✐♥tr♦✲ ❞✉③✐r ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ✉s❛r ❛ ❞✐ss❡rt❛çã♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♣❛r❛ ✉♠ ♣♦ssí✈❡❧ tr❛❜❛❧❤♦ ❢✉t✉r♦ s♦❜r❡ ❝❛t❡❣♦r✐❛s t❡♥s♦r✐❛✐s q✉❡ sã♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ❡ ♠♦♥♦✐❞❛✐s✳

❯♠❛ ❞❛s ♠♦t✐✈❛çõ❡s ♣❛r❛ r❡❛❧✐③❛r♠♦s ❡ss❡ tr❛❜❛❧❤♦ ❢♦r❛♠ ❞✉❛s ♦❜✲ s❡r✈❛çõ❡s ❢❡✐t❛s ❡♠ ✭❬✺❪✿ ❘❡♠❛r❦ ✷✳✽✳✻ ❡ ✷✳✽✳✼✮✳ P❛r❛ s✐t✉❛r♠♦s ♦ ❧❡✐t♦r✱ ♦s ❞♦✐s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❡st✉❞❛❞♦s ❛q✉✐ sã♦ q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡s✲ q✉❡❧ét✐❝❛ ❡ q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✱ ❡st❡ é ♦ ❝♦♥❤❡❝✐❞♦ ▼❛❝ ▲❛♥❡✬s str✐❝t✲ ♥❡ss t❤❡♦r❡♠✳ ◆❛ ♦❜s❡r✈❛çã♦ ✷✳✽✳✻✱ ❡♥❝♦♥tr❛♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ q✉❡ ♥ã♦ é ❡str✐t❛✱ ♠❛s q✉❡ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ▼❛❝ ▲❛♥❡ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✳

P♦r ♦✉tr♦ ❧❛❞♦✱ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛✲ ❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡sq✉❡❧ét✐❝❛ ❡ ♣❡❧♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♦❜s❡r✈❛çã♦ ✷✳✽✳✻ é ♣♦ssí✈❡❧ ❝♦♥❝❧✉✐r♠♦s q✉❡ ❛ ❝❛t❡❣♦r✐❛ ❞♦ ❡①❡♠♣❧♦ r❡❢❡✲ r✐❞♦ ❛❝✐♠❛✱ ♥ã♦ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ q✉❡ s❡❥❛ ❡sq✉❡❧ét✐❝❛ ❡ ❡str✐t❛ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ✐ss♦ é ❞✐t♦ ❡①❛t❛♠❡♥t❡ ♥♦ ✜♥❛❧ ❞❛ ♦❜s❡r✈❛çã♦ ✷✳✽✳✼✳ ❖ q✉❡ ✜❝❛ ♣♦r ❞❡trás ❞❡ss❡ ❢❛t♦ é q✉❡✱ s❡❣✉♥❞♦ ❬✺❪✱ ♣❛r❛ t♦r♥❛r♠♦s ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✱ é ♥❡❝❡s✲ sár✐♦ ❛❞✐❝✐♦♥❛r ♥♦✈♦s ♦❜❥❡t♦s ❛ ❡❧❛ ✭♦❜❥❡t♦s ❡st❡s ✐s♦♠♦r❢♦s✱ ♠❛s ♥ã♦ ✐❣✉❛✐s ❛♦s ❥á ❡①✐st❡♥t❡s✮✳ ❖ ❞❡s❡❥♦ ❞❡ ❡✈✐t❛r ❛❞✐❝✐♦♥❛r t❛✐s ♦❜❥❡t♦s ♥♦s ❢❛③ tr❛❜❛❧❤❛r ❝♦♠ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ♥ã♦ ❡str✐t❛s ✭♦✉ s❡❥❛✱a✱ l ❡ r

♥ã♦ s❡♥❞♦ ♦s ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ✐❞❡♥t✐❞❛❞❡✮ ♠✉✐t♦ ❡♠❜♦r❛ ♦ t❡♦✲ r❡♠❛ ❞❡ ▼❛❝ ▲❛♥❡ ❞✐❣❛ q✉❡ ✐ss♦ ♥ã♦ s❡❥❛ ♥❡❝❡ssár✐♦✳ ❉❡ ❛❧❣✉♠❛ ❢♦r♠❛

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❛ss❡❣✉r❛✲s❡ q✉❡ ❛❧❣✉♠❛s ❝❛t❡❣♦r✐❛s s❡❥❛♠ ✏♠❛✐s ❡str✐t❛s✑ ❞♦ q✉❡ ❛ ❝❛✲ t❡❣♦r✐❛ ❛♣r❡s❡♥t❛❞❛ ♥♦ ❡①❡♠♣❧♦ ❞❛❞♦✱ ❛ s❛❜❡r✱ V ecω

G✳ ❊st✉❞❛♠♦s ❡ss❛

❝❛t❡❣♦r✐❛ ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✱ ♣♦ré♠ ❛ ❞❡♥♦t❛♠♦s ♣♦r C(G, ω)✳

◆♦ss❛ ♣r♦♣♦st❛ ❞❡ tr❛❜❛❧❤♦ ❢♦✐ ❡♥t❡♥❞❡r ❜❡♠ ❛s ♣r♦✈❛s ❞♦s ❞♦✐s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❝✐t❛❞♦s ❛❝✐♠❛ ❡ ❛ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ❛ ❞✐s♣♦s✐çã♦ ❞♦s ❝❛♣ít✉❧♦s ❞❡ss❛ ❞✐ss❡rt❛çã♦✳

◆♦ ❈❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ♦s ♣ré✲r❡q✉✐s✐t♦s s♦❜r❡ t❡♦r✐❛ ❞❡ ❝❛t❡✲ ❣♦r✐❛s✳ ❊♥tr❡ ❡❧❡s✱ ❡stã♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❝❛t❡❣♦r✐❛s✱ ❢✉♥t♦r❡s ❡ tr❛♥s❢♦r✲ ♠❛çõ❡s ♥❛t✉r❛✐s✳ ❆❧é♠ ❞✐ss♦✱ ❞❡✜♥✐♠♦s ❡q✉✐✈❛❧ê♥❝✐❛s ❡♥tr❡ ❝❛t❡❣♦r✐❛s ❡ ❞❡♠♦♥str❛♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ ❞✉❛s ❝❛t❡❣♦r✐❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st✐r ✉♠ ❢✉♥t♦r ✜❡❧✱ ♣❧❡♥♦ ❡ ❞❡♥s♦ ❡♥tr❡ ❡st❛s✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❡st✉❞❛♠♦s ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s✳ ❉❡✜♥✐♠♦s ♦❜❥❡✲ t♦s ✐♥✐❝✐❛✐s✱ ✜♥❛✐s ❡ ♥✉❧♦s✱ ❝❛t❡❣♦r✐❛s ♣ré✲❛❞✐t✐✈❛s✱ ❛❞✐t✐✈❛s ❡ ❛❜❡❧✐❛♥❛s✳ ❯♠ ❞♦s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s é q✉❡ ❡♠ ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ✈❛❧❡ ♦ ❚❡♦r❡♠❛ ❞♦ ✐s♦♠♦r✜s♠♦✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ❡st✉❞❛♠♦s ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✳ ❆♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❝❧áss✐❝❛ ❞❡st❛s ❝❛t❡❣♦r✐❛s ❡ ♣r♦✈❛♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦✲ ♣r✐❡❞❛❞❡s✳ ❆♦ ❞❡✜♥✐r♠♦s ❝❛t❡❣♦r✐❛s ❡sq✉❡❧ét✐❝❛s✱ q✉❡ ♣♦ss✉❡♠ ✉♠❛ ❡str✉t✉r❛ ♠❛✐s s✐♠♣❧❡s✱ ♠♦str❛♠♦s q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦✲ ♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡sq✉❡❧ét✐❝❛✳

◆♦ ❈❛♣ít✉❧♦ ✹✱ ❞❡♠♦♥str❛♠♦s ♦ ✏▼❛❝ ▲❛♥❡✬s ❙tr✐❝t♥❡ss ❚❤❡♦r❡♠✑✱ q✉❡ ❛✜r♠❛ q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✳

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## Pré✲r❡q✉✐s✐t♦s

◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❡st✉✲ ❞❛r♠♦s ❝❛t❡❣♦r✐❛s✳ ❉❡✜♥✐♠♦s ❝❛t❡❣♦r✐❛✱ ❢✉♥t♦r✱ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✱ ♠♦♥♦♠♦r✜s♠♦ ❡ ❡♣✐♠♦r✜s♠♦✳

### ✶✳✶ ❈❛t❡❣♦r✐❛s

❉❡✜♥✐çã♦ ✶✳✶✳✶ ❯♠❛ ❝❛t❡❣♦r✐❛C ❝♦♥s✐st❡ ❞❡

✭✐✮ ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♦❜❥❡t♦sOb(C)❀

✭✐✐✮ ♣❛r❛ ❝❛❞❛ ♣❛r (X, Y) ❞❡ ♦❜❥❡t♦s ❡♠ C✱ ✉♠❛ ❝♦❧❡çã♦HomC(X, Y)

❞❡ ♠♦r✜s♠♦s ❞❡X ♣❛r❛Y❀

✭✐✐✐✮ ♣❛r❛ q✉❛❧q✉❡r ♦❜❥❡t♦X❡♠Ob(C)✱ ✉♠ ♠♦r✜s♠♦idX❡♠HomC(X, X)✱

❝❤❛♠❛❞♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞❡X❀

✭✐✈✮ ♣❛r❛ q✉❛✐sq✉❡rX, Y, Z ♦❜❥❡t♦s ❡♠Ob(C)✱ ✉♠❛ ❢✉♥çã♦

HomC(X, Y)×HomC(Y, Z) → HomC(X, Z)

(f, g) 7→ g◦f

❝❤❛♠❛❞❛ ❝♦♠♣♦s✐çã♦✱ q✉❡ s❛t✐s❢❛③ ♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s✿

✭❛✮ ♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦sX ❡Y ❡♠Ob(C)✱ ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡idX

❡♠HomC(X, X) s❛t✐s❢❛③

f◦idX=f ❡ idX◦g=g,

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♣❛r❛ q✉❛✐sq✉❡r f ❡♠ HomC(X, Y)❡ g ❡♠HomC(Y, X)❀

✭❜✮ ❞❛❞♦s ♦❜❥❡t♦sX, Y, Z, W ❡♠Ob(C)❡ ♠♦r✜s♠♦sf ❡♠HomC(X, Y)✱

g ❡♠HomC(Y, Z)✱h❡♠HomC(Z, W)✱ ❛ ❝♦♠♣♦s✐çã♦ é ❛ss♦❝✐❛t✐✈❛✱ ♦✉

s❡❥❛✱

h◦(g◦f) = (h◦g)◦f.

❆ r❡❢❡rê♥❝✐❛[✽]é ❜ás✐❝❛ ♥♦ ❡st✉❞♦ ❞❛ ❚❡♦r✐❛ ❞❡ ❈❛t❡❣♦r✐❛s✳ ◆❡ss❛

r❡❢❡rê♥❝✐❛ sã♦ ❛♣r❡s❡♥t❛❞❛s três ❞❡✜♥✐çõ❡s ❞❡ ❝❛t❡❣♦r✐❛✱ ❛ q✉❡ ❛♣r❡s❡♥✲ t❛♠♦s✱ ✉♠❛ q✉❡ ❡♥✈♦❧✈❡ ❛♣❡♥❛s ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♠♦r✜s♠♦s ❡ ✉♠❛ ♦✉tr❛ q✉❡ ❝♦♥s✐❞❡r❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠♦ ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ ❝♦♠ ❞✉❛s ❢✉♥çõ❡s✱ ✐❞❡♥t✐❞❛❞❡ ❡ ❝♦♠♣♦s✐çã♦✳ ❊s❝♦❧❤❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❛❞❛ ♣♦r s❡r ❛ ♠❛✐s ❝♦♠✉♠ ❡ ❝ô♠♦❞❛ ♣❛r❛ tr❛❜❛❧❤❛r✳

❱❛❧❡ ♥♦t❛r q✉❡ ✉♠❛ ♣❛❧❛✈r❛ ✐♠♣♦rt❛♥t❡ ♥❛ ❞❡✜♥✐çã♦ ❞❛❞❛ é ✏❝♦❧❡✲ çã♦✑✳ ❊✈✐t❛✲s❡ ❡s❝r❡✈❡r ✏❝♦♥❥✉♥t♦ ❞❡ ♦❜❥❡t♦s✑ ❡ ✏❝♦♥❥✉♥t♦ ❞❡ ♠♦r✜s♠♦s✑✱ ♣♦✐s ❛s ❝♦❧❡çõ❡s ❞❡ ♦❜❥❡t♦s ❡ ♠♦r✜s♠♦s ♥ã♦ ❝♦st✉♠❛♠ s❡r ❝♦♥❥✉♥t♦s✱ ♠❡s♠♦ ♥❛s ❝❛t❡❣♦r✐❛s ♠❛✐s ❝♦♠✉♥s✳ ◆❛ ✈❡r❞❛❞❡✱ ❡①✐st❡♠ ✈ár✐❛s q✉❡s✲ tõ❡s ✐♥t❡r❡ss❛♥t❡s ❞❛ ❚❡♦r✐❛ ❞❡ ❈♦♥❥✉♥t♦s ❡ ❞♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ ♠❛t❡✲ ♠át✐❝❛ ❡♥✈♦❧✈✐❞♦s ♥♦ ❡st✉❞♦ ❞❡ ❝❛t❡❣♦r✐❛s✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ ❢♦❝❛r❡♠♦s ♥❡st❡s ❛s♣❡❝t♦s✱ ❛♣r❡s❡♥t❛♥❞♦ ❛♣❡♥❛s ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s ❛ r❡s♣❡✐t♦ ❞❡❧❡s✳ P❛r❛ ♦ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦✱ ❛ ❥á ❝✐t❛❞❛ r❡❢❡rê♥❝✐❛[✽]❛♣r❡s❡♥t❛ ✉♠

q✉❛❞r♦ ❣❡r❛❧ ❡ ❢♦r♥❡❝❡ ót✐♠❛s ✐♥❞✐❝❛çõ❡s ♣❛r❛ ❡♥t❡♥❞❡r ♠❛✐s ♣r♦❢✉♥❞❛✲ ♠❡♥t❡ ❡ss❛s q✉❡stõ❡s✳

❆♣❡s❛r ❞❛s ❝♦❧❡çõ❡s ❡♥✈♦❧✈✐❞❛s ❡♠ ❝❛t❡❣♦r✐❛s ♥ã♦ s❡r❡♠ s❡♠♣r❡ ❝♦♥❥✉♥t♦s✱ ✈❛♠♦s ✉s❛r ♦s sí♠❜♦❧♦s ❡ t❡r♠♦s ❥á ❝♦♥❤❡❝✐❞♦s✱ ❝♦♠♦ ∈✱⊆✱

✏❢✉♥çã♦✑✱ ✏❛♣❧✐❝❛çã♦✑✱ ♣❛r❛ r❡❧❛❝✐♦♥❛r ❝♦❧❡çõ❡s ❡ s❡✉s ❡❧❡♠❡♥t♦s✳ ❙❛✲ ❜❡♥❞♦ ❞✐ss♦✱ ✜①❛♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ♥♦t❛çõ❡s✳

❉❡♥♦t❛♠♦s ✉♠ ♠♦r✜s♠♦ f ❡♠ HomC(X, Y) ♣♦r f : X → Y ♦✉

f

X →Y✳ ❆❧é♠ ❞✐ss♦✱ X ❡ Y sã♦ ❝❤❛♠❛❞♦s ❞♦♠í♥✐♦ ❡ ❝♦❞♦♠í♥✐♦ ❞♦

♠♦r✜s♠♦ f✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊s❝r❡✈❡♠♦s f ∈ HomC(X, Y) ❡✱ ♣♦r

❛❜✉s♦ ❞❡ ♥♦t❛çã♦✱ ❡s❝r❡✈❡♠♦s ✏X ∈C✑ ♣❛r❛ ❞❡s✐❣♥❛r ✉♠ ♦❜❥❡t♦X ❡♠ Ob(C)✳

❆♣❡s❛r ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ s❡r ❝♦♥st✐t✉í❞❛ ♣♦r ♦❜❥❡t♦s✱ ♠♦r✜s♠♦s✱ ♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡ ❡ ✉♠❛ ❝♦♠♣♦s✐çã♦✱ ❣❡r❛❧♠❡♥t❡ s❡ ❛♣r❡s❡♥t❛♠ ❛♣❡♥❛s ♦s ♦❜❥❡t♦s ❡ ♠♦r✜s♠♦s✱ ✜❝❛♥❞♦ s✉❜❡♥t❡♥❞✐❞♦s ♦s ♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡ ❡ ❛ ❝♦♠♣♦s✐çã♦✳ ➱ ❞❡ss❛ ❢♦r♠❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛ ♠❛✐♦r✐❛ ❞♦s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r✳

❊①❡♠♣❧♦ ✶✳✶✳✷ ❆ ❝❛t❡❣♦r✐❛Seté ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❝♦♥❥✉♥✲

t♦s ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s ❢✉♥çõ❡s✳

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❊①❡♠♣❧♦ ✶✳✶✳✸ ❆ ❝❛t❡❣♦r✐❛Relé ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❝♦♥❥✉♥✲

t♦s ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s r❡❧❛çõ❡s✳

▲❡♠❜r❛♥❞♦✱ ♣❛r❛ X, Y ❝♦♥❥✉♥t♦s✱ ✉♠❛ r❡❧❛çã♦ R ❡♥tr❡ X ❡ Y é

✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ X ×Y✳ ❆❣♦r❛✱ ♣❛r❛ X, Y, Z

❝♦♥❥✉♥t♦s ❡R ⊆X×Y✱ S ⊆Y ×Z r❡❧❛çõ❡s✱ ❞❡✜♥✐♠♦s ❛ ❝♦♠♣♦s✐çã♦ S◦R⊆X×Z ❝♦♠♦ ❛ s❡♥❞♦ ❛ r❡❧❛çã♦

S◦R:={(x, z)∈X×Z : ❡①✐st❡ y∈Y t❛❧ q✉❡ (x, y)∈R,(y, z)∈S}.

◆❡ss❡ ❝❛s♦✱idX⊆X×X é ❛ r❡❧❛çã♦idX ={(x, x) :x∈X}✳

❊①❡♠♣❧♦ ✶✳✶✳✹ ❆ ❝❛t❡❣♦r✐❛Grpé ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❣r✉♣♦s

❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s✳

❊①❡♠♣❧♦ ✶✳✶✳✺ ❆ ❝❛t❡❣♦r✐❛Ab é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❣r✉♣♦s

❛❜❡❧✐❛♥♦s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s✳

❊①❡♠♣❧♦ ✶✳✶✳✻ ❆ ❝❛t❡❣♦r✐❛Div é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❣r✉♣♦s

❞✐✈✐sí✈❡✐s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s✳

▲❡♠❜r❛♥❞♦✱ s❡Gé ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦Gé ❞✐✈✐sí✈❡❧ s❡✱ ♣❛r❛

t♦❞♦x∈G❡ t♦❞♦ ✐♥t❡✐r♦ ♥ã♦✲♥✉❧♦n✱ ❡①✐st❡y∈Gt❛❧ q✉❡x=ny✳ ❆❧✲

❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❣r✉♣♦s ❞✐✈✐sí✈❡✐s sã♦Q✱Q/Z✱ ♦ ❣r✉♣♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦sC∗❡ ♦ ❣r✉♣♦ ❞❡ Prü❢❡r Z(p)

❊①❡♠♣❧♦ ✶✳✶✳✼ ❆ ❝❛t❡❣♦r✐❛Ring é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❛♥é✐s

❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❛♥é✐s✳

❆ ❝❛t❡❣♦r✐❛ring é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❛♥é✐s ❝♦♠ ✉♥✐❞❛❞❡ ❡

♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❛♥é✐s q✉❡ ♣r❡s❡r✈❛♠ ❛ ✉♥✐❞❛❞❡✳ ❆ ❝❛t❡❣♦r✐❛ Cring é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❛♥é✐s ❝♦♠✉t❛t✐✲

✈♦s ❝♦♠ ✉♥✐❞❛❞❡ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❛♥é✐s q✉❡ ♣r❡s❡r✈❛♠ ❛ ✉♥✐❞❛❞❡✳

❊①❡♠♣❧♦ ✶✳✶✳✽ ❙❡❥❛ R ✉♠ ❛♥❡❧✳ ❉❡♥♦t❛♠♦s ♣♦r RM ✭r❡s♣❡❝t✐✈❛✲

♠❡♥t❡ MR✮ ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛

✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡

R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮✳

❊①❡♠♣❧♦ ✶✳✶✳✾ ❙❡❥❛k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r V ectk ❛ ❝❛t❡❣♦r✐❛

❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦sk✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s tr❛♥s❢♦r✲

♠❛çõ❡sk✲❧✐♥❡❛r❡s✳

❉❡♥♦t❛♠♦s ♣♦r vectk ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s k✲❡s♣❛ç♦s

✈❡t♦r✐❛✐s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s tr❛♥s❢♦r♠❛çõ❡s k✲

❧✐♥❡❛r❡s✳

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❊①❡♠♣❧♦ ✶✳✶✳✶✵ ❙❡❥❛ k✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r Algk ❛ ❝❛t❡❣♦r✐❛

❝✉❥♦s ♦❜❥❡t♦s sã♦ ❛s k✲á❧❣❡❜r❛s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s

❞❡k✲á❧❣❡❜r❛s✳

❊①❡♠♣❧♦ ✶✳✶✳✶✶ ❙❡❥❛A✉♠❛k✲á❧❣❡❜r❛✳ ❉❡♥♦t❛♠♦s ♣♦rAM✭r❡s♣❡❝✲

t✐✈❛♠❡♥t❡MA✮ ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛

✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡

A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮✳

❆ ❝❛t❡❣♦r✐❛ Am ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ mA✮ é ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s

sã♦ ♦sA✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮ ❞❡ ❞✐♠❡♥sã♦

✜♥✐t❛ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛

✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮✳

P❛r❛ ♦s ❞♦✐s ♣ró①✐♠♦s ❡①❡♠♣❧♦s✱ ❧❡♠❜r❛♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❡ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❡ ❞❡ ❜✐á❧❣❡❜r❛✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✷ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❯♠❛ k✲á❧❣❡❜r❛ ❞❡ ▲✐❡ é ✉♠ ♣❛r (L,[−,−])✱ ❡♠ q✉❡Lé ✉♠k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡[−,−] :L⊗L→Lé ✉♠❛

❛♣❧✐❝❛çã♦ k✲❧✐♥❡❛r✱ ❝❤❛♠❛❞❛ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✱ q✉❡ s❛t✐s❢❛③ às s❡❣✉✐♥t❡s

♣r♦♣r✐❡❞❛❞❡s✿

✭✐✮[x, x] = 0✱ ♣❛r❛ t♦❞♦x∈L❀

✭✐✐✮ [x,[y, z]] + [y,[z, x]] + [z,[x, y]] = 0✱ ♣❛r❛ q✉❛✐sq✉❡r x, y, z∈L✳

❊ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ■❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✳ ❊①❡♠♣❧♦ ✶✳✶✳✶✸ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r Liek ❛ ❝❛t❡❣♦r✐❛

❝✉❥♦s ♦❜❥❡t♦s sã♦ ❛sk✲á❧❣❡❜r❛s ❞❡ ▲✐❡ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✲

✜s♠♦s ❞❡k✲á❧❣❡❜r❛s ❞❡ ▲✐❡✱ ♦✉ s❡❥❛✱ ❛♣❧✐❝❛çõ❡sk✲❧✐♥❡❛r❡s q✉❡ ♣r❡s❡r✈❛♠

♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✹ ❙❡❥❛k ✉♠ ❝♦r♣♦✳ ❯♠❛k✲❜✐á❧❣❡❜r❛ é ✉♠❛ q✉í♥t✉♣❧❛ (H, M, µ,∆, ε)✱ ❡♠ q✉❡(H, M, µ)é ✉♠❛k✲á❧❣❡❜r❛✱(H,∆, ε)é ✉♠❛ k✲

❝♦á❧❣❡❜r❛ ❡ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮M ❡µsã♦ ❤♦♠♦♠♦r✜s♠♦s ❞❡ k✲❝♦á❧❣❡❜r❛s❀

✭✐✐✮ ∆❡ εsã♦ ❤♦♠♦♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s✳

❊①❡♠♣❧♦ ✶✳✶✳✶✺ ❙❡❥❛k✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦rBialgk❛ ❝❛t❡❣♦r✐❛

❝✉❥♦s ♦❜❥❡t♦s sã♦ ❛sk✲❜✐á❧❣❡❜r❛s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s

❞❡ k✲❜✐á❧❣❡❜r❛s✱ ♦✉ s❡❥❛✱ ❛♣❧✐❝❛çõ❡s k✲❧✐♥❡❛r❡s q✉❡ sã♦ ❤♦♠♦♠♦r✜s♠♦s

❞❡k✲á❧❣❡❜r❛s ❡k✲❝♦á❧❣❡❜r❛s✳

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❊①❡♠♣❧♦ ✶✳✶✳✶✻ ❆ ❝❛t❡❣♦r✐❛T opé ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❡s♣❛ç♦s

t♦♣♦❧ó❣✐❝♦s ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✳

❊①❡♠♣❧♦ ✶✳✶✳✶✼ ❆ ❝❛t❡❣♦r✐❛Dif f é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ❛s ✈❛✲

r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❊①❡♠♣❧♦ ✶✳✶✳✶✽ ❙❡❥❛A✉♠❛k✲á❧❣❡❜r❛✳ ❉❡♥♦t❛♠♦s ♣♦rA❛ ❝❛t❡❣♦r✐❛

❝♦♠ ✉♠ ú♥✐❝♦ ♦❜❥❡t♦∗❡HomA(∗,∗) =A✳ ❆ ❝♦♠♣♦s✐çã♦ é ❞❛❞❛ ♣❡❧♦

♣r♦❞✉t♦ ❞❡A❡id∗= 1A✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✾ ❙❡❥❛♠ C✱ D ❝❛t❡❣♦r✐❛s✱ D é ❞✐t❛ ✉♠❛ s✉❜❝❛t❡❣♦r✐❛

❞❡Cs❡Ob(D)⊆Ob(C)✱HomD(X, Y)⊆HomC(X, Y)✱ ♣❛r❛ q✉❛✐sq✉❡r

X✱Y ∈D✱ ❡ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s ❡♠ D é ❛ ❝♦♠♣♦s✐çã♦ ❝♦♠♦

❡♠C✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✵ ❯♠❛ s✉❜❝❛t❡❣♦r✐❛D❞❡Cé ❞✐t❛ ♣❧❡♥❛ s❡HomD(X, Y) =

HomC(X, Y)✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈D✳

❊①❡♠♣❧♦ ✶✳✶✳✷✶ ◆♦t❡♠♦s q✉❡✱ ♣❛r❛ X✱Y ❝♦♥❥✉♥t♦s ❡ f : X → Y

❢✉♥çã♦✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛rf ❝♦♠♦ ❛ r❡❧❛çã♦

{(x, f(x)) :x∈X} ⊆X×Y.

❉❡ss❛ ❢♦r♠❛✱Set é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❞❡Rel✱ ♠❛s ♥ã♦ é ✉♠❛ s✉❜✲

❝❛t❡❣♦r✐❛ ♣❧❡♥❛✱ ♣♦✐s ♥❡♠ t♦❞❛ r❡❧❛çã♦ é ✉♠❛ ❢✉♥çã♦✳

❊①❡♠♣❧♦ ✶✳✶✳✷✷ ❆ ❝❛t❡❣♦r✐❛ Div é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❡ Ab✱

q✉❡ ♣♦r s✉❛ ✈❡③ é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❡Grp✳

❊①❡♠♣❧♦ ✶✳✶✳✷✸ ❆ ❝❛t❡❣♦r✐❛ ring é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❞❡ Ring q✉❡

♥ã♦ é ♣❧❡♥❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ R ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦

❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s

f : R → R×R r 7→ (r,0).

❚❛❧ ❤♦♠♦♠♦r✜s♠♦ é ✉♠ ♠♦r✜s♠♦ ❡♠Ring✱ ♠❛s ♥ã♦ ❡♠ ring✳

❊①❡♠♣❧♦ ✶✳✶✳✷✹ P❛r❛k✉♠ ❝♦r♣♦✱vectké ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❡ V ectk✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣❛r❛A✉♠❛ k✲á❧❣❡❜r❛✱ Am é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛

♣❧❡♥❛ ❞❡AM✳

❊①❡♠♣❧♦ ✶✳✶✳✷✺ ❆ ❝❛t❡❣♦r✐❛ Dif f é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❞❡ T op q✉❡

♥ã♦ é ♣❧❡♥❛✳

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❉❡✜♥✐çã♦ ✶✳✶✳✷✻ ❯♠❛ ❝❛t❡❣♦r✐❛ é ❞✐t❛ ♣❡q✉❡♥❛ s❡ ❛s ❝♦❧❡çõ❡s ❞❡ ♦❜✲ ❥❡t♦s ❡ ♠♦r✜s♠♦s ❢♦r❡♠ ❝♦♥❥✉♥t♦s✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✼ ❯♠❛ ❝❛t❡❣♦r✐❛ Cé ❞✐t❛ ❧♦❝❛❧♠❡♥t❡ ♣❡q✉❡♥❛ s❡✱ ♣❛r❛

q✉❛✐sq✉❡r X, Y ∈C✱HomC(X, Y)é ✉♠ ❝♦♥❥✉♥t♦✳

❆s ♣r✐♥❝✐♣❛✐s ❝❛t❡❣♦r✐❛s q✉❡ ✈❛♠♦s ❡st✉❞❛r sã♦ ❧♦❝❛❧♠❡♥t❡ ♣❡q✉❡✲ ♥❛s✳ P♦r ❡ss❛ r❛③ã♦✱ ❞❛q✉✐ ❡♠ ❞✐❛♥t❡ ✈❛♠♦s ❝♦♥s✐❞❡r❛r t♦❞❛s ❛s ❝❛t❡✲ ❣♦r✐❛s ❝♦♠♦ ❧♦❝❛❧♠❡♥t❡ ♣❡q✉❡♥❛s✳

❆♣r❡s❡♥t❛♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ❝♦♥str✉çõ❡s ❜ás✐❝❛s q✉❡ ♥♦s ♣❡r♠✐t❡♠ ♦❜t❡r ♥♦✈❛s ❝❛t❡❣♦r✐❛s ❛ ♣❛rt✐r ❞❡ ❝❛t❡❣♦r✐❛s ❥á ❝♦♥❤❡❝✐❞❛s✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✽ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛✳ ❉❡♥♦t❛♠♦s ♣♦r Cop ❛ ❝❛t❡✲

❣♦r✐❛ ♦♣♦st❛ ❛ C✱ ❞❡✜♥✐❞❛ ❝♦♠♦ s❡❣✉❡✿

✭✐✮Ob(Cop) =Ob(C)

✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡rX, Y ∈Cop

HomCop(X, Y) =HomC(Y, X);

✭✐✐✐✮ ♣❛r❛ ♠♦r✜s♠♦s f ∈HomCop(X, Y)✱ g∈HomCop(Y, Z)✱ ❛ ❝♦♠♣♦✲

s✐çã♦ é ❞❛❞❛ ♣♦r

g◦opf =fg.

◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡(Cop)op=C✳ ❆ ❝❛t❡❣♦r✐❛ ♦♣♦st❛ é ✐♠♣♦rt❛♥t❡

♣❛r❛ ❡st✉❞❛r ❞✉❛❧✐❞❛❞❡ ❡ ❞❡✜♥✐r ❢✉♥t♦r❡s ❝♦♥tr❛✈❛r✐❛♥t❡s✳ ◆♦ ❡st✉❞♦ ❞❡ ❝❛t❡❣♦r✐❛s✱ ❝❛❞❛ ❝♦♥❝❡✐t♦ é ❛❝♦♠♣❛♥❤❛❞♦ ❞♦ s❡✉ ❝♦♥❝❡✐t♦ ❞✉❛❧✱ ♦❜t✐❞♦ ✏✐♥✈❡rt❡♥❞♦ ❛s ✢❡❝❤❛s✑ ♥❛ ❞❡✜♥✐çã♦ ❞♦ ❝♦♥❝❡✐t♦ ♦r✐❣✐♥❛❧✳ ■ss♦ ✈❛✐ ✜❝❛r ♠❛✐s ❝❧❛r♦ ♥❛s s❡çõ❡s s❡❣✉✐♥t❡s✳ P❛r❛ ♠❛✐s ✐♥❢♦r♠❛çõ❡s✱ ♦ ❧❡✐t♦r ♣♦❞❡ ♣❡sq✉✐s❛r ❡♠ ✭❬✷❪✱ s❡❝t✐♦♥ ✸✳✶✮✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✾ ❙❡❥❛♠C✱D❝❛t❡❣♦r✐❛s✳ ❉❡♥♦t❛♠♦s ♣♦rC×D❛ ❝❛✲

t❡❣♦r✐❛ ♣r♦❞✉t♦ ❞❡C ❡D✱ ❞❡✜♥✐❞❛ ❝♦♠♦ s❡❣✉❡✿

✭✐✮Ob(C×D) =Ob(C)×Ob(D)❀

✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r(X, Y),(X′, Y)C×D

HomC×D((X, Y),(X′, Y′)) =HomC(X, X′)×HomD(Y, Y′);

✭✐✐✐✮ ♣❛r❛ ❝❛❞❛ ♣❛r (X, Y) ❡♠ Ob(C×D)✱ id(X,Y) = (idX, idY) é ♦

♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❡♠ HomC×D((X, Y),(X, Y))❀

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✭✐✈✮ ♣❛r❛ ♠♦r✜s♠♦s (f, g)∈HomC(X, X′)×HomD(Y, Y′)✱ (f′, g′)∈

HomC(X′, X′′)×HomD(Y′, Y′′)✱ ❛ ❝♦♠♣♦s✐çã♦ é ❞❛❞❛ ♣♦r

(f′, g)(f, g) = (ff, gg).

❆ ❝❛t❡❣♦r✐❛ ♣r♦❞✉t♦ ✈❡♠ ❛❝♦♠♣❛♥❤❛❞❛ ❞❡ ❢✉♥t♦r❡s ❞❡ ♣r♦❥❡çã♦✱ ❝♦♠♦ s❡rá ✈✐st♦ ♥❛ s❡çã♦ s♦❜r❡ ❢✉♥t♦r❡s✳ ❋✉♥t♦r❡s ❞❡✜♥✐❞♦s ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ♣r♦❞✉t♦ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❜✐❢✉♥t♦r❡s✱ q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❞❡✜♥✐r ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✳

### ✜s♠♦s

❋✉♥çõ❡s ✐♥❥❡t♦r❛s ❡ s♦❜r❡❥❡t♦r❛s ♣♦❞❡♠ s❡r ❞❡✜♥✐❞❛s ❡♠ t❡r♠♦s ❞❡ ❡❧❡♠❡♥t♦s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ é ❛♣r❡s❡♥t❛❞❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳ ❉❡✜♥✐çã♦ ✶✳✷✳✶ ❙❡❥❛♠X, Y ❝♦♥❥✉♥t♦s✳ ❯♠❛ ❢✉♥çã♦f :X →Y é

✭✐✮ ✐♥❥❡t♦r❛ s❡✱ ♣❛r❛x, y∈X t❛✐s q✉❡f(x) =f(y)✱ ❡♥tã♦ x=y❀

✭✐✐✮ s♦❜r❡❥❡t♦r❛ s❡ ♣❛r❛ ❝❛❞❛y∈Y✱ ❡①✐st❡ x∈X t❛❧ q✉❡f(x) =y✳

❊♠ ✉♠❛ ❝❛t❡❣♦r✐❛ q✉❛❧q✉❡r✱ ♥❡♠ s❡♠♣r❡ ♦s ♦❜❥❡t♦s sã♦ ❝♦♥❥✉♥t♦s✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r ♥ã♦ ❢❛r✐❛ s❡♥t✐❞♦ ♣❛r❛ ♠♦r✜s♠♦s✳ P♦r ❡ss❛ r❛③ã♦✱ ♥❛ t❡♥t❛t✐✈❛ ❞❡ ❣❡♥❡r❛❧✐③❛r ❡ss❡s ❝♦♥❝❡✐t♦s✱ é ♣r❡❝✐s♦ ❡♥t❡♥❞❡r ❝♦♠♦ ❛s ❢✉♥çõ❡s ✐♥❥❡t♦r❛s ❡ s♦❜r❡❥❡t♦r❛s s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ❛s ♦✉tr❛s ❢✉♥çõ❡s✳ ❯♠ r❡s✉❧t❛❞♦ ❝♦♥❤❡❝✐❞♦ é ❛♣r❡s❡♥t❛❞♦ ♣❡❧❛ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✷ ❙❡❥❛♠ X, Y ❝♦♥❥✉♥t♦s✳ ❯♠❛ ❢✉♥çã♦f :X→Y é

✭✐✮ ✐♥❥❡t♦r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ g : Y → X t❛❧ q✉❡ g◦f =idX

✭✐✐✮ s♦❜r❡❥❡t♦r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦g:Y →X t❛❧ q✉❡ f◦g=idY

➱ s❛❜✐❞♦ q✉❡ ❛ ❢✉♥çã♦ g :Y →X ❡♠ ❛♠❜♦s ✐t❡♥s ✭✐✮ ❡ ✭✐✐✮ ♥ã♦ é

❝❛♥♦♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛✳ ❊ss❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❢✉♥çõ❡s ✐♥❥❡t♦r❛s ❡ s♦❜r❡❥❡t♦r❛s s❡r✐❛♠ ♠✉✐t♦ r❡str✐t✐✈❛s ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ q✉❛❧q✉❡r✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ❝❛t❡❣♦r✐❛ RM✳ ❙❡❥❛♠ M ∈ RM✱ N ✉♠ R✲

s✉❜♠ó❞✉❧♦ ❞❡M ❡f :N →M ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✳ ❊♥tã♦✱ é ♣♦ssí✈❡❧

♠♦str❛r q✉❡ ❡①✐st❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡R✲♠ó❞✉❧♦sg:M →Nt❛❧ q✉❡ g◦f =idN s❡✱ ❡ s♦♠❡♥t❡ s❡✱ N é ✉♠ s♦♠❛♥❞♦ ❞✐r❡t♦ ❞❡M✳ P♦r ❡ss❛

r❛③ã♦✱ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❣❡r❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❛❞❛ é ❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛❣♦r❛✳

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❉❡✜♥✐çã♦ ✶✳✷✳✸ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ f : X →Y ✉♠ ♠♦r✜s♠♦

❡♠ C✳ ❊♥tã♦✱ ♦ ♠♦r✜s♠♦f é ❞✐t♦ ✉♠

✭✐✮ ♠♦♥♦♠♦r✜s♠♦ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♠♦r✜s♠♦sg, h:Z→X t❛✐s q✉❡ f ◦g=f◦h✱ t❡♠✲s❡g=h❀

✭✐✐✮ ❡♣✐♠♦r✜s♠♦ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♠♦r✜s♠♦s g, h:Y →Z t❛✐s q✉❡ g◦f =h◦f✱ t❡♠✲s❡g=h❀

✭✐✐✐✮ ✐s♦♠♦r✜s♠♦ s❡ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ g:Y →X t❛❧ q✉❡g◦f =idX

❡ f◦g=idY

❆❧é♠ ❞✐ss♦✱X ❡Y sã♦ ❞✐t♦s ✐s♦♠♦r❢♦s✱ ❡ ❞❡♥♦t❛♠♦s ♣♦rX≃Y✱ s❡

❡①✐st✐r ✉♠ ✐s♦♠♦r✜s♠♦ f :X →Y✳

◆♦t❡♠♦s q✉❡ ♠♦♥♦♠♦r✜s♠♦ ❡ ❡♣✐♠♦r✜s♠♦ sã♦ ❝♦♥❝❡✐t♦s ❞✉❛✐s✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦s ♠♦♥♦♠♦r✜s♠♦s ❞❡ Csã♦ ❡①❛t❛♠❡♥t❡ ♦s ❡♣✐♠♦r✜s✲

♠♦s ❞❡ Cop✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ✐s♦♠♦r✜s♠♦ é ❛✉t♦✲❞✉❛❧✳ ❚❛♠❜é♠✱ t♦❞♦

✐s♦♠♦r✜s♠♦ é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡ ✉♠ ❡♣✐♠♦r✜s♠♦✳ ❆ r❡❝í♣r♦❝❛ ♥❡♠ s❡♠♣r❡ é ✈❡r❞❛❞❡✐r❛✱ ❝♦♠♦ s❡rá ♠♦str❛❞♦ ♥♦s ❡①❡♠♣❧♦s✳ ❆q✉✐✱ ❢❛③❡✲ ♠♦s ♦ ❝♦♠❡♥tár✐♦ ❞❡ q✉❡ q✉❛♥❞♦ C é ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✱ ♦❜❥❡t♦

❞❡ ❡st✉❞♦ ❞♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ❛ r❡❝í♣r♦❝❛ é ✈❡r❞❛❞❡✐r❛✳

◆❛ ❝❛t❡❣♦r✐❛ Set✱ ❢✉♥çõ❡s ✐♥❥❡t♦r❛s ✭s♦❜r❡❥❡t♦r❛s✮ sã♦ ❡①❛t❛♠❡♥t❡

♦s ♠♦♥♦♠♦r✜s♠♦s ✭❡♣✐♠♦r✜s♠♦s✮✳ ❊ss❡s ❢❛t♦s s❡❣✉❡♠ ❞✐r❡t❛♠❡♥t❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✷✳✷✳ ❆❜❛✐①♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ♠♦♥♦♠♦r✜s♠♦s ♥ã♦ ✐♥✲ ❥❡t♦r❡s ❡ ❡♣✐♠♦r✜s♠♦s ♥ã♦ s♦❜r❡❥❡t♦r❡s✳

❊①❡♠♣❧♦ ✶✳✷✳✹ ❊♠ Div✱ ❛ ♣r♦❥❡çã♦ π : Q→ Q/Z é ✉♠ ♠♦♥♦♠♦r✲ ✜s♠♦ ♥ã♦ ✐♥❥❡t♦r✳

❉❡ ❢❛t♦✱ s❡❥❛♠G ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❞✐✈✐sí✈❡❧ ❡ g, h: G →Q ❤♦✲ ♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s t❛✐s q✉❡ π◦g = π◦h✳ ■ss♦ q✉❡r ❞✐③❡r q✉❡ g(x), h(x)♣❡rt❡♥❝❡♠ à ♠❡s♠❛ ❝❧❛ss❡ ❡♠Q/Z✱ ♦✉ s❡❥❛✱g(x)−h(x)∈Z✱ ♣❛r❛ t♦❞♦ x∈ G✳ ❈❤❛♠❛♠♦s k: G→Q ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ❞❛❞♦ ♣♦rk=g−h✳

❙❡❥❛x∈Gt❛❧ q✉❡ k(x)≥0✳ ❈♦♠♦Gé ❞✐✈✐sí✈❡❧✱ ♣❛r❛n=k(x) + 1>0✱ ❡①✐st❡y∈Gt❛❧ q✉❡x=ny✳ P♦rt❛♥t♦✱k(x) =k(ny) =nk(y) =

(k(x) + 1)k(y)✳ ❊♥tã♦

0≤ k(x)

k(x) + 1=k(y)<1.

❈♦♠♦k(y)∈Z✱ s❡❣✉❡ q✉❡k(y) = 0✳ ❈♦♠♦x=ny✱ t❡♠♦sk(x) = 0✳

❙❡k(x)<0✱ ✉s❛♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦✱ ❝❤❡❣❛♠♦s à ❞❡s✐❣✉❛❧❞❛❞❡

0< k(x)

k(x) + 1=k(y)<1,

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♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐sk(y)∈Z✳ ■ss♦ ✐♠♣❧✐❝❛ k= 0✱ ❧♦❣♦g=h❡π

é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ ❈❧❛r❛♠❡♥t❡✱π♥ã♦ é ✐♥❥❡t♦r✳

❊①❡♠♣❧♦ ✶✳✷✳✺ ❊♠Ring✱ ❛ ✐♥❝❧✉sã♦i:Z→Qé ✉♠ ❡♣✐♠♦r✜s♠♦ ♥ã♦ s♦❜r❡❥❡t♦r✳

❉❡ ❢❛t♦✱ s❡❥❛♠ R ✉♠ ❛♥❡❧ ❡g, h:Q→R❤♦♠♦♠♦r✜s♠♦s ❞❡ ❛♥é✐s

t❛✐s q✉❡g◦i=h◦i✳ ■ss♦ q✉❡r ❞✐③❡r q✉❡g ❡h❝♦✐♥❝✐❞❡♠ ♥♦s ✐♥t❡✐r♦s✱

♦✉ s❡❥❛✱g(n) =h(n)✱ ♣❛r❛ t♦❞♦n∈Z✳

❆❣♦r❛✱ ❞❛❞♦ q ∈ Q✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q =nm−1✱ ♣❛r❛ n, m Z

m6= 0✳ ❊♥tã♦

g(q) = g(nm−1)

= g(nm−11)

= g(n)g(m−1)g(1) = h(n)g(m−1)h(1) = h(n)g(m−1)h(mm−1) = h(n)g(m−1)h(m)h(m−1) = h(n)g(m−1)g(m)h(m−1)

= h(n)g(m−1m)h(m−1) = h(n)g(1)h(m−1) = h(n)h(1)h(m−1) = h(n1m−1)

= h(nm−1)

= h(q).

P♦rt❛♥t♦✱g=h❡ié ✉♠ ❡♣✐♠♦r✜s♠♦ q✉❡ ♥ã♦ é s♦❜r❡❥❡t♦r✳

❊①❡♠♣❧♦ ✶✳✷✳✻ P♦❞❡ ❛✐♥❞❛ ❛❝♦♥t❡❝❡r ❞❡ ✉♠❛ ❜✐❥❡çã♦ ♥ã♦ s❡r ✉♠ ✐s♦✲ ♠♦r✜s♠♦✳ ❊♠ T op✱ ❛ ❢✉♥çã♦ f : [0,2π) → S1 ❞❛❞❛ ♣♦r f(t) = eit t∈ [0,2π)✱ é ✉♠❛ ❜✐❥❡çã♦ ❝♦♥tí♥✉❛✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ✐♥✈❡rs❛ ❞❡ f ♥ã♦ é

❝♦♥tí♥✉❛✱ ❧♦❣♦f ♥ã♦ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠T op✳

❆❜❛✐①♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s ❡♠ q✉❡ ♠♦♥♦♠♦r✜s♠♦s sã♦ ✐♥❥❡t♦r❡s ❡ ❡♣✐♠♦r✜s♠♦s sã♦ s♦❜r❡❥❡t♦r❡s✳

❊①❡♠♣❧♦ ✶✳✷✳✼ ❊♠Grp✱ ♦s ♠♦♥♦♠♦r✜s♠♦s sã♦ ✐♥❥❡t♦r❡s✳

❉❡ ❢❛t♦✱ s❡❥❛♠f :G→H ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠Grp ❡K={x∈

G:f(x) =eH} ♦ s❡✉ ♥ú❝❧❡♦✳ ❙❡❥❛♠g, h:K→G❞❡✜♥✐❞♦s ♣♦r

g(x) =x ❡ h(x) =eG, ♣❛r❛ t♦❞♦ x∈K,

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♦✉ s❡❥❛✱ ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛ ❡ ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦f ◦g=f◦h✳ ❈♦♠♦f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦sg=h❡ ✐ss♦

✐♠♣❧✐❝❛ K={eG}✳ ▲♦❣♦✱f é ✐♥❥❡t♦r✳

Pr♦✈❛s ❛♥á❧♦❣❛s ❛♦ ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ♠♦str❛♠ q✉❡ ♦s ♠♦♥♦♠♦r✲ ✜s♠♦s sã♦ ✐♥❥❡t♦r❡s ♥❛s ❝❛t❡❣♦r✐❛sRing❡RM✳

❊①❡♠♣❧♦ ✶✳✷✳✽ ❊♠ring✱ ♦s ♠♦♥♦♠♦r✜s♠♦s sã♦ ✐♥❥❡t♦r❡s✳

❱❛♠♦s ♠♦str❛r ❛ ❝♦♥tr❛♣♦s✐t✐✈❛✱ ♦✉ s❡❥❛✱ ✉♠ ♠♦r✜s♠♦ ♥ã♦ ✐♥❥❡t♦r ❡♠ ring ♥ã♦ é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ ❙❡❥❛♠ f : R → S ✉♠ ♠♦r✜s♠♦

❡♠ ring ❡r, s ∈ R✱ r =6 s✱ t❛✐s q✉❡ f(r) = f(s)✳ ❙❡ R[x] é ♦ ❛♥❡❧ ❞❡

♣♦❧✐♥ô♠✐♦s s♦❜r❡ R✱ ❡①✐st❡♠ g, h: R[x]→ R ♠♦r✜s♠♦s ❡♠ring t❛✐s

q✉❡ g(x) = r ❡h(x) = s✳ P♦rt❛♥t♦✱ f ◦g =f ◦h✱ ♠❛s g 6= h✱ ♦ q✉❡

♣r♦✈❛ q✉❡f ♥ã♦ é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳

❊①❡♠♣❧♦ ✶✳✷✳✾ ❙❡❥❛R✉♠ ❛♥❡❧✳ ❊♠ RM✱ ♦s ❡♣✐♠♦r✜s♠♦s sã♦ s♦❜r❡✲

❥❡t♦r❡s✳

❉❡ ❢❛t♦✱ s❡❥❛ f : M → N ✉♠ ❡♣✐♠♦r✜s♠♦ ❡♠ RM✳ ❙❡❥❛♠ g, h : N →N/f(M)❞❛❞❛s ♣♦r

g(n) =n+f(M) ❡ h(n) = 0 +f(M), ♣❛r❛ t♦❞♦ n∈N,

♦✉ s❡❥❛✱ ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛ ❡ ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦ g◦f =h◦f✳ ❈♦♠♦ f é ✉♠ ❡♣✐♠♦r✜s♠♦✱ t❡♠♦sg =h ❡ ✐ss♦

✐♠♣❧✐❝❛ f(M) =N✳ ▲♦❣♦✱f é s♦❜r❡❥❡t♦r✳

❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♠♦str❛ q✉❡ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦♥♦♠♦r✜s♠♦s ❡ ❡♣✐♠♦r✜s♠♦s é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡ ✉♠ ❡♣✐♠♦r✜s♠♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ Pr♦♣♦s✐çã♦ ✶✳✷✳✶✵ ❙❡❥❛♠ C✉♠❛ ❝❛t❡❣♦r✐❛ ❡f :X→Y✱f′ :Y Z

♠♦r✜s♠♦s ❡♠ C✳

✭✐✮ ❙❡ f′ é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ ❡♥tã♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦ s❡✱ ❡ s♦✲

♠❡♥t❡ s❡✱ f′f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳

✭✐✐✮ ❙❡f é ✉♠ ❡♣✐♠♦r✜s♠♦✱ ❡♥tã♦f′ é ✉♠ ❡♣✐♠♦r✜s♠♦ s❡✱ ❡ s♦♠❡♥t❡

s❡✱f′f é ✉♠ ❡♣✐♠♦r✜s♠♦✳

❉❡♠♦♥str❛çã♦✿ ✭✐✮ ✭⇒✮ ❙❡❥❛♠g, h:W →X ♠♦r✜s♠♦s ❡♠Ct❛✐s q✉❡

f′fg=ffh✳ ❈♦♠♦fé ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦sfg=fh

❈♦♠♦f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦s g=h✳

✭⇐✮ ❙❡❥❛♠g, h:W →X♠♦r✜s♠♦s ❡♠Ct❛✐s q✉❡f◦g=f◦h✳ ❊♥✲

tã♦f′fg=ffh❡ ❝♦♠♦ff é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦sg=h

✭✐✐✮ ❙❡❣✉❡ ❞♦ ✐t❡♠ ✭✐✮ ♣❛r❛Cop✳

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### ✶✳✸ ❋✉♥t♦r❡s

❆♦ ❞❡✜♥✐r♠♦s ✉♠❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛✱ é ♥❛t✉r❛❧ q✉❡ s❡❥❛♠ ❞❡✜♥✐✲ ❞♦s t❛♠❜é♠ ♦s ♠♦r✜s♠♦s q✉❡ ♣r❡s❡r✈❛♠ t❛❧ ❡str✉t✉r❛✳ ◆♦ ♥♦ss♦ ❝❛s♦✱ ❞❡✜♥✐♠♦s ❛❣♦r❛ ♦s ✏♠♦r✜s♠♦s✑ ❡♥tr❡ ❝❛t❡❣♦r✐❛s✱ ❝❤❛♠❛❞♦s ❢✉♥t♦r❡s✳ ❉❡✜♥✐çã♦ ✶✳✸✳✶ ❙❡❥❛♠C✱D ❝❛t❡❣♦r✐❛s✳ ❯♠ ❢✉♥t♦r ❡♥tr❡ C ❡D✱ ❞❡✲

♥♦t❛❞♦ ♣♦rF :C→D✱ ❝♦♥s✐st❡ ❞❡ ❞✉❛s ❛♣❧✐❝❛çõ❡s✿

✭✐✮ ✉♠❛ ❛♣❧✐❝❛çã♦F :Ob(C)→Ob(D) q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ♦❜❥❡t♦X ∈C

❛ ✉♠ ♦❜❥❡t♦F(X)∈D❀

✭✐✐✮ ✉♠❛ ❛♣❧✐❝❛çã♦ F :HomC(X, Y)→HomD(F(X), F(Y))q✉❡ ❛ss♦✲

❝✐❛ ❝❛❞❛ ♠♦r✜s♠♦ f :X → Y ❡♠ C ❛ ✉♠ ♠♦r✜s♠♦ F(f) : F(X)→

F(Y)❡♠ D t❛❧ q✉❡

F(idX) =idF(X) ❡ F(g◦f) =F(g)◦F(f),

♣❛r❛X ∈C❡f, g ♠♦r✜s♠♦s ❡♠C t❛❧ q✉❡ ❛ ❝♦♠♣♦s✐çã♦g◦f ❡①✐st❛✳

❖❜s❡r✈❛çã♦ ✶✳✸✳✷ ❖ ❢✉♥t♦r ❞❡✜♥✐❞♦ ❛❝✐♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❢✉♥t♦r ❝♦✈❛r✐❛♥t❡✳ ❯♠ ❢✉♥t♦r ❝♦♥tr❛✈❛r✐❛♥t❡ é ❝♦♠♣❧❡t❛♠❡♥t❡ ❛♥á❧♦❣♦✱ ❡①❝❡t♦ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ✏✐♥✈❡rt❡ ✢❡❝❤❛s✑✱ ♦✉ s❡❥❛✱ s❡f :X→Y é ✉♠ ♠♦r✜s♠♦

❡♠ C✱ ❡♥tã♦ F(f) : F(Y) → F(X) é ✉♠ ♠♦r✜s♠♦ ❡♠ D✳ P♦rt❛♥t♦✱

♣❛r❛f, g♠♦r✜s♠♦s ❡♠Ct❛❧ q✉❡ ❛ ❝♦♠♣♦s✐çã♦g◦f ❡①✐st❛✱ t❡♠✲s❡ F(g◦f) =F(f)◦F(g).

P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠ ❢✉♥t♦r ❝♦♥tr❛✈❛r✐❛♥t❡ F:C→D❝♦♠♦ ✉♠

❢✉♥t♦r ❝♦✈❛r✐❛♥t❡F :Cop D✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❛♥❞♦f, g ♠♦r✜s♠♦s

❡♠Cop✱ ❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r s❡ t♦r♥❛

F(f◦opg) =F(gf) =F(f)F(g).

➱ ♣♦r ❡ss❛ r❛③ã♦ q✉❡ é s✉✜❝✐❡♥t❡ ❡st✉❞❛r♠♦s ❢✉♥t♦r❡s ❝♦✈❛r✐❛♥t❡s✳ ❉❡✜♥✐çã♦ ✶✳✸✳✸ ❙❡❥❛♠ C✱D✱E ❝❛t❡❣♦r✐❛s ❡ F : C → D✱ G : D → E

❢✉♥t♦r❡s✳ ❆ ❝♦♠♣♦s✐çã♦G◦F :C→Eé ❞❡✜♥✐❞❛ ♣♦r

(G◦F)(X) =G(F(X)) ❡ (G◦F)(f) =G(F(f)),

♣❛r❛X ∈C❡f ✉♠ ♠♦r✜s♠♦ ❡♠C✳

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P❛r❛C,D❝❛t❡❣♦r✐❛s✱ ❞❡♥♦t❛♠♦s ♣♦rF un(C,D)❛ ❝♦❧❡çã♦ ❞❡ ❢✉♥t♦✲

r❡s ❡♥tr❡C❡D✳ ◆❛ ♣ró①✐♠❛ s❡çã♦✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛ ❝♦❧❡çã♦F un(C,D)

❝♦♠♦ ✉♠❛ ❝❛t❡❣♦r✐❛✳ P❛r❛ ✉♠❛ ❝❛t❡❣♦r✐❛C✱F un(C,C) =End(C)✳

❆♣r❡s❡♥t❛♠♦s ❛❣♦r❛ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❢✉♥t♦r❡s✳ ❈♦♠❡ç❛♠♦s ❛♣r❡✲ s❡♥t❛♥❞♦ ❢✉♥t♦r❡s q✉❡ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ❡♠ q✉❛❧q✉❡r ❝❛t❡❣♦r✐❛✳ ❊①❡♠♣❧♦ ✶✳✸✳✹ ❚♦❞❛ ❝❛t❡❣♦r✐❛ C ♣♦ss✉✐ ✉♠ ❢✉♥t♦r ✐❞❡♥t✐❞❛❞❡ IdC : C → C❞❡✜♥✐❞♦ ♣♦r IdC(X) = X ❡ IdC(f) =f✱ ♣❛r❛ X ∈ C❡ f ✉♠

♠♦r✜s♠♦ ❡♠C✳

❊①❡♠♣❧♦ ✶✳✸✳✺ ❙❡❥❛♠ C✉♠❛ ❝❛t❡❣♦r✐❛ ❡ D ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❞❡C✳

❖ ❢✉♥t♦r ✐♥❝❧✉sã♦ID:D→C é ❞❡✜♥✐❞♦ ♣♦rID(X) =X ❡ID(f) =f✱

♣❛r❛X ∈D ❡f ✉♠ ♠♦r✜s♠♦ ❡♠D✳

❊①❡♠♣❧♦ ✶✳✸✳✻ ❙❡❥❛♠ C,D ❝❛t❡❣♦r✐❛s ❡ Z ∈ D✳ ❖ ❢✉♥t♦r ❝♦♥st❛♥t❡

CZ :C→D é ❞❡✜♥✐❞♦ ♣♦rCZ(X) =Z ❡CZ(f) =idZ✱ ♣❛r❛X ∈C❡

f ✉♠ ♠♦r✜s♠♦ ❡♠C✳

❊①❡♠♣❧♦ ✶✳✸✳✼ ❙❡❥❛♠ C,D ❝❛t❡❣♦r✐❛s✳ ❖ ❢✉♥t♦r PC : C×D → C✱

❝❤❛♠❛❞♦ ♣r♦❥❡çã♦ s♦❜r❡C✱ é ❞❡✜♥✐❞♦ ♣♦rPC(X, Y) =X❡PC(f, g) =f✱

♣❛r❛(X, Y)∈C×D❡(f, g)✉♠ ♠♦r✜s♠♦ ❡♠C×D✳

❆♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❢✉♥t♦r PD : C×D → D✱

❝❤❛♠❛❞♦ ♣r♦❥❡çã♦ s♦❜r❡D✳

❊①❡♠♣❧♦ ✶✳✸✳✽ ❙❡❥❛♠ C✉♠❛ ❝❛t❡❣♦r✐❛ ❡X ∈C✳ ❖ ❢✉♥t♦rLX :C→

Set é ❞❡✜♥✐❞♦ ♣♦r

LX(Y) =HomC(X, Y), ♣❛r❛ Y ∈C,

❡ ♣❛r❛f :Y →Z ✉♠ ♠♦r✜s♠♦ ❡♠C✱

LX(f) : HomC(X, Y) → HomC(X, Z)

α 7→ f ◦α.

❖ ❢✉♥t♦rLX é ❝❤❛♠❛❞♦ ❢✉♥t♦r r❡♣r❡s❡♥t❛❞♦ ♣♦rX✳

❊①❡♠♣❧♦ ✶✳✸✳✾ ❆♥á❧♦❣♦ ❛♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♦ ❢✉♥t♦rRX:C→Set

é ❞❡✜♥✐❞♦ ♣♦r

RX(Y) =HomC(Y, X), ♣❛r❛ Y ∈C,

❡ ♣❛r❛f :Y →Z ✉♠ ♠♦r✜s♠♦ ❡♠C✱

RX(f) : HomC(Z, X) → HomC(Y, X)

α 7→ α◦f.

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◆❡ss❡ ❝❛s♦✱ ♦ ❢✉♥t♦r RX é ❝♦♥tr❛✈❛r✐❛♥t❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ Y ∈C ❡

♣❛r❛ t♦❞♦α∈HomC(Y, X)✱ t❡♠♦s

RX(idY)(α) = α◦idY

= α

= idRX(Y)(α).

❙❡❥❛♠f :Y →Z❡g:Z→W ♠♦r✜s♠♦s ❡♠C❡α∈HomC(W, X)✳

❊♥tã♦

RX(g◦f)(α) = α◦(g◦f)

= (α◦g)◦f

= (RX(g)(α))◦f

= RX(f)(RX(g)(α))

= (RX(f)◦RX(g))(α).

❯♠❛ ♥♦t❛çã♦ ♠❛✐s ❝♦♥❤❡❝✐❞❛ ♣❛r❛LX✭r❡s♣❡❝t✐✈❛♠❡♥t❡RX✮ éHomC(X,−)

✭r❡s♣❡❝t✐✈❛♠❡♥t❡HomC(−, X)✮✳ ◆❡ss❛ ♥♦t❛çã♦✱ ❡s❝r❡✈❡✲s❡ f∗ ✭r❡s♣❡❝✲

t✐✈❛♠❡♥t❡f∗✮ ♣❛r❛Hom

C(X,−)(f)✭r❡s♣❡❝t✐✈❛♠❡♥t❡HomC(−, X)(f)✮✳

❊ss❡s ❞♦✐s ❢✉♥t♦r❡s ♣♦❞❡♠ ❛✐♥❞❛ s❡r ❝♦♠❜✐♥❛❞♦s✱ ❝♦♠♦ ♠♦str❛ ♦ ♣ró✲ ①✐♠♦ ❡①❡♠♣❧♦✳

❊①❡♠♣❧♦ ✶✳✸✳✶✵ ❙❡❥❛♠C✉♠❛ ❝❛t❡❣♦r✐❛ ❡X ∈C✳ ❖ ❢✉♥t♦rHomC(−,−) : Cop×CSeté ❞❡✜♥✐❞♦ ♣♦r

HomC(−,−)(X, Y) =HomC(X, Y), ♣❛r❛ (X, Y)∈Cop×C,

❡ ♣❛r❛(f, g) : (X, Y)→(X′, Y)✉♠ ♠♦r✜s♠♦ ❡♠Cop×C HomC(−,−)(f, g) : HomC(X, Y) → HomC(X′, Y′)

α 7→ g◦α◦f.

◆♦t❡♠♦s q✉❡ HomC(−,−) é ✉♠ ❜✐❢✉♥t♦r ❡ HomC(−,−)(f, g) =

f∗g

∗=g∗◦f∗✳

❆❣♦r❛✱ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❢✉♥t♦r ❝♦♥tr❛✈❛r✐❛♥t❡RX✱ ❞❡✜♥✐❞♦ ♥♦

❊①❡♠♣❧♦ ✶✳✸✳✾✳

❊①❡♠♣❧♦ ✶✳✸✳✶✶ ❉❡♥♦t❛♠♦s ♣♦r D : V ectk → V ectk ♦ ❢✉♥t♦r Rk✳

P♦rt❛♥t♦✱ t❡♠♦s

D(V) =V∗ ❡ D(T) =T∗,

♣❛r❛V ✉♠k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡T :V →W ✉♠❛ tr❛♥s❢♦r♠❛çã♦k✲❧✐♥❡❛r✱

❡♠ q✉❡V∗=Homk(V, k)é ♦k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞✉❛❧ ❞❡ V

T∗: W V

f 7→ f◦T

é ✉♠❛ tr❛♥s❢♦r♠❛çã♦k✲❧✐♥❡❛r✱ ❝❤❛♠❛❞❛ tr❛♥s♣♦st❛ ❞❡T :V →W✳

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❖s ♣ró①✐♠♦s ❡①❡♠♣❧♦s sã♦ ❝❤❛♠❛❞♦s ❢✉♥t♦r❡s ❞❡ ❡sq✉❡❝✐♠❡♥t♦✳ ❙ã♦ ❝❤❛♠❛❞♦s ❛ss✐♠ ♣♦rq✉❡ ♦ ❡❢❡✐t♦ ❞❡❧❡s s♦❜r❡ ♦s ♦❜❥❡t♦s ❝♦♥s✐st❡ ❡♠ ❡s✲ q✉❡❝❡r ♣❛rt❡ ❞❛ ❡str✉t✉r❛ ♦✉ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♦❜❥❡t♦✱ ❡♥q✉❛♥t♦ ♦s ♠♦r✜s♠♦s sã♦ ♣r❡s❡r✈❛❞♦s✳ P♦rt❛♥t♦✱ ❡ss❡s ❢✉♥t♦r❡s sã♦✱ ✐♥t✉✐t✐✈❛✲ ♠❡♥t❡✱ ❞❛ ❢♦r♠❛ U :C→D✱ U(X) =X ❡U(f) =f✱ ♣❛r❛X ∈C ❡f

✉♠ ♠♦r✜s♠♦ ❡♠C✳

❊①❡♠♣❧♦ ✶✳✸✳✶✷ ❖ ❢✉♥t♦r ❞❡ ❡sq✉❡❝✐♠❡♥t♦U :Grp→Set❡sq✉❡❝❡ ❛

❡str✉t✉r❛ ❞❡ ❣r✉♣♦✳

❊①❡♠♣❧♦ ✶✳✸✳✶✸ ❖ ❢✉♥t♦r ❞❡ ❡sq✉❡❝✐♠❡♥t♦ U : Ab→Grp ❡sq✉❡❝❡ ❛

❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❣r✉♣♦s ❛❜❡❧✐❛♥♦s✳ ◆♦t❡ q✉❡ ❡ss❡ ❢✉♥t♦r é ❡①❛t❛✲ ♠❡♥t❡ ♦ ❢✉♥t♦r ✐♥❝❧✉sã♦ ❞❡Ab❡♠Grp✳

❊①❡♠♣❧♦ ✶✳✸✳✶✹ ❙❡❥❛ R ✉♠ ❛♥❡❧✳ ❖ ❢✉♥t♦r ❞❡ ❡sq✉❡❝✐♠❡♥t♦ U : RM→Ab❡sq✉❡❝❡ ❛ ❛çã♦ ❞❡Rs♦❜r❡ ♦s R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳

❊①❡♠♣❧♦ ✶✳✸✳✶✺ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❖ ❢✉♥t♦r ❞❡ ❡sq✉❡❝✐♠❡♥t♦ U : Algk →V ectk ❡sq✉❡❝❡ ♦ ♣r♦❞✉t♦ ❞❛sk✲á❧❣❡❜r❛s✳

❯s❛♥❞♦ ♦s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s q✉❡ ❛♣r❡s❡♥t❛♠♦s✱ ♣♦❞❡rí❛♠♦s ❝♦♥s✐❞❡r❛r ❛✐♥❞❛ ✈ár✐♦s ♦✉tr♦s ❡①❡♠♣❧♦s ❞❡ ❢✉♥t♦r❡s ❞❡ ❡sq✉❡❝✐♠❡♥t♦✳ ❆♣❡s❛r ❞❡ s❡r❡♠ s✐♠♣❧❡s✱ ❡st❡s ❢✉♥t♦r❡s sã♦ ✐♠♣♦rt❛♥t❡s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ q✉❛♥❞♦ s❡ t❡♥t❛ ❝♦♥s✐❞❡r❛r ❢✉♥t♦r❡s ❛ss♦❝✐❛❞♦s ❛ ❡❧❡s ♥❛ ❞✐r❡çã♦ ❝♦♥✲ trár✐❛✳ ❊♠ ♠✉✐t♦s ❝❛s♦s✱ ♦❜té♠✲s❡ ❢✉♥t♦r❡s q✉❡ ❛ss♦❝✐❛♠ ♦❜❥❡t♦s ❛ ♦❜❥❡t♦s ❝❤❛♠❛❞♦s ❧✐✈r❡s ♦✉ ✉♥✐✈❡rs❛✐s✳ P❛r❛ ♣♦❞❡r♠♦s ❞❛r ♦ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦✱ ❢❛③❡♠♦s ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ❣r✉♣♦s ❧✐✈r❡s ❣❡r❛❞♦s ♣♦r ✉♠ ❝♦♥❥✉♥t♦✳

❉❡✜♥✐çã♦ ✶✳✸✳✶✻ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦✳ ❯♠ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦rX

é ✉♠ ♣❛r (FX, ιX)✱ ❡♠ q✉❡ FX é ✉♠ ❣r✉♣♦ ❡ ιX : X → FX é ✉♠❛

❢✉♥çã♦✱ s❛t✐s❢❛③❡♥❞♦ à s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✿ ♣❛r❛ q✉❛❧q✉❡r

♣❛r (G, f)✱ ❡♠ q✉❡ G é ✉♠ ❣r✉♣♦ ❡f :X → Gé ✉♠❛ ❢✉♥çã♦✱ ❡①✐st❡

✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s g:FX→Gt❛❧ q✉❡g◦ιX =f✳

➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦X✱ ❡①✐st❡ ✉♠ ❣r✉♣♦

❧✐✈r❡FX ❣❡r❛❞♦ ♣♦rX✱ ú♥✐❝♦✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ P♦r ❡ss❛ r❛③ã♦✱

❞✐③❡♠♦s q✉❡ FX é ♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r X✳ ❉❡✈✐❞♦ à ♣r♦♣r✐❡❞❛❞❡

✉♥✐✈❡rs❛❧ ❞♦s ❣r✉♣♦s ❧✐✈r❡s✱ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f : X → Y✱ ❡①✐st❡ ✉♠

ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s f¯:FX FY t❛❧ q✉❡f¯ιX =ιY f

P❛r❛ ♠❛✐s ✐♥❢♦r♠❛çõ❡s✱ ✈❡❥❛ ✭❬✼❪✱ ♣á❣s✳ ✻✹✲✻✻✮✳

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