Universidade Federal de Santa Catarina Curso de Pós-Graduação em Matemática Pura e Aplicada

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

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

❊q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ❝❛t❡❣♦r✐❛s

❦✲❧✐♥❡❛r❡s

▲✉✐s ❆✉❣✉st♦ ❯❧✐❛♥❛

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢✳➟ ❉r❛✳ ❱✐r❣í♥✐❛ ❙✐❧✈❛ ❘♦❞r✐❣✉❡s

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

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

❊q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ❝❛t❡❣♦r✐❛s ❦✲❧✐♥❡❛r❡s

❉✐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❛✳

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❊q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ❝❛t❡❣♦r✐❛s ❦✲❧✐♥❡❛r❡s

♣♦r

▲✉✐s ❆✉❣✉st♦ ❯❧✐❛♥❛✶

❊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❛ ✲ ❯❋❙❈✮

Pr♦❢✳ ❉r✳ ❆❜❞❡❧♠♦✉❜✐♥❡ ❆♠❛r ❍❡♥♥✐ ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮

Pr♦❢✳ ❉r✳ ❏✉❛♥ ▼❛rtí♥ ▼♦♠❜❡❧❧✐ ✭❯♥✐✈❡rs✐❞❛❞ ◆❛❝✐♦♥❛❧ ❞❡ ❈ór❞♦❜❛ ✲ ❯◆❈✮

Pr♦❢✳ ❉r✳ ❱✐t♦r ❞❡ ❖❧✐✈❡✐r❛ ❋❡rr❡✐r❛ ✭❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ✲ ❯❙P✮

❋❧♦r✐❛♥ó♣♦❧✐s✱ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✺✳

❇♦❧s✐st❛ ❞❛ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ✲

❈❆P❊❙

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■❡s✉s ❞✐①✐t✱ ✏s✐ q✉✐s ✈✉❧t ♣♦st ♠❡ s❡q✉✐✱ ❞❡♥❡❣❡t s❡ ✐♣s✉♠ ❡t t♦❧❧❛t ❝r✉❝❡♠ s✉❛♠ ❡t s❡q✉❛t✉r ♠❡✳✑ ▼❝ ✽✱✸✹

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ à ❉❡✉s✱ ♣♦r t❡r ♠❡ ❢❡✐t♦ ❡♥①❡r❣❛r ♣♦r ♠❡✐♦ ❞❡ ◆♦ss❛ ❙❡♥❤♦r❛ ❡ ❙❛♥t❛ ❇❡r♥❛❞❡tt❡ ❛ ✈❡r❞❛❞❡✐r❛ ❜❡❧❡③❛ ❞♦ ♠✉♥❞♦✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♠✐♥❤❛ ♠ã❡✱ ✐r♠ã✱ ♠❡✉ t✐♦ ❍❛rr② ❡ ♠✐♥❤❛ t✐❛ ❱❡r❛ ♣♦r t❡r❡♠ ❛❥✉❞❛❞♦ ❞✉r❛♥t❡ ❡ss❡ ♣❡rí♦❞♦✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ♣♦r t❡r t✐❞♦ ♠✉✐t❛ ❞❡❞✐❝❛çã♦ ♥♦ ❞❡✲ s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛ ♣♦r t❡r❡♠ ❛❝❡✐t♦ ♦ ❝♦♥✈✐t❡ ❞❡ ♣❛rt✐❝✐♣❛r ❡ ♣♦r t❡r❡♠ ❢❡✐t♦ ❝♦rr❡çõ❡s ♣❛r❛ ❞❡✐①❛r ❡st❡ tr❛❜❛❧❤♦ ♠❡❧❤♦r✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ♣r♦❢❡ss♦r❡s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡ss❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✳

❆❣r❛❞❡ç♦ à ❊❧✐s❛✱ s❡❝r❡tár✐❛ ❞❛ ♣ós✱ ♣♦r t❡r ♠❡ ❛❥✉❞❛❞♦ ♥♦s ♣❡rí♦❞♦s ❞❡ ♠❛trí❝✉❧❛ ❡ ❝♦♠ t♦❞❛ ❛ ♣❛rt❡ ❜✉r♦❝rát✐❝❛ ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s✳

❆❣r❛❞❡ç♦ à ❈❆P❊❙ ✭❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r✮ ♣❡❧❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s ❢♦r♥❡❝✐❞❛✱ s❡♠ ❛ q✉❛❧ ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ❡st❛ ❞✐ss❡rt❛çã♦✳

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❘❡s✉♠♦

❆ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s é ❛♣r❡s❡♥t❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ ✶✾✹✺✱ ♥♦ tr❛❜❛❧❤♦ ❡♥t✐t✉❧❛❞♦ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ◆❛t✉r❛❧ ❊q✉✐✈❛❧❡♥❝❡s✳ ◆❛ ♣✉❜❧✐❝❛çã♦ ❞❡ ✶✾✺✵✱ ❡♥t✐t✉❧❛❞❛ ❉✉❛❧✐t② ❢♦r ●r♦✉♣s✱ ▼❛❝▲❛♥❡ ✐♥tr♦❞✉③ ♣♦r ♠❡✐♦ ❛①✐♦♠át✐❝♦ ❛ ♥♦çã♦ ❞❡ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✳

❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❛❧❣✉♠❛s ❝♦♥str✉çõ❡s ❢❡✐t❛s ❡♠ ❝❛t❡❣♦r✐❛s k✲❧✐♥❡❛r❡s ✭q✉❡ sã♦ ❛❜❡❧✐❛♥❛s✮✳ P❛ss❛♠♦s ♣♦r t♦❞❛s ❛s ❞❡✜✲ ♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s ♥❛ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s ♣❛r❛ ♣♦❞❡r♠♦s ❞❡✜♥✐r ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦ ✜♥✐t♦ G ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❞❡✜♥✐r ❛ ❡q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛k✲❧✐♥❡❛r✳

❈♦♠♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✱ ♠♦str❛♠♦s q✉❡ ❛ ❡q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r é✱ t❛♠❜é♠✱ k✲❧✐♥❡❛r✳ P❛r❛ ❡ss❡ ❡st✉❞♦✱ ✉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 ❞♦ ♣r♦❢✳ ❉r✳ ▼❛rtí♥ ▼♦♠❜❡❧❧✐✳

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❆❜str❛❝t

❚❤❡ ❝❛t❡❣♦r② t❤❡♦r② ✐s ✐♥tr♦❞✉❝❡❞ ❢♦r t❤❡ ✜rst t✐♠❡ ✐♥ ✶✾✹✺ ✐♥ ❛ r❡s❡❛r❝❤ ❡♥t✐t✉❧❛t❡❞ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ◆❛t✉r❛❧ ❊q✉✐✈❛❧❡♥❝❡s✳ ■♥ ✶✾✺✵ ✐♥ ❛ ♣✉❜❧✐❝❛t✐♦♥ ❝❛❧❧❡❞ ❉✉❛❧✐t② ❢♦r ●r♦✉♣s✱ ▼❛❝▲❛♥❡ ✐♥tr♦❞✉❝❡ tr♦✉❣❤ ❛①✐♦♠s t❤❡ ♥♦t✐♦♥ ♦❢ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r②✳

❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s r❡s❡❛r❝❤ ✐s st✉❞②✐♥❣ s♦♠❡ ❝♦♥tr✉❝t✐♦♥s ❞♦♥❡ ✐♥ k✲❧✐♥❡❛r ❝❛t❡❣♦r✐❡s ✭✇❤✐❝❤ ❛r❡ ❛❜❡❧✐❛♥s✮✳ ❲❡ ❤❛✈❡ st✉❞✐❡❞ ❛❧❧ ♥❡❝❡ss❛r② ❞❡✜♥✐t✐♦♥s ❛♥❞ r❡s✉❧ts ✐♥ t❤❡ ❝❛t❡❣♦r✐❡s t❤❡♦r② s♦ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❛❝t✐♦♥ ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G✐♥ ❛k✲❧✐♥❡❛r ❝❛t❡❣♦r② ❛♥❞ ❛❢t❡r t❤❛t ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❡q✉✐✈❛r✐❛♥t✐③❛t✐♦♥ ♦❢ ❛k✲❧✐♥❡❛r ❝❛t❡❣♦r②✳

❆s t❤❡ ♠❛✐♥ r❡s✉❧t ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❡ ❡q✉✐✈❛r✐❛♥t✐③❛t✐♦♥ ♦❢ ❛ k✲ ❧✐♥❡❛r ❝❛t❡❣♦r② ✐s✱ ❛❧s♦✱ k✲❧✐♥❡❛r✳ ❲❡ st✉❞② ❛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❤❡ ♣r♦❢✳ ❉r✳ ▼❛rtí♥ ▼♦♠❜❡❧❧✐✳

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❙✉♠ár✐♦

✶ ❈❛t❡❣♦r✐❛s ✺

✶✳✶ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ◆ú❝❧❡♦s ❡ ❝♦♥ú❝❧❡♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ▼♦♥♦♠♦r✜s♠♦s✱ ❡♣✐♠♦r✜s♠♦s ❡ ✐s♦♠♦r✜s♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✹ Pr♦❞✉t♦s ❡ ❝♦♣r♦❞✉t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷ ❋✉♥t♦r❡s ✷✵

✷✳✶ ❋✉♥t♦r❡s ❡ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❋✉♥t♦r❡s ❛❞❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✸ ❈❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ✺✸

✹ ❊q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ❝❛t❡❣♦r✐❛s k✲❧✐♥❡❛r❡s ✼✶ ❆ ➪❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ✾✹ ❆✳✶ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ❆✳✷ ➪❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻

❇ ❈♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s ❡ ❝♦❝❛❞❡✐❛s ✾✾

❈ ❈♦♥str✉çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞♦ ●r✉♣♦ ❞❡ ❚r❛♥ç❛s ✶✵✷ ❈✳✶ ▲✐♥❦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ❈✳✷ ❚❛♥❣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺ ❈✳✸ ❚r❛♥ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷

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■♥tr♦❞✉çã♦

❊♠ ♠❡❛❞♦s ❞❡ ✶✾✹✵ ❙❛✉♥❞❡rs ▼❛❝▲❛♥❡ ❡ ❙❛♠✉❡❧ ❊✐❧❡♥❜❡r❣ tr❛❜❛✲ ❧❤❛r❛♠ ❝♦♥❥✉♥t❛♠❡♥t❡✱ ♥❡ss❡ ♣❡rí♦❞♦ ♣✉❜❧✐❝❛r❛♠ ♦ tr❛❜❛❧❤♦[✸]♥♦ q✉❛❧ ✉t✐❧✐③❛✈❛♠ ♦ t❡r♠♦ ✏✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✑ ♣❛r❛ ❞❡s✐❣♥❛r ❝❡rt♦s t✐♣♦s ❞❡ ✐s♦♠♦r✜s♠♦s✱ ❝✉❥♦s ❛✉t♦r❡s r❡❢❡r❡♠ ❝♦♠♦ s❡♥❞♦ ✉♠ ✏❢❡♥ô♠❡♥♦✑ q✉❡ ♦❝♦rr✐❛ ❡♠ ✈ár✐♦s ❝♦♥t❡①t♦s ❞❛ ♠❛t❡♠át✐❝❛✳

❆ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s é ❛♣r❡s❡♥t❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ ✶✾✹✺✱ ♥♦ tr❛❜❛❧❤♦ [✹] ❡♥t✐t✉❧❛❞♦ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ◆❛t✉r❛❧ ❊q✉✐✈❛❧❡♥❝❡s✳ ◆♦ ❡♥t❛♥t♦✱ é ♦❜s❡r✈❛❞♦ q✉❡ ❛ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s s✉r❣❡ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ tr❛❜❛❧❤❛r ❡ t♦r♥❛r ♣r❡❝✐s❛ ❛ ♥♦çã♦ ❞❡ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ■♥✐❝✐❛❧✲ ♠❡♥t❡ ❛ t❡♦r✐❛ ❝❤❡❣❛ ❛ s❡r ❝❤❛♠❛❞❛ ❞❡ ✏❛❜str❛çã♦ s❡♠ s❡♥t✐❞♦✑✱ ❝♦♠♦ ♦❜s❡r✈❛❞♦ ❡♠[✻]✱ ♥♦ ❡♥t❛♥t♦✱ ❞❡♣♦✐s ❞♦ tr❛❜❛❧❤♦ ❞❡ ●r♦t❤❡♥❞✐❡❝❦✱ ❉❛✲ ♥✐❡❧ ❑❛♥ ❡ ♦✉tr♦s ❛ t❡♦r✐❛ ❣❛♥❤❛ ✉♠ ❡s♣❛ç♦ ❡ r❡s♣❡✐t♦ ❞❡♥tr♦ ❞❡ t♦❞❛ ❛ ♠❛t❡♠át✐❝❛✳

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ♣❛rt❡ ❞❛s ♥♦t❛s ❞❡ ❛✉❧❛[✶✻]❡s✲ ❝r✐t❛s ♣❡❧♦ Pr♦❢✳ ❉r✳ ▼❛rtí♥ ▼♦♠❜❡❧❧✐✱ s❡❣✉✐♠♦s ❜❛s✐❝❛♠❡♥t❡ ❛ ♠❡s♠❛ ♦r❣❛♥✐③❛çã♦ ❡ ♦r❞❡♠ ❞♦s ❛ss✉♥t♦s ❞❡ss❛ r❡❢❡rê♥❝✐❛✳ ❖ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❢❛③ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s✱ q✉❡ sã♦ ✉t✐❧✐③❛❞♦s ♥♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥t❡s✳ ❈♦♠♦ r❡❢❡rê♥❝✐❛s ❛❞✐❝✐♦♥❛✐s ❝✐t❛♠♦s[✶✹]✱[✶]❡[✶✵]✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥t♦r ❡ tr❛♥s❢♦r✲ ♠❛çã♦ ♥❛t✉r❛❧✳ ❆ ♥♦çã♦ ❞❡ ❢✉♥t♦r ❛♣❛r❡❝❡ ❢♦r♠❛❧♠❡♥t❡ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ [✸] ♥♦ ❝♦♥t❡①t♦ ❞❡ ❣r✉♣♦s ❡ é ♦❜s❡r✈❛❞♦ q✉❡ t❛❧ ♥♦çã♦ ♣♦❞❡✲ r✐❛ ❢❛❝✐❧♠❡♥t❡ s❡r ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ♦✉tr♦s ✏❝♦♥t❡①t♦s✑✳ ❊♠ [✸]✱ ❝✉r✐✲ ♦s❛♠❡♥t❡ ❡♥❝♦♥tr❛♠♦s ❝♦♠♦ ❡①❡♠♣❧♦ ❞❡ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✱ ♦ ✐s♦✲ ♠♦r✜s♠♦ Hom(−, Hom(−,−)) ≃ Hom(− ⊗ −,−)✱ q✉❡ r❡❧❛❝✐♦♥❛ ♦s

❢✉♥t♦r❡sHom❡⊗✱ s❡♠ ❢❛③❡r r❡❢❡rê♥❝✐❛ à ♥♦çã♦ ❞❡ ❛❞❥✉♥çã♦✳

P♦❞❡♠♦s ❝✐t❛r ♦ ▲❡♠❛ ❞❡ ❨♦♥❡❞❛ ❝♦♠♦ ✉♠ ❞♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❞♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❡st❡ q✉❡ é ✉♠ ❞♦s r❡s✉❧t❛❞♦s ♠❛✐s ❝♦♥❤❡❝✐❞♦s ❞❛ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s✳ ❈✉r✐♦s❛♠❡♥t❡✱ ♦ ❧❡♠❛ ♥ã♦ ❢♦✐ ♣r♦✈❛❞♦ ♥♦ ❛rt✐❣♦ ❞❡ ❨♦♥❡❞❛✱ ♦❜s❡r✈❛♠♦s ♦ q✉❡ ❞✐③ P❡t❡r ❋r❡②❞ ❡♠ ✭[✼]✱ ♣✳✶✹✮

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❚❤❡ ❨♦♥❡❞❛ ▲❡♠♠❛ t✉r♥s ♦✉t ♥♦t t♦ ❜❡ ✐♥ ❨♦♥❡❞❛✬s ♣❛♣❡r✳ ❲❤❡♥✱ s♦♠❡ t✐♠❡ ❛❢t❡r ❜♦t❤ ♣r✐♥t✐♥❣s ♦❢ t❤❡ ❜♦♦❦ ❛♣♣❡❛✲ r❡❞✱ t❤✐s ✇❛s ❜r♦✉❣❤t t♦ ♠② ✭♠✉❝❤ ❝❤❛❣r✐♥❡❞✮ ❛tt❡♥t✐♦♥✱ ■ ❜r♦✉❣❤t ✐t t❤❡ ❛tt❡♥t✐♦♥ ♦❢ t❤❡ ♣❡rs♦♥ ✇❤♦ ❤❛❞ t♦❧❞ ♠❡ t❤❛t ✐t ✇❛s t❤❡ ❨♦♥❡❞❛ ▲❡♠♠❛✳ ❍❡ ❝♦♥s✉❧t❡❞ ❤✐s ♥♦t❡s ❛♥❞ ❞✐s✲ ❝♦✈❡r❡❞ t❤❛t ✐t ❛♣♣❡❛r❡❞ ✐♥ ❛ ❧❡❝t✉r❡ t❤❛t ▼❛❝▲❛♥❡ ❣❛✈❡ ♦♥ ❨♦♥❡❞❛✬s tr❡❛t♠❡♥t ♦❢ t❤❡ ❤✐❣❤❡r ❊①t ❢✉♥❝t♦rs✳ ❚❤❡ ♥❛♠❡ ✏❨♦♥❡❞❛ ▲❡♠♠❛✑ ✇❛s ♥♦t ❞♦♦♠❡❞ t♦ ❜❡ r❡♣❧❛❝❡❞✳

❊♠ ✶✾✺✽✱ ❉❛♥✐❡❧ ❑❛♥ ❡♠ s❡✉ ❛rt✐❣♦[✶✶]❡♥t✐t✉❧❛❞♦ ❆❞❥♦✐♥t ❋✉♥❝t♦rs ✐♥tr♦❞✉③ ❛ ♥♦çã♦ ❞❡ ❛❞❥✉♥çã♦✱ ✉t✐❧✐③❛♥❞♦ ❝♦♠♦ ♠♦t✐✈❛çã♦ ♦ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❞♦ ❛rt✐❣♦ ❞❡ ✶✾✹✷ ❝✐t❛❞♦ ❛❝✐♠❛✳ ◆♦ ❝❛♣ít✉❧♦ ❞♦✐s✱ ♠♦str❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❝♦♥❝r❡t♦s ❞❡ ❛❞❥✉♥çã♦ ❡ ♣r♦✈❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣❡rt✐♥❡♥t❡s✳ P❛r❛ ❡ss❡ ❝❛♣ít✉❧♦✱ ✉t✐❧✐③❛♠♦s ❝♦♠♦ ❜❛s❡[✶✻]✱ [✶✹]❡[✶]✳

❊♠ ✭[✶✺]✱ ♣✳ ✸✸✽✮ ❙❛✉♥❞❡rs ▼❛❝▲❛♥❡ ❡s❝r❡✈❡

❚❤❡ ♥❡①t st❡♣ ✐♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❝❛t❡❣♦r② t❤❡♦r② ✇❛s t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❝❛t❡❣♦r✐❡s ✇✐t❤ str✉❝t✉r❡✳ ❆❜♦✉t ✶✾✹✼✱ ■ ♥♦t✐❝❡ t❤❛t t❤❡ ❊✐❧❡♥❜❡r❣ ❙t❡❡♥r♦❞ ❛①✐♦♠❛t✐❝ ❤♦♠♦❧♦❣② t❤❡♦r② ❝♦♥❝❡r♥❡❞ ❢✉♥❝t♦rs ❢r♦♠ ❛ ❝❛t❡❣♦r② ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s t♦ ✈❛r✐♦✉s ❝❛t❡❣♦r✐❡s ✇✐t❤ ❛♥ ✏❛❞❞✐t✐✈❡✑ str✉❝t✉r❡ ✲ ❝❛t❡❣♦r✐❡s ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✱ ♦r ♦❢ R✲♠♦❞✉❧❡s ❢♦r ✈❛r✐♦✉s r✐♥❣sR✳ ■ ❝♦♥s❡q✉❡♥t❧② s❡t ❛❜♦✉t t♦ ❞❡s❝r✐❜❡ ❛①✐♦♠❛t✐❝❛❧❧② t❤❡s❡ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r✐❡s✳

◆❛ ♣✉❜❧✐❝❛çã♦ [✶✸] ❞❡ ✶✾✺✵✱ ❡♥t✐t✉❧❛❞❛ ❉✉❛❧✐t② ❢♦r ●r♦✉♣s✱ ▼❛✲ ❝▲❛♥❡ ✐♥tr♦❞✉③ ♣♦r ♠❡✐♦ ❛①✐♦♠át✐❝♦ ❛ ♥♦çã♦ ❞❡ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✳ ◆❡ss❡ ❛rt✐❣♦ ❞❡ ✶✾✺✵✱ ▼❛❝▲❛♥❡ ♦❜s❡r✈❛ ❛ ❞✉❛❧✐❞❛❞❡ ✭❞❛s ❞❡♠♦♥str❛✲ çõ❡s✮ ❡①✐st❡♥t❡ ❡♥tr❡ ❝❡rt❛s ♥♦çõ❡s ❝♦♠♦ ♣r♦❞✉t♦ ❡ ❝♦♣r♦❞✉t♦✱ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ♣r♦❝✉r❛♠♦s ❢❛❧❛r s♦❜r❡ ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s ❡ ❛❜❡❧✐❛♥❛s ❡ ♣r♦✈❛r ♦s r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✳ ❯♠ r❡s✉❧t❛❞♦ q✉❡ ♣r♦✈❛♠♦s✱ ♠♦str❛ ❛ r❡❧❛çã♦ ❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡①✐st❡♥t❡ ❡♥tr❡ ❛s ♥♦çõ❡s ❞❡ ♣r♦❞✉t♦ ❡ ❝♦♣r♦❞✉t♦ ❡♠ ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✳

◆♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐♠♦s ❛ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦G❡♠ ✉♠❛ ❝❛t❡✲ ❣♦r✐❛k✲❧✐♥❡❛rC✳ ❆ ❡str✉t✉r❛ ❞❡ ❛❞✐t✐✈✐❞❛❞❡ ❞❛ ❝❛t❡❣♦r✐❛ ❡ ❛ ❛❞✐t✐✈✐❞❛❞❡ ❞♦s ❢✉♥t♦r❡s sã♦ ♥❡❝❡ssár✐♦s ❡ ❢❛③❡♠ ✉♠ ♣❛r❛❧❡❧♦ ❝♦♠ ❛ ♥♦çã♦ ❞❡ ❛çã♦ ❞❡ ❣r✉♣♦ ❡♠ ❝♦♥❥✉♥t♦s✳

❚❡r♠✐♥❛♠♦s ❞❡✜♥✐♥❞♦ ✉♠❛ ♥♦✈❛ ❝❛t❡❣♦r✐❛✱ ❞❡♥♦t❛❞❛CG✱ ❝❤❛♠❛❞❛ ❡q✉✐✈❛r✐❛♥t✐③❛çã♦✳ P❛r❛ ❝♦♥str✉✐r ❡ss❛ ♥♦✈❛ ❝❛t❡❣♦r✐❛✱ t✉❞♦ ♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥♦ tr❛❜❛❧❤♦ é ✉t✐❧✐③❛❞♦✳ Pr♦✈❛♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛s ❝❛t❡✲ ❣♦r✐❛s (Am)G ❡A

kkGm✱ q✉❡ ✐❧✉str❛ ✉♠ ❢❛t♦ ♦❜s❡r✈❛❞♦ ❡♠[✻]✱ ❞❡ q✉❡

♣❛r❛ t♦❞❛ ❝❛t❡❣♦r✐❛k✲❧✐♥❡❛r ✜♥✐t❛D✱ ❡①✐st❡ ✉♠❛ á❧❣❡❜r❛At❛❧ q✉❡Dé

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❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ❞❡A✲♠ó❞✉❧♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ ✉♠ ❝♦r♣♦k✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧✱ ♣r♦✈❛♠♦s q✉❡✱ ❝❛s♦Cs❡❥❛k✲❧✐♥❡❛r✱ ❡♥tã♦CG ék✲❧✐♥❡❛r✳

■♥❢♦r♠❛♠♦s ❛♦ ❧❡✐t♦r q✉❡ ♦ ❡st✉❞♦ ❞❡ ❡q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ❝❛t❡✲ ❣♦r✐❛s ♣♦r ✉♠ ❣r✉♣♦ ♣♦❞❡ s❡r ❛♠♣❧✐❛❞♦ ♣❛r❛ ♦✉tr❛s ❝❛t❡❣♦r✐❛s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❝❛t❡❣♦r✐❛s t❡♥s♦r✐❛✐s ✜♥✐t❛s q✉❡ é ❛ ♦r❞❡♠ ❝r♦♥♦❧ó❣✐❝❛ ❞❡ [✶✻]✳ ❊ss❡ ❡st✉❞♦ é ❛♣❧✐❝❛❞♦ ♥❛ t❡♦r✐❛ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❝❛t❡❣♦r✐❛s t❡♥s♦r✐❛✐s✳

❖ ❆♣ê♥❞✐❝❡ ❆ ❝♦♥té♠ t♦❞❛s ❛s ♥♦t❛çõ❡s ❡ r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s ♣❛r❛ q✉❡ ♣✉❞éss❡♠♦s ❢❛❧❛r ❞♦s ❢✉♥t♦r❡sU:Liek Algk ❡L:Algk

Liek✳ ❯t✐❧✐③❛♠♦s t❛❧ ❛♣ê♥❞✐❝❡ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ♣r♦✈❛r✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ t❛✐s ❢✉♥t♦r❡s sã♦ ❛❞❥✉♥t♦s✳

❖s ❆♣ê♥❞✐❝❡s ❇ ❡ ❈ ❝♦♥tê♠ ❛ ❝♦♥str✉çã♦ ❡ ❛ ❞❡✜♥✐çã♦ ❞♦s ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛ ❡ ❞♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉✉r❛♥t❡ ❡st❡ tr❛✲ ❜❛❧❤♦✱ ❢♦r❛♠ ❢❡✐t♦s s❡♠✐♥ár✐♦s s❡♠❛♥❛✐s ♥♦s q✉❛✐s ❢♦✐ ❛♣r❡s❡♥t❛❞❛ t♦❞❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ❞♦ ♠❡s♠♦ ❡ ✐ss♦ ❥✉st✐✜❝❛ ♦ ❆♣ê♥❞✐❝❡ ❈✳

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❈❛♣ít✉❧♦ ✶

❈❛t❡❣♦r✐❛s

◆❡ss❡ ❝❛♣ít✉❧♦ ❢❛❧❛♠♦s s♦❜r❡ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s✱ ❝♦♥❝❡✐t♦s ❢✉♥❞❛✲ ♠❡♥t❛✐s ❡ t♦❞❛ ❛ ♥♦♠❡❝❧❛t✉r❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥✲ t❡s✳ ■♥✐❝✐❛♠♦s ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝❛t❡❣♦r✐❛ ❡ ❞❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s✳ ❉❡♣♦✐s ♣❛rt✐♠♦s ♣❛r❛ ❛❧❣✉♠❛s ♥♦çõ❡s ❝♦♠♦✿ ♥ú❝❧❡♦✱ ♣r♦❞✉t♦ ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❞✉❛✐s✳ ❚♦❞❛s ❛s ❞❡✜♥✐çõ❡s ❡ ♥♦çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠[✶✹]✳

✶✳✶ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s

❉❡✜♥✐çã♦ ✶✳✶ ❯♠❛ ❝❛t❡❣♦r✐❛C ❝♦♥s✐st❡ ❞❡ ✭✐✮ ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♦❜❥❡t♦s Ob(C)❀

✭✐✐✮ ♣❛r❛ t♦❞♦ ♣❛r (U, V)❞❡ ♦❜❥❡t♦s ❡♠C ❤á ✉♠❛ ❝♦❧❡çã♦HomC(U, V)

❞❡ ♠♦r✜s♠♦s ❞❡ U ♣❛r❛V❀

✭✐✐✐✮ ♣❛r❛ q✉❛❧q✉❡r ♦❜❥❡t♦ W ❡♠ Ob(C) ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ IW ❡♠ HomC(W, W)❝❤❛♠❛❞♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡❀

✭✐✈✮ ♣❛r❛ q✉❛✐sq✉❡r U, V ❡W ♦❜❥❡t♦s ❡♠Ob(C) ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ HomC(U, V)×HomC(V, W) → HomC(U, W)

(f, g) 7→ g◦f

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

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✭❛✮ ♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦sU ❡V ❡♠Ob(C)✱ ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡IU ❡♠HomC(U, U)é t❛❧ q✉❡f◦IU =f ❡IU◦g=g✱ ♣❛r❛ q✉❛✐sq✉❡rf ❡♠

HomC(U, V)❡g ❡♠ HomC(V, U)❀

✭❜✮ ❞❛❞♦s ♦❜❥❡t♦sU, V, W ❡Z❡♠Ob(C)❡ ♠♦r✜s♠♦sf ❡♠HomC(U, V)✱

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

✐✳❡✳✱h◦(g◦f) = (h◦g)◦f✳

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

f

U →V✳ ❆❧é♠ ❞✐ss♦✱ U é ❝❤❛♠❛❞♦ ❞♦♠í♥✐♦ ❞♦ ♠♦r✜s♠♦f ❡V é ❝❤❛✲ ♠❛❞♦ ❝♦❞♦♠í♥✐♦ ❞❡f✳

❖❜s❡r✈❛♠♦s q✉❡✱ ❝♦♠♦ ❝❛❞❛ ♦❜❥❡t♦ ♣♦❞❡ s❡r ❛ss♦❝✐❛❞♦ ❛♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡✱ é t❡❝♥✐❝❛♠❡♥t❡ ♣♦ssí✈❡❧ tr❛❜❛❧❤❛r ❝♦♠ ✉♠❛ ❞❡✜♥✐çã♦ ❞❡ ❝❛t❡❣♦r✐❛ q✉❡ t❡♥❤❛ ❛♣❡♥❛s ♠♦r✜s♠♦s ❝♦♠♦ é ♦❜s❡r✈❛❞♦ ❡♠ [✶✹]✳ ◆♦ ❡♥t❛♥t♦✱ é ♠❛✐s ❝ô♠♦❞♦ tr❛❜❛❧❤❛r ✉s❛♥❞♦ ❡ss❛ ❞❡✜♥✐çã♦ ❝♦♠ ♦❜❥❡t♦s✳ ❆ s❡❣✉✐r✱ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s✳

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

❉❡ ❢❛t♦✱ s❡❥❛X ❡♠Ob(Set)✳ ❈♦♥s✐❞❡r❛♠♦s ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡IX ❡♠HomSet(X, X)❝♦♠♦ s❡♥❞♦ ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡✳ ❆ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s é ❛ss♦❝✐❛t✐✈❛ ❡ ♣♦rt❛♥t♦✱Seté ✉♠❛ ❝❛t❡❣♦r✐❛✳

❊①❡♠♣❧♦ ✶✳✸ ❆ ❝❛t❡❣♦r✐❛ Grp ❝♦♠♦ s❡♥❞♦ ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❣r✉♣♦s ❡ ♦s ♠♦r✜s♠♦s ❡♥tr❡ ♦❜❥❡t♦s sã♦ ♦s ♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s✳ ❊①❡♠♣❧♦ ✶✳✹ ❆ ❝❛t❡❣♦r✐❛Abé ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ❡ ❝✉❥♦s ♠♦r✜s♠♦s ❡♥tr❡ ♦❜❥❡t♦s sã♦ ♦s ♠♦r✜s♠♦s ❞❡ ❣r✉♣♦✳ ❊①❡♠♣❧♦ ✶✳✺ ❆ ❝❛t❡❣♦r✐❛Ringé ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❛♥é✐s ❡ ♦s ♠♦r✜s♠♦s ❡♥tr❡ ♦❜❥❡t♦s sã♦ ♦s ♠♦r✜s♠♦s ❞❡ ❛♥é✐s✳

❊①❡♠♣❧♦ ✶✳✻ ❙❡❥❛k✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦rV ectk ❛ ❝❛t❡❣♦r✐❛ ❝✉✲ ❥♦s ♦❜❥❡t♦s sã♦ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s✳

❉❡♥♦t❛♠♦svectk ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ ♦s ♠♦r✜s♠♦s ❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s✳

❊①❡♠♣❧♦ ✶✳✼ ❙❡❥❛k✉♠ ❝♦r♣♦✳ ❉❡✜♥✐♠♦sAlgk ❝♦♠♦ s❡♥❞♦ ❛ ❝❛t❡❣♦✲ r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ❛sk✲á❧❣❡❜r❛s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s✳

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❊①❡♠♣❧♦ ✶✳✽ ❙❡❥❛R✉♠ ❛♥❡❧✳ ❉❡♥♦t❛♠♦s ♣♦rRM✭MR✮ ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭à ❞✐r❡✐t❛✮✳ ❖s ♠♦r✜s♠♦s sã♦ ♦s ♠♦r✜s♠♦s ❞❡R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭à ❞✐r❡✐t❛✮✳

❊①❡♠♣❧♦ ✶✳✾ ❙❡❥❛ A ✉♠❛ k✲á❧❣❡❜r❛✳ ❉❡♥♦t❛♠♦s ♣♦r AM ✭MA✮ ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦sk✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♠✉♥✐❞♦s ❞❡ ✉♠❛ ❛çã♦ q✉❡ ♦s t♦r♥❛ A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭à ❞✐r❡✐t❛✮✳ ❖s ♠♦r✜s♠♦s sã♦ ♦s ♠♦r✜s♠♦s ❞❡k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❡ ❞❡A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭à ❞✐r❡✐t❛✮✳ ❊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❡♠♣❧♦ ✶✳✶✶ ❙❡❥❛A ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❝♦♠ ✉♥✐❞❛❞❡✳ ❆ ❝❛t❡✲ ❣♦r✐❛ A é ❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ✉♠ ú♥✐❝♦ ♦❜❥❡t♦✱ ❛ s❛❜❡r ❛ á❧❣❡❜r❛ A✱ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❡❧❡♠❡♥t♦s ❞❡ A✳ ❆ ❝♦♠♣♦s✐çã♦ é ♦ ♣r♦❞✉t♦ ❞❡A✳ ❊①❡♠♣❧♦ ✶✳✶✷ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r Liek ❛ ❝❛t❡❣♦r✐❛ ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ s♦❜r❡ ♦ ❝♦r♣♦ k✳ ❖s ♠♦r✜s♠♦s ❞❡ss❛ ❝❛t❡❣♦r✐❛✱ sã♦ ♦s ♠♦r✜s♠♦s ❞❡ ➪❧❣❡❜r❛s ❞❡ ▲✐❡✱ ♦✉ s❡❥❛✱ ❛♣❧✐❝❛çõ❡s k✲❧✐♥❡❛r❡s q✉❡ ♣r❡s❡r✈❛♠ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✱ ❝✐t❛♠♦s ♦ ❆♣ê♥❞✐❝❡ A✳

❊①❡♠♣❧♦ ✶✳✶✸ ❙❡❥❛R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡✳ ❉❡♥♦t❛♠♦s ♣♦r Ch(R)❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛ s♦❜r❡

♦ ❛♥❡❧ R✱ ✐✳❡✳✱ ♣❛r❡s (C, d)❡♠ q✉❡ C é ✉♠ R✲♠ó❞✉❧♦Z✲❣r❛❞✉❛❞♦ ❡ d é ✉♠ ❡♥❞♦♠♦r✜s♠♦ ❞❡ ❣r❛✉ ✶ t❛❧ q✉❡ d2= 0✳ ❆ ❝♦♠♣♦s✐çã♦ ❞❡✜♥✐❞❛

❡♥tr❡ ❞♦✐s ♠♦r✜s♠♦s s❛t✐s❢❛③ ♦s ❛①✐♦♠❛s ❞❡ ❝❛t❡❣♦r✐❛✳ ❊ss❡ ❡①❡♠♣❧♦ é ❛♣r❡s❡♥t❛❞♦ ❞❡ ❢♦r♠❛ ♠❛✐s ❞❡t❛❧❤❛❞❛ ♥♦ ❆♣ê♥❞✐❝❡B✳

❊①❡♠♣❧♦ ✶✳✶✹ ❉❡ ♠❛♥❡✐r❛ ♠❛✐s ❣❡r❛❧ q✉❡ ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♣♦✲ ❞❡♠♦s ❝♦♥s✐❞❡r❛r ChN(R) ♣❛r❛ ✉♠ N N ❝♦♠ N > 2 ❛ ❝❛t❡❣♦r✐❛ ❞♦sN✲❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛✱ ♦♥❞❡ ♦s ♦❜❥❡t♦s sã♦ ♣❛r❡s(C, d)❝♦♠♦ ♥❛ ❝❛t❡❣♦r✐❛ ❛♥t❡r✐♦r ❝♦♠ ❛ r❡ss❛❧✈❛ ❞❡ q✉❡dN = 0✳

❯♠ ❞♦s ❡①❡♠♣❧♦s ✐♥t❡r❡ss❛♥t❡s ❞❡ ✉♠❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ q✉❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ❝❛t❡❣♦r✐❛ é ♦ ❝❤❛♠❛❞♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s✳ ❚♦❞❛s ❛s ♥♦t❛çõ❡s✱ ❛ ♣r♦✈❛ ❞❛ ❡①✐stê♥❝✐❛ ❡ ❛s r❡❢❡rê♥❝✐❛s sã♦ ❝✐t❛❞❛s ❡ ❢❡✐t❛s ♥♦ ❆♣ê♥❞✐❝❡C✳ ❆q✉✐✱ ❛♣r❡s❡♥t❛♠♦s s✉❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❡ s❡♠♣r❡ q✉❡ ❢♦r ❢❡✐t❛ ❛❧❣✉♠❛ ♦❜s❡r✈❛çã♦ ❣❡♦♠étr✐❝❛✱ ❢❛③❡♠♦s r❡❢❡rê♥❝✐❛ ❛♦ ♠♦❞❡❧♦ ❝♦♥str✉í❞♦ ♥❡ss❡ ❛♣ê♥❞✐❝❡✳

❙❡❥❛n∈N✜①♦✳ ❉❡✜♥✐♠♦s ♦ ❣r✉♣♦ ❞❡ n✲tr❛♥ç❛s✱ ❞❡♥♦t❛❞♦ ♣♦rBn✱

✈✐❛ r❡❧❛çõ❡s s♦❜r❡ ❣❡r❛❞♦r❡s ❝♦♠♦ ❛❜❛✐①♦✳

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❉❡✜♥✐çã♦ ✶✳✶✺ ❖ ❣r✉♣♦ Bn é ♦ ❣r✉♣♦ ❣❡r❛❞♦ ♣♦r σ1✱ · · · ✱ σn−1 s✉✲

❥❡✐t♦s às r❡❧❛çõ❡s✿

✭✐✮ ❙❡♠♣r❡ q✉❡ 3≤n❡1≤i, j≤n−1 ❝♦♠|i−j|>1 t❡♠♦s

σiσj=σjσi.

✭✐✐✮ ❚❛♠❜é♠ é ✈á❧✐❞♦ q✉❡

σiσi+1σi=σi+1σiσi+1

♣❛r❛ t♦❞♦i∈ {1,· · · , n−1}✳

➱ ✐♥t❡r❡ss❛♥t❡ ♦❜s❡r✈❛r♠♦s q✉❡ ♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s✱ ❞❡ ❝❡rt❛ ♠❛✲ ♥❡✐r❛✱ ❣❡♥❡r❛❧✐③❛ ♦ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çõ❡s✳ P♦❞❡♠♦s ❞❡✜♥✐r ♦ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çõ❡s ❞❡ n ❡❧❡♠❡♥t♦s Sn ✈✐❛ r❡❧❛çõ❡s s♦❜r❡ ❣❡r❛❞♦r❡s ❡♠ q✉❡ σi= (i, i+ 1)sã♦ ♦s ❣❡r❛❞♦r❡s✱ ❛s ❝❤❛♠❛❞❛s tr❛♥s♣♦s✐çõ❡s✱ q✉❡ s❛t✐s❢❛✲ ③❡♠ ♥ã♦ s♦♠❡♥t❡ ❛s r❡❧❛çõ❡s ❞❡ tr❛♥ç❛(i)❡(ii)♠❛s t❛♠❜é♠ ❛ ❝♦♥❞✐çã♦ σ2

i = 1 ♣❛r❛ t♦❞♦1 ≤i≤n−1✳ ■ss♦ ♥ã♦ ♦❝♦rr❡ ♥♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s ♣❛r❛n >1❡✱ ♥❡ss❡ s❡♥t✐❞♦✱ ♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s é ♠❛✐s ❣❡r❛❧✳

❚❛♠❜é♠ ❡①✐st❡ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ q✉❡ ♦❝♦rr❡ ♥♦ ♣r♦✲ ❞✉t♦σ2

i✳ ❙❡❣✉✐♥❞♦ ❛ ♥♦t❛çã♦ ❛❞♦t❛❞❛ ♥♦ ❆♣ê♥❞✐❝❡ C♣♦❞❡♠♦s r❡♣r❡✲ s❡♥t❛rσ2

i ♣❡❧♦ ❞✐❛❣r❛♠❛

❖ q✉❡ ♦❝♦rr❡ ♥❡ss❡ ♣r♦❞✉t♦ é q✉❡ ❛♦ ✐♥✈és ❞❡ ✈♦❧t❛r ♣❛r❛ ❛ ❝♦♥✲ ✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❛s ♣❡r♠✉t❛çõ❡s✱ q✉❛♥❞♦ ❢❛③❡♠♦s σ2

i ♦❜t❡♠♦s ✉♠❛ tr❛♥ç❛ ♥♦ s❡♥t✐❞♦ ❝♦♠✉♠ ❞❛ ♣❛❧❛✈r❛✱ ✐♠❛❣✐♥❛♥❞♦ q✉❡ ♦s ❛r❝♦s ♣♦❧✐❣♦♥❛✐s sã♦ ✏❝♦r❞❛s✑✳ ❙❡ ❝♦♥t✐♥✉❛r♠♦s ♦ ♣r♦❝❡ss♦ ❡ ✜③❡r♠♦s✱ ♣♦r ❡①❡♠♣❧♦σ3

i✱ ♦❜t❡♠♦s ♠❛✐s ✉♠❛ ✈♦❧t❛✱ ✉♠ ✏♥ó✑ ❡♥tr❡ ❛s ♣♦❧✐❣♦♥❛✐s ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❙❡❣✉✐♥❞♦ ❡ss❡ r❛❝✐♦❝í♥✐♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r t❛♠✲ ❜é♠ q✉❡ ♦ ❣r✉♣♦ ❞❡ n✲tr❛♥ç❛s ♣❛r❛ n > 1 t❡♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❡♥q✉❛♥t♦ ♦ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çõ❡s é ✜♥✐t♦✳

(20)

❊①❡♠♣❧♦ ✶✳✶✻ ❉❡♥♦t❛♠♦s ♣♦r B ❛ ❝❛t❡❣♦r✐❛ ❞❡ tr❛♥ç❛s✳ ❊ss❛ ❝❛t❡✲

❣♦r✐❛ t❡♠ ❝♦♠♦ ♦❜❥❡t♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ❞❛❞♦s m, n∈ Nt❡♠♦s

q✉❡

HomB(m, n) =

∅, s❡m6=n

Bn, s❡m=n.

P❛r❛ ❡ss❡ ❝♦♥❥✉♥t♦ ❞❡ ♠♦r✜s♠♦s ❞❡✜♥✐♠♦s ❛ ❝♦♠♣♦s✐çã♦ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ❞♦ ❣r✉♣♦✳ P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ m = n s❡❣✉❡✱ ❞♦ ❢❛t♦ ❞❡Bn

s❡r ❣r✉♣♦✱ q✉❡ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s é ❛ss♦❝✐❛t✐✈❛ ❡ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ 1n✳ P❛r❛ ♦ ❝❛s♦ m 6= n ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ♣♦r ✈❛❝✉✐❞❛❞❡✳

❖❜s❡r✈❛♠♦s q✉❡ ♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ ♣♦❞❡r✐❛ t❡r s✐❞♦ ❝♦♥str✉í❞♦ ✉t✐❧✐✲ ③❛♥❞♦ q✉❛❧q✉❡r ❝♦❧❡çã♦ ❞❡ ❣r✉♣♦s ✐♥❞❡①❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞♦s ♥❛t✉r❛✐s✳ ❏✉st✐✜❝❛♠♦s ❛ ✉t✐❧✐③❛çã♦ ❞♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s✱ ♣♦✐s ❞✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✲ ✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✱ ❢♦✐ ❞❛❞❛ ✉♠❛ ❛t❡♥çã♦ ❡s♣❡❝✐❛❧ à ❝♦♥str✉çã♦ ❞❡ss❡ ❣r✉♣♦ ❡ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s s❡♠✐♥ár✐♦s s❡♠❛♥❛✐s q✉❡ ❞❡r❛♠ ♦r✐❣❡♠ ❛♦ ❆♣ê♥❞✐❝❡ ❈✳

❉❡✜♥✐çã♦ ✶✳✶✼ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛✳ ❉✐③❡♠♦s q✉❡ D é ✉♠❛ s✉❜❝❛✲ t❡❣♦r✐❛ ❞❡Cs❡ Dé ✉♠❛ ❝❛t❡❣♦r✐❛ t❛❧ q✉❡ t♦❞♦ ♦❜❥❡t♦ ❞❡ Dé ♦❜❥❡t♦ ❞❡ C❡✱ ♣❛r❛ q✉❛✐sq✉❡rUV ❡♠Ob(C)✱ ♦s ♠♦r✜s♠♦s ❞❡U ♣❛r❛V ❡♠D sã♦ ♠♦r✜s♠♦s ❞❡ U ♣❛r❛ V ❡♠ C✳ ❆ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s ❡♠D é ❛ ♠❡s♠❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s q✉❡ ❡♠ C✳

❉❡✜♥✐çã♦ ✶✳✶✽ ❉✐③❡♠♦s q✉❡Dé ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❡ Cs❡D é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ t❛❧ q✉❡ HomD(U, V) =HomC(U, V) ♣❛r❛ q✉❛✐sq✉❡r

U ❡ V ❡♠ Ob(D)✳

❊①❡♠♣❧♦ ✶✳✶✾ ❆ ❝❛t❡❣♦r✐❛Abé ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❛ ❝❛t❡❣♦r✐❛ Grp ❡vectk é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❛ ❝❛t❡❣♦r✐❛V ectk✳

❊①❡♠♣❧♦ ✶✳✷✵ ❈♦♥s✐❞❡r❡♠♦s ❛ ❝❛t❡❣♦r✐❛ring❞♦s ❛♥é✐s ❝♦♠ ✉♥✐❞❛❞❡✱ ❝✉❥♦s ♠♦r✜s♠♦s sã♦ ♦s ♠♦r✜s♠♦s ❞❡ ❛♥é✐s q✉❡ ♣r❡s❡r✈❛♠ ❛ ✉♥✐❞❛❞❡✳ ❊ss❛ é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❞❡Ring q✉❡ ♥ã♦ é ♣❧❡♥❛✳ ❉❡ ❢❛t♦✱ ❞❡✜♥✐♠♦s ♦ ♠♦r✜s♠♦ ❞❡ ❛♥é✐s

f : R → M2×2(R)

x 7→

x 0 0 0

❚❛❧ ♠♦r✜s♠♦ ♣❡rt❡♥❝❡ ❛ ❝❛t❡❣♦r✐❛Ring ♠❛s ♥ã♦ ♣❡rt❡♥❝❡ à ❝❛t❡✲ ❣♦r✐❛ ring ♣♦✐s ♥ã♦ ♣r❡s❡r✈❛ ✉♥✐❞❛❞❡✳

(21)

❉❡✜♥✐çã♦ ✶✳✷✶ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❝❛t❡❣♦r✐❛ Cé ♣❡q✉❡♥❛ s❡ ❛ ❝♦❧❡çã♦ ❞♦s s❡✉s ♦❜❥❡t♦s ❢♦r ✉♠ ❝♦♥❥✉♥t♦ ❡ s❡ ♣❛r❛ q✉❛❧q✉❡r ♣❛r ❞❡ ♦❜❥❡t♦s ❛ ❝♦❧❡çã♦ ❞♦s ♠♦r✜s♠♦s ❡♥tr❡ ❡ss❡s ♦❜❥❡t♦s t❛♠❜é♠ é ✉♠ ❝♦♥❥✉♥t♦✳ ❉❡✜♥✐çã♦ ✶✳✷✷ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❝❛t❡❣♦r✐❛Cé ❧♦❝❛❧♠❡♥t❡ ♣❡q✉❡♥❛ s❡ ❞❛❞♦sX ❡Y ❡♠ Ct✐✈❡r♠♦s q✉❡HomC(X, Y)é ✉♠ ❝♦♥❥✉♥t♦✳

P❛r❛ ❛ ♠❛✐♦r ♣❛rt❡ ❞♦s r❡s✉❧t❛❞♦s q✉❡ ✈❛♠♦s ♠♦str❛r ♥❡ss❡ tr❛❜❛❧❤♦✱ é s✉✜❝✐❡♥t❡ q✉❡ ❛ ❝❛t❡❣♦r✐❛ s❡❥❛ ❧♦❝❛❧♠❡♥t❡ ♣❡q✉❡♥❛✳ ❊♠ ❞❡tr✐♠❡♥t♦ ❞✐ss♦✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ ❡ss❛s ❝❛t❡❣♦r✐❛s ❡♠ t♦❞♦ ♦ tr❛❜❛❧❤♦✳ ❊s❝r❡✈❡✲ ♠♦sf ∈HomC(U, V)♣❛r❛ ❞❡s✐❣♥❛r q✉❛❧q✉❡r ♠♦r✜s♠♦ ❞❡U ♣❛r❛V ❡✱

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

❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❝♦♥tr✉çõ❡s ✐♠♣♦rt❛♥t❡s ✉t✐❧✐③❛❞❛s ♠❛✐s ❛❞✐❛♥t❡✳

❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛✳ ❉❡♥♦t❛♠♦s ♣♦r Cop ❛ ❝❛t❡❣♦r✐❛ t❛❧ q✉❡ Ob(Cop) =Ob(C)❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡rX, Y Ob(Cop)✱Hom

Cop(X, Y) =

HomC(Y, X)✳

❆ss✐♠✱ ❞❛❞♦s f ∈HomCop(X, Y)❡g∈HomCop(Y, Z)✱ ❞❡✜♥✐♠♦s ❛

❝♦♠♣♦s✐çã♦

g◦opf =fg

q✉❡ é ❛ss♦❝✐❛t✐✈❛✳ ❉❡ ❢❛t♦✱ s❡❥❛♠f❡g❝♦♠♦ ❛❝✐♠❛ ❡h∈HomCop(Z, W)✳

❊♥tã♦

h◦op(gopf) = (gopf)h = (f ◦g)◦h = f ◦(g◦h) = (g◦h)◦opf = (h◦opg)opf.

❖ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞❛ ❝❛t❡❣♦r✐❛ C é ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ♥❛ ❝❛t❡❣♦r✐❛Cop✳ ❆ ❝❛t❡❣♦r✐❛Cop é ❝❤❛♠❛❞❛ ❝❛t❡❣♦r✐❛ ♦♣♦st❛ ❞❡ C✳

❙❡❥❛♠ C ❡ D ❞✉❛s ❝❛t❡❣♦r✐❛s✳ ❉❡♥♦t❛♠♦s ♣♦r C×D ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ♣❛r❡s(X, Y)❝♦♠X ∈Ob(C)Y Ob(D)✳ P❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♦❜❥❡t♦s(X, Y),(U, V)∈Ob(C×D)t❡♠♦s

HomC×D((X, Y),(U, V)) = (HomC(X, U), HomD(Y, V)).

❊♥tã♦ é ♣♦ssí✈❡❧ ❞❡✜♥✐r ❛ s❡❣✉✐♥t❡ ♦♣❡r❛çã♦✿

(HomC(X, U), HomD(Y, V))×(HomC(U, Z), HomD(V, W))

((f, g),(r, s))

(22)

♣♦r (f, g)◦(r, s) = (r◦f, s◦g) ∈ (HomC(X, Z), HomD(Y, W)). ❚❛❧

♦♣❡r❛çã♦ é ❛ss♦❝✐❛t✐✈❛✳ ❉❡ ❢❛t♦✱

((f, g)◦(r, s))◦(u, v) = (r◦f, s◦g)◦(u, v) = (u◦(r◦f), v◦(s◦g)) = ((u◦r)◦f,(v◦s)◦g) = (f, g)◦(u◦r, v◦s) = (f, g)◦((r, s)◦(u, v)).

❉❛❞♦ ✉♠ ♦❜❥❡t♦(X, Y)♦ ♠♦r✜s♠♦ (IX, IY)é ♦ ❞❡ ♠♦r✜s♠♦ ✐❞❡♥✲ t✐❞❛❞❡ ♥❡ss❛ ❝❛t❡❣♦r✐❛✳

❊ss❛ ❝♦♥str✉çã♦ ♥♦s ♣❡r♠✐t❡ ♦❜t❡r ♥♦✈❛s ❝❛t❡❣♦r✐❛s ❛ ♣❛rt✐r ❞❡ ❝❛✲ t❡❣♦r✐❛s ❥á ❝♦♥❤❡❝✐❞❛s✳

✶✳✷ ◆ú❝❧❡♦s ❡ ❝♦♥ú❝❧❡♦s

❉❡✜♥✐çã♦ ✶✳✷✸ ❙❡❥❛ C✉♠❛ ❝❛t❡❣♦r✐❛✳ ❯♠ ♦❜❥❡t♦Z Ob(C)❝❤❛♠❛✲ s❡ ♦❜❥❡t♦ ③❡r♦ ✭♦✉ ♦❜❥❡t♦ ♥✉❧♦✮ s❡ ♣❛r❛ t♦❞♦ X∈Ob(C)❡①✐st❡♠ ú♥✐❝♦s ♠♦r✜s♠♦sφX :X →Z ❡ψX :Z→X✱ ♦✉ s❡❥❛✱HomC(X, Z) ={φX}

❡ HomC(Z, X) ={ψX}✳

❆♥t❡s ❞❡ ❡♥✉♥❝✐❛r♠♦s ❛ ♦❜s❡r✈❛çã♦ s❡❣✉✐♥t❡✱ r❡❝♦♠❡♥❞❛♠♦s ❛♦ ❧❡✐✲ t♦r q✉❡ t♦♠❡ ❝✐ê♥❝✐❛ ❞❛ ❉❡✜♥✐çã♦1.37✱ ❛ q✉❛❧ ❡♥❝♦♥tr❛✲s❡ ♥❛ ♣ró①✐♠❛ s❡çã♦✳

Pr♦♣♦s✐çã♦ ✶✳✷✹ ❖ ♦❜❥❡t♦ ③❡r♦ é ú♥✐❝♦✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s q✉❡Z❡W s❡❥❛♠ ♦❜❥❡t♦s ③❡r♦s✳ ❊♥tã♦ ❡①✐st❡♠ ú♥✐❝♦s φW : W →Z ❡ ψW : Z →W ❡ ♣♦rt❛♥t♦ φW ◦

ψW ∈HomC(Z, Z) ={IZ}✳ ❆♥❛❧♦❣❛♠❡♥t❡ ❝♦♥❝❧✉í♠♦s q✉❡ψW◦φW =

IW✳ ▲♦❣♦✱Z ≃W✳

❊①❡♠♣❧♦ ✶✳✷✺ ◆❛ ❝❛t❡❣♦r✐❛Grp✱ ♦ ❣r✉♣♦ tr✐✈✐❛❧{e}é ✉♠ ♦❜❥❡t♦ ③❡r♦✳ ❊①❡♠♣❧♦ ✶✳✷✻ ❆ ❝❛t❡❣♦r✐❛Set ♥ã♦ ♣♦ss✉✐ ♦❜❥❡t♦ ③❡r♦✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡ Z s❡❥❛ ✉♠ ♦❜❥❡t♦ ③❡r♦ ❡♠ Set✳ ❙❡ ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ Z é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❞♦ q✉❡ ✷✱ ❡♥tã♦ ❞❛❞♦ ♦ ❝♦♥❥✉♥t♦ ✉♥✐tár✐♦ {∅}✱ ♣♦❞❡♠♦s ❞❡✜♥✐r✱ ♣❡❧♦ ♠❡♥♦s✱ ❞✉❛s ❢✉♥çõ❡s ❞✐st✐♥t❛s ❞❡ss❡ ❝♦♥❥✉♥t♦ ❡♠ Z ❡ t❡♠♦s ✉♠ ❛❜s✉r❞♦✳ ❙❡Z é ✉♥✐tár✐♦ ❡♥tã♦✱ ♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❝♦♠ ❞♦✐s ❡❧❡♠❡♥t♦s✱ ♣♦❞❡♠♦s t♦♠❛r ❞✉❛s ❢✉♥çõ❡s ❞✐st✐♥t❛s ❞❡Z ♥❡ss❡ ❝♦♥❥✉♥t♦ ❡ t❡♠♦s ❛ss✐♠ ✉♠ ❛❜s✉r❞♦✳ ◆♦ ❝❛s♦ ❡♠ q✉❡Z =∅♥ã♦ t❡♠♦s

❢✉♥çã♦ ❝♦♠ ❝♦♥tr❛❞♦♠í♥✐♦Z✳

(23)

❉❡✜♥✐çã♦ ✶✳✷✼ ❙❡❥❛C✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦Z✳ P❛r❛ q✉❛✐s✲ q✉❡rX, Y ∈Ob(C)❞❡✜♥✐♠♦s ♦ ♠♦r✜s♠♦ ♥✉❧♦0X

Y :X →Y ❝♦♠♦ s❡♥❞♦ ♦ ♠♦r✜s♠♦ q✉❡ ❝♦♠✉t❛ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛

X 0 X Y / / φX ! ! B B B B B B B B Y Z. ψY > > } } } } } } }

❯t✐❧✐③❛♠♦s s❡♠♣r❡ ❛ ♥♦t❛çã♦ ❛❝✐♠❛ ♣❛r❛ ♥♦s r❡❢❡r✐r♠♦s ❛♦s ú♥✐❝♦s ♠♦r✜s♠♦s ❡♠ r❡❧❛çã♦ ❛ ✉♠ ♦❜❥❡t♦ ③❡r♦ ❝♦♥s✐❞❡r❛❞♦ ♥❛ ❝❛t❡❣♦r✐❛✱ s❡♠ ♠❡♥❝✐♦♥❛r t❛❧ ♦❜❥❡t♦✱ ✜❝❛♥❞♦ ♣♦rt❛♥t♦ ✐♠♣❧í❝✐t♦✳

Pr♦♣♦s✐çã♦ ✶✳✷✽ ❖ ♠♦r✜s♠♦ ♥✉❧♦ ❞❡✜♥✐❞♦ ❛❝✐♠❛ ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ ♦❜❥❡t♦ ③❡r♦ ❞❛ ❝❛t❡❣♦r✐❛✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ❞♦✐s ♦❜❥❡t♦sX❡Y ❡♠C✳ ❙❡❥❛♠ZZ′ ❞♦✐s ♦❜❥❡t♦s ③❡r♦s✳ ❙❡❣✉❡ ❞❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛ q✉❡Z≃Z′ ❡ ❞❡♥♦t❛♠♦s

♣♦rφ:Z→Z′t❛❧ ✐s♦♠♦r✜s♠♦✳ ❙❡❥❛0X

Y :X→Y ♠♦r✜s♠♦ ♥✉❧♦ ♦❜t✐❞♦ ❡♠ r❡❧❛çã♦ ❛♦ ♦❜❥❡t♦ ③❡r♦Z✱ ♦✉ s❡❥❛✱0X

Y =ψY ◦φX✳

❱❡r✐✜q✉❡♠♦s q✉❡0X

Y =ψ

Y ◦φ

X ❡♠ q✉❡ ψ

Y :Z′ →Y ❡φ

X :X →

Z′✳ ❉❡ ❢❛t♦✱

ψY ◦φX = ψ′Y ◦φ◦φX = ψ′Y ◦φ◦φ−1φ

X = ψ′Y ◦φ′X.

Pr♦♣♦s✐çã♦ ✶✳✷✾ ❙❡❥❛♠C✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦✱X, Y, Z, W Ob(C)f HomC(Y, Z)✳ ❊♥tã♦ f0X

Y = 0XZ ❡ 0ZW ◦f = 0YW✳

❉❡♠♦♥str❛çã♦✿ P♦r ❞❡✜♥✐çã♦ t❡♠♦s q✉❡f◦0X

Y =f◦ψY ◦φX✳ ❆❧é♠ ❞✐ss♦✱ψZ =f◦ψY✳ P♦rt❛♥t♦✱

f◦0X

Y = f ◦ψY ◦φX

= ψZ◦φX = 0X

Z.

❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ♦❜t❡♠♦s ❛ ♦✉tr❛ ✐❣✉❛❧❞❛❞❡✳ Pr♦♣♦s✐çã♦ ✶✳✸✵ ❙❡ ❳ ♥ã♦ é ✉♠ ♦❜❥❡t♦ ③❡r♦ ❡♥tã♦0X

X 6=IX✳

(24)

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡0X

X=IX ❡ s❡❥❛ Y ✉♠ ♦❜❥❡t♦ q✉❛❧q✉❡r ❡♠ C✳ ▼♦str❡♠♦s q✉❡ HomC(X, Y) = {0X

Y} ❡ q✉❡ HomC(Y, X) ={0YX}✳ ❙❡❥❛f ∈HomC(X, Y)✳ ❊♥tã♦

f = f◦IX = f◦0X X = 0X

Y,

❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ✈❛♠♦s t❡rHomC(Y, X) ={0YX}✳ P♦rt❛♥t♦✱X é ✉♠ ♦❜❥❡t♦ ③❡r♦✱ ♦ q✉❡ ❝♦♥tr❛✲

❞✐③ ❛ ❤✐♣ót❡s❡✳ ▲♦❣♦✱0X

X6=IX✳

❉❡✜♥✐çã♦ ✶✳✸✶ ❙❡❥❛♠C✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦✱X, Y Ob(C) ❡ f ∈ HomC(X, Y)✳ ❯♠ ♥ú❝❧❡♦ ❞❡ f é ✉♠ ♣❛r (Ker(f), k) ❡♠ q✉❡

Ker(f) ∈ Ob(C)k : Ker(f) X é ✉♠ ♠♦r✜s♠♦ t❛❧ q✉❡ f k = 0KerY (f). ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ♣❛r(K′, k)❝♦♠k:KX

t❛❧ q✉❡ f ◦k′ = 0K′

Y , ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u :K

Ker(f) t❛❧

q✉❡k◦u=k′✱ ✐✳❡✳✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

K′ k′ $ $ H H H H H H H H H u

0K′ Y

$

$

X f //Y

Ker(f). 0

Ker(f)

Y : : k ; ; w w w w w w w w w

Pr♦♣♦s✐çã♦ ✶✳✸✷ ❙❡ ✉♠ ♠♦r✜s♠♦ ❛❞♠✐t❡ ♥ú❝❧❡♦✱ ❡st❡ é ú♥✐❝♦ ❛ ♠❡✲ ♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ (K, k) ❡ (K′, k) ❞♦✐s ♥ú❝❧❡♦s ❞❡ f✳ P❡❧❛ ❞❡✲

✜♥✐çã♦✱ ❡①✐st❡♠ ú♥✐❝♦s ♠♦r✜s♠♦s u:K′ K v :K Kt❛✐s q✉❡

k◦u=k′ kv=k✳ ❙❡❣✉❡ ❞❛ ✉♥✐❝✐❞❛❞❡ q✉❡uv=IK vu=IK ′✳

P♦rt❛♥t♦✱K≃K′

❉❡✜♥✐çã♦ ✶✳✸✸ ❙❡❥❛♠C✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦✱X, Y Ob(C) ❡f ∈HomC(X, Y)✳ ❯♠ ❝♦♥ú❝❧❡♦ ❞❡f é ✉♠ ♣❛r(CoKer(f), q)❡♠ q✉❡

CoKer(f)∈Ob(C)q:Y CoKer(f)é ✉♠ ♠♦r✜s♠♦ t❛❧ q✉❡qf = 0X

CoKer(f)✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ♣❛r(Q, q

)❝♦♠q:Y Q

t❛❧ q✉❡ q′f = 0X

(25)

q✉❡q′=uq✱ ✐✳❡✳✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ Q X 0X Q e e JJJJ JJJJ JJJ 0X CoKer(f)

z z tttt tttt tt f / /Y q′ l l q s s CoKer(f). u K K

❈♦♠♦ ❛❝✐♠❛✱ s❡ ✉♠ ♠♦r✜s♠♦ ♣♦ss✉✐ ❝♦♥ú❝❧❡♦✱ ❡st❡ é ú♥✐❝♦ s❛❧✈♦ ✐s♦♠♦r✜s♠♦✳

❊①❡♠♣❧♦ ✶✳✸✹ ❙❡❥❛C✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ ❡ s❡❥❛♠X, Y Ob(C)✳ ❊♥tã♦✱ ✉♠ ♥ú❝❧❡♦ ♣❛r❛ ♦ ♠♦r✜s♠♦ 0X

Y é ♦ ♣❛r (X, IX)❡ ✉♠ ❝♦♥ú❝❧❡♦ ♣❛r❛0X

Y é ♦ ♣❛r(Y, IY)✳ ❉❡ ❢❛t♦✱ é ❝❧❛r♦ q✉❡0X

Y ◦IX= 0XY ❡ ❞❛❞♦ q✉❛❧q✉❡r ♦✉tr♦ ♣❛r(K, φ) ❡♠ q✉❡ φ : K → X ❡ 0X

Y ◦φ = 0KY ❡♥tã♦ ❝❧❛r❛♠❡♥t❡ φ ❝♦♠✉t❛ ♦ ❞✐❛❣r❛♠❛

K

φ

φAAAA

A A

A 0K

Y # # X 0 X Y / /Y X 0 X Y ; ; IX > > } } } } } } }

♣♦✐sIX◦φ=φ❡φé ♦ ú♥✐❝♦ q✉❡ ♦ ❢❛③✳ ◆♦t❡♠♦s q✉❡ s❡ψ:K→X é t❛❧ q✉❡IX◦ψ=φ❡♥tã♦✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ψ=φ✳

❊①❡♠♣❧♦ ✶✳✸✺ ❊♠ ❝❛t❡❣♦r✐❛s ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦RM❛ ♥♦çã♦ ❞❡ ♥ú✲ ❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦ ❛❧❣é❜r✐❝♦ s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❞❛s ❞❡✜♥✐çõ❡s1.31❡ 1.33 r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦✉ s❡❥❛✱ ❞❛❞♦ ✉♠ ♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s f : X → Y ♦ ♣❛r (Ker(f), i) ❡♠ q✉❡ Ker(f) = {m ∈ X : f(m) = 0}

❡ i : Ker(f) → X é ❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦ é ✉♠ ♥ú❝❧❡♦ ❞❡ f ❡ ♦ ♣❛r (Coker(f), π) ❡♠ q✉❡ Coker(f) =Y /Im(f) ❡π : Y → Y /Im(f) é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛ é ✉♠ ❝♦♥ú❝❧❡♦ ❞❡f✳

✶✳✸ ▼♦♥♦♠♦r✜s♠♦s✱ ❡♣✐♠♦r✜s♠♦s ❡ ✐s♦♠♦r✲

✜s♠♦s

P❛r❛ ❡ss❛ s❡çã♦✱ ❝❛s♦ ♦ ❧❡✐t♦r q✉❡✐r❛ ❛♣r♦❢✉♥❞❛r✱ ❝✐t❛♠♦s ❝♦♠♦ r❡✲ ❢❡rê♥❝✐❛[✾]✳

(26)

❉❡✜♥✐çã♦ ✶✳✸✻ ❙❡❥❛ ❛ ❝❛t❡❣♦r✐❛ Set✳ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦f ❡♥tr❡ ❝♦♥✲ ❥✉♥t♦sX ❡Y✳ ❉✐③❡♠♦s q✉❡

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

❊st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❞❛r ♥♦çõ❡s ♠❛✐s ❣❡r❛✐s✱ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝❛t❡❣♦r✐❛s✱ q✉❡ s❡ ❛♣r♦①✐♠❡♠ ❞❛ ✐❞❡✐❛ ❞❡ ✐♥❥❡t✐✈✐❞❛❞❡ ❡ s♦❜r❡❥❡t✐✈✐❞❛❞❡✳

❙❡❥❛♠C✉♠❛ ❝❛t❡❣♦r✐❛ ❡f HomC(X, Y)✳

❉❡✜♥✐çã♦ ✶✳✸✼ f é ❞✐t♦ ✉♠ ✐s♦♠♦r✜s♠♦ s❡ ❡①✐st❡ g : Y → X ♠♦r✲ ✜s♠♦ t❛❧ q✉❡ f ◦g=IY ❡ g◦f =IX✳ ❖s ♦❜❥❡t♦sX, Y sã♦ ❞✐t♦s ✐s♦✲ ♠♦r❢♦s ❡ ❞❡♥♦t❛♠♦s ♣♦rX≃Y✱ s❡ ❡①✐st✐r ✉♠ ✐s♦♠♦r✜s♠♦f :X →Y✳ ❉❡✜♥✐çã♦ ✶✳✸✽ f é ❞✐t♦ ✉♠ ♠♦♥♦♠♦r✜s♠♦ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♠♦r✲ ✜s♠♦s g, h:Z →X t❛✐s q✉❡ f◦g=f ◦h❡♥tã♦ g=h✳

❉❡✜♥✐çã♦ ✶✳✸✾ f é ❞✐t♦ ✉♠ ❡♣✐♠♦r✜s♠♦ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♠♦r✜s✲ ♠♦s g, h:Y →Z t❛✐s q✉❡ g◦f =h◦f ❡♥tã♦ g=h✳

Pr♦♣♦s✐çã♦ ✶✳✹✵ ❙❡❥❛ (Ker(f), k)✉♠ ♥ú❝❧❡♦ ♣❛r❛ f✳ ❊♥tã♦ k é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠g, h:U →Ker(f)♠♦r✜s♠♦s t❛✐s q✉❡k◦h= k◦g✳ ❖❜s❡r✈❡♠♦s q✉❡ f◦(k◦g) = 0YKer(f)◦g= 0U

Y✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u: U → Ker(f)t❛❧ q✉❡ k◦u= k◦g =k◦h ❡ ❛ss✐♠✱u=g=h✳

❊①❡♠♣❧♦ ✶✳✹✶ ❈♦♥s✐❞❡r❡ ❛ ❝❛t❡❣♦r✐❛ Ring✳ ❊♥tã♦ ♦ ♠♦r✜s♠♦ ✐♥❝❧✉✲ sã♦i:Z→Qé ✉♠ ❡♣✐♠♦r✜s♠♦ q✉❡ ♥ã♦ é s♦❜r❡❥❡t♦r✳

❉❡ ❢❛t♦✱ s❡❥❛♠ R ✉♠ ❛♥❡❧ ❡ g, h: Q→ R ♠♦r✜s♠♦s ❞❡ ❛♥é✐s t❛✐s q✉❡g◦i=h◦i✳ ❙❡❥❛z∈Z♥ã♦✲♥✉❧♦✳ ❊♥tã♦

h(1) = g(1) = g(z

z)

= g(z)g(1

z)

= h(z)g(1

z).

▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♣♦rh(1

z)♦❜t❡♠♦s h(1z) = h(z1)h(1)

= h(z1)h(z)g(1z) = h(1)g(1z) = g(1)g(1z) = g(1

z).

(27)

❆ss✐♠✱ ❞❛❞♦ q∈ Q ♣♦❞❡♠♦s ❡s❝r❡✈❡r q= a

b ❝♦♠a, b ∈ Z✱ b 6= 0✳

❉❛íg(q) =g(a

b) =g(a)g(

1

b) =h(a)h(

1

b) =h(q)∀q∈Q✳ ▲♦❣♦✱g=h✳ ❊♠ ✭[✾]✱ ❈❤❛♣t❡r ❳✱ ♣✳✹✽✶✮✱ ❤á ✉♠ ❡①❡♠♣❧♦ ♦♥❞❡ é ❝♦♥s✐❞❡r❛❞❛ ❛ ❝❛t❡❣♦r✐❛ ❞❡ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ❞✐✈✐sí✈❡✐s✱ ♥♦ q✉❛❧ é ❡①✐❜✐❞♦ ✉♠ ♠♦♥♦✲ ♠♦r✜s♠♦ q✉❡ ♥ã♦ é ✐♥❥❡t♦r✳

❆q✉✐✱ s✉r❣❡♠ q✉❡stõ❡s ♥❛t✉r❛✐s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❤á ❛❧❣✉♠❛ ❝❛✲ t❡❣♦r✐❛ ❡♠ q✉❡ ❛s ♥♦çõ❡s ❞❡ ♠♦♥♦♠♦r✜s♠♦ ❡ ✐♥❥❡t✐✈✐❞❛❞❡ ❝♦✐♥❝✐❞❡♠❄ ❆ ♠❡s♠❛ ♣❡r❣✉♥t❛ ♣❛r❛ ❛s ♥♦çõ❡s ❞❡ ❡♣✐♠♦r✜s♠♦ ❡ s♦❜r❡❥❡t✐✈✐❞❛❞❡✳ ❆ s❡❣✉✐r✱ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❝❛t❡❣♦r✐❛ ♦♥❞❡ ❛s ♥♦çõ❡s ❝♦✐♥❝✐❞❡♠✳

Pr♦♣♦s✐çã♦ ✶✳✹✷ ❙❡❥❛ ❘ ✉♠ ❛♥❡❧✳ ❯♠ ♠♦r✜s♠♦ f ❡♠ RMé ♠♦♥♦✲ ♠♦r✜s♠♦ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ❡♣✐♠♦r✜s♠♦✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f é ✐♥❥❡t♦r ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ s♦❜r❡❥❡t♦r✮✳

❉❡♠♦♥str❛çã♦✿ ✭⇐✮ ❈♦♥s✐❞❡r❡♠♦s f : X → Y ✉♠ ♠♦r✜s♠♦ ♥❛ ❝❛t❡❣♦r✐❛RM✳ ❙✉♣♦♥❤❛♠♦s f ✐♥❥❡t♦r ❡ s❡❥❛♠g, h:Z →X ♠♦r✜s♠♦s ❞❡ R✲♠ó❞✉❧♦s t❛✐s q✉❡ f ◦g = f ◦h✳ ❚❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ k:Y →X t❛❧ q✉❡ k◦f =IX✳ ❆ss✐♠✱ k◦(f◦g) = k◦(f◦h)♦ q✉❡ ✐♠♣❧✐❝❛g=h✳

❙✉♣♦♥❤❛♠♦s f s♦❜r❡❥❡t♦r ❡ s❡❥❛♠ g, h : Y → Z ♠♦r✜s♠♦s ❞❡ R✲ ♠ó❞✉❧♦s t❛✐s q✉❡g◦f =h◦f✳ ❚❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦k:Y →X t❛❧ q✉❡f◦k=IY ❡ ♣♦rt❛♥t♦✱g=h✳

✭⇒✮ ❙✉♣♦♥❤❛♠♦s q✉❡ f ♥ã♦ s❡❥❛ ✐♥❥❡t♦r✱ ✐✳❡✳ Ker(f)6={0}✳ ❈♦♥✲

s✐❞❡r❡♠♦s ❛ ✐♥❝❧✉sã♦i:Ker(f)→X✳ ❊♥tã♦f ◦i:Ker(f)→Y é ✉♠ ♠♦r✜s♠♦✳ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ♠♦r✜s♠♦ h: Ker(f)→X ❞❡✜♥✐❞♦ ♣♦rh(x) = 0♣❛r❛ t♦❞♦ x∈Ker(f)✳ ❖❜s❡r✈❛♠♦s q✉❡ f◦i=f◦h❡✱ ♥♦ ❡♥t❛♥t♦✱i6=h✱ ♦✉ s❡❥❛✱f ♥ã♦ é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳

❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ f ♥ã♦ s❡❥❛ s♦❜r❡❥❡t♦r✳ ❈♦♠♦ Im(f) é ✉♠ R✲s✉❜♠ó❞✉❧♦ ❞❡Y✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ♠ó❞✉❧♦ q✉♦❝✐❡♥t❡Y /Im(f) q✉❡✱ ♥❡ss❡ ❝❛s♦✱ é ❞✐❢❡r❡♥t❡ ❞♦ ♠ó❞✉❧♦ ♥✉❧♦✳

❯s❛♥❞♦ ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛π:Y →Y /Im(f)❞❡✜♥✐❞❛ ♣♦r π(y) = y+Im(f)❡ ♦ ♠♦r✜s♠♦h:Y →Y /Im(f)❞❡✜♥✐❞♦ ♣♦rh(y) = 0+Im(f) t❡♠♦s π◦f = h◦f✳ ◆♦ ❡♥t❛♥t♦✱ π 6= h ❡ ♣♦rt❛♥t♦✱ f ♥ã♦ é ✉♠ ❡♣✐♠♦r✜s♠♦✳

✶✳✹ Pr♦❞✉t♦s ❡ ❝♦♣r♦❞✉t♦s

❆ ♥♦çã♦ ❞❡ ♣r♦❞✉t♦ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ♣r♦✲ ❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❡ ❝♦♥❥✉♥t♦s✳ ❊ss❡ é ♣♦ss✐✈❡❧♠❡♥t❡ ✉♠ ❞♦s ♣r✐♠❡✐r♦s

(28)

❡①❡♠♣❧♦s ❞♦ ✉s♦ ❞❡ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s ♣❛r❛ ❞❡✜♥✐r ✉♠❛ ♥♦çã♦ ♠❛t❡✲ ♠át✐❝❛✱ s❡♥❞♦ q✉❡ ♦ t❡r♠♦ ✏❞❡✜♥✐r✑ ❛q✉✐ s✐❣♥✐✜❝❛ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣♦r ♠❡✐♦ ❞❡ ♠♦r✜s♠♦s ❡♥tr❡ ♦❜❥❡t♦s✳

❉❡✜♥✐çã♦ ✶✳✹✸ ❙❡❥❛♠ C✉♠❛ ❝❛t❡❣♦r✐❛ ❡ X, Y Ob(C)✳ ❯♠ ♣r♦❞✉t♦ ❞❡X ❡Y é ✉♠❛ tr✐♣❧❛(P, πX, πY)t❛❧ q✉❡ P ∈Ob(C)✱ πX :P X ❡ πY : P →Y sã♦ ♠♦r✜s♠♦s ♥❡ss❛ ❝❛t❡❣♦r✐❛✳ ❆❧é♠ ❞✐ss♦✱ s❡ ❡①✐st✐r❡♠ ♦✉tr♦ ♦❜❥❡t♦ Q ❡ ♠♦r✜s♠♦s qX : Q →X ❡ qY : Q→ Y ❡♠ C ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦φ:Q→P t❛❧ q✉❡πX◦φ=qX❡πY◦φ=qY✳

❊ss❛ ❞❡✜♥✐çã♦ ♣♦❞❡ s❡r ✈✐s✉❛❧✐③❛❞❛ ❛tr❛✈és ❞♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦

X

Q

qX

3

3

qY

+

+

φ

/

/P

πX

?

?

~ ~ ~ ~ ~ ~ ~ ~

πY

@ @ @ @ @ @ @ @

Y.

Pr♦♣♦s✐çã♦ ✶✳✹✹ ❖ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ♦❜❥❡t♦s X ❡Y ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ C✱ s❡ ❡①✐st❡✱ é ú♥✐❝♦ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ (P, πX, πY)❡(Q, qX, qY)❞♦✐s ♣r♦❞✉t♦s ❞❡X ❡ Y ❡♠ C✳ ❈♦♠♦ (P, πX, πY) é ♣r♦❞✉t♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ φ:Q→P t❛❧ q✉❡ πX◦φ=qX ❡πY ◦φ=qY✳

❚❛♠❜é♠✱ ❞♦ ❢❛t♦ ❞❡(Q, qX, qY)s❡r ♣r♦❞✉t♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✲ ✜s♠♦ ψ : P →Q t❛❧ q✉❡ qX◦ψ = πX ❡qY ◦ψ =πY. ◆♦t❡♠♦s q✉❡ ψ◦φ:Q→Q❡φ◦ψ:P →P sã♦ ♠♦r✜s♠♦s t❛✐s q✉❡qX◦(ψ◦φ) = πX◦φ=qX✱qY ◦(ψ◦φ) =πY ◦φ=qY✱πX◦(φ◦ψ) =qX◦ψ=πX ❡πY ◦(φ◦ψ) =qY ◦ψ=πY✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦s ♠♦r✜s♠♦s s❡❣✉❡ q✉❡ ψ◦φ=IQ ❡φ◦ψ=IP✳

❊①❡♠♣❧♦ ✶✳✹✺ ❖ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡B ♥❛ ❝❛t❡❣♦r✐❛ Set é ❛ tr✐♣❧❛ (A×B, πA, πB)✱ ♦✉ s❡❥❛✱ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❝♦♠ ❛s r❡s✲ ♣❡❝t✐✈❛s ♣r♦❥❡çõ❡s ❝❛♥ô♥✐❝❛s✳ ❉❡ ❢❛t♦✱ s❡❥❛ (D, PA, PB) ♦✉tr❛ tr✐♣❧❛✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛

(29)

D

φ

PB

PA

A×B

πB

#

#

F F F F F F F F F

πA

|

|

xxxx xxxx

x

A B

❡♠ q✉❡ ❞❡✜♥✐♠♦sφ:D→A×B ♣♦rφ(d) = (PA(d), PB(d))♣❛r❛ t♦❞♦ d∈D✳ ◆❡ss❡ ❝❛s♦✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ é ❝❧❛r❛ ❡ ❛ ✉♥✐❝✐❞❛❞❡

❞❡φt❛♠❜é♠✳

❊①❡♠♣❧♦ ✶✳✹✻ ❉❛❞♦s ❞♦✐s ❣r✉♣♦sG❡H ♥❛ ❝❛t❡❣♦r✐❛Grp✱ ♦ ♣r♦❞✉t♦ ❞❡ss❡s ♦❜❥❡t♦s é ❛ tr✐♣❧❛(G×H, πG, πH)t❛❧ q✉❡(G×H,·)é ✉♠ ❣r✉♣♦ ❡♠ q✉❡G×H é ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞♦s ❝♦♥❥✉♥t♦s ❡·é ✉♠❛ ♦♣❡r❛çã♦ ❞❡✜♥✐❞❛ ♣♦r ·((g, h),(r, s)) = (gr, hs)✳ ❆s ♣r♦❥❡çõ❡s sã♦ ♠♦r✜s♠♦s ❞❡ ❣r✉♣♦ ❡ ❛ ✈❡r✐✜❝❛çã♦ é ❛♥á❧♦❣❛ ❛♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳

❉❡✜♥✐çã♦ ✶✳✹✼ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ X, Y Ob(C)✳ ❯♠ ❝♦♣r♦✲ ❞✉t♦ ❞❡X ❡Y é ✉♠❛ tr✐♣❧❛(Q, iX, iY)t❛❧ q✉❡Q∈Ob(C)✱iX:X Q ❡iY :Y →Qsã♦ ♠♦r✜s♠♦s ♥❡ss❛ ❝❛t❡❣♦r✐❛✳ ❆❧é♠ ❞✐ss♦✱ s❡ ❡①✐st✐r❡♠ ♦✉tr♦ ♦❜❥❡t♦ Q′ ❡ ♠♦r✜s♠♦s jX :X Q jY :Y Q❡♠ C❡♥tã♦

❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ψ:Q→Q′ t❛❧ q✉❡ψiX =jX ψiY =jY

P♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ❡ss❛ ❞❡✜♥✐çã♦ ✈✐❛ ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦

X

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w

w

jX

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Q′oo ψ Q

Y.

iY

g

g

jY

c

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❖❜s❡r✈❛çã♦ ✶✳✹✽ ❖ ❝♦♣r♦❞✉t♦ é ú♥✐❝♦ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ ❢❛t♦ ♣♦❞❡ s❡r ❢❡✐t❛ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ ❝❛s♦ ❞♦ ♣r♦❞✉t♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦❜s❡r✈❛♠♦s q✉❡ ♦ ❝♦♣r♦❞✉t♦ ❞❡ X ❡ Y ♥❛ ❝❛t❡❣♦r✐❛Cé ♦ ♣r♦❞✉t♦ ❞❡XY ♥❛ ❝❛t❡❣♦r✐❛Cop✳

❉❡ ❢❛t♦✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ✐♠♣❧✐❝❛ ♥❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♠♦r✲ ✜s♠♦ψ:Q→Q′✱ ♦✉ s❡❥❛✱ ψHom

Cop(Q′, Q)t❛❧ q✉❡ψ◦iX =jX ❡

ψ◦iY =jY✱ ✐✳ ❡✳✱ iX◦opψ=jX iY opψ =jY✳ P♦❞❡♠♦s ♣♦rt❛♥t♦

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

❊①❡♠♣❧♦ ✶✳✹✾ ❖ ❝♦♣r♦❞✉t♦ ♥❛ ❝❛t❡❣♦r✐❛ Set ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B é ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❞❡st❡s ❝♦♠ ❛s r❡s♣❡❝t✐✈❛s ✐♥❝❧✉sõ❡s✳

❉❡ ❢❛t♦✱ ❞❡♥♦t❛♠♦s ♣♦rA⊔B❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❞❡A❡B✳ ❱❡r✐✜q✉❡✲ ♠♦s q✉❡ ❛ tr✐♣❧❛(A⊔B, iA, iB)é ❞❡ ❢❛t♦ ♦ ❝♦♣r♦❞✉t♦✳ ❙❡❥❛(Q, jA, jB) ♦✉tr❛ tr✐♣❧❛✳ ❉❡✜♥✐♠♦s ψ: A⊔B →Q ♣♦rψ(a) =jA(a)s❡ a∈A ❡ ψ(b) =jB(b) s❡b ∈B✳ ➱ ✐♠❡❞✐❛t♦ q✉❡ ψ◦iA =jA ❡ψ◦iB = jB ❡ ❝❧❛r❛♠❡♥t❡ψ é ú♥✐❝❛✳

❖❜s❡r✈❛çã♦ ✶✳✺✵ ❊ss❛s ♥♦çõ❡s ❞❡ ♣r♦❞✉t♦ ❡ ❝♦♣r♦❞✉t♦ ♣♦❞❡♠ s❡r ❣❡♥❡r❛❧✐③❛❞❛s✳ ❙❡❥❛♠C✉♠❛ ❝❛t❡❣♦r✐❛ ❡{Xi}iI ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♦❜❥❡t♦s ❡♠ C✳ ❉✐③❡♠♦s q✉❡ (X,{πi}iI) é ♦ ♣r♦❞✉t♦ ❞❛ ❢❛♠í❧✐❛ {Xi}iI s❡ X ∈ Ob(C){πi : X Xi}iI é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♠♦r✜s♠♦s ♥❡ss❛ ❝❛t❡❣♦r✐❛ t❛❧ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ♦❜❥❡t♦Y ❡ ❢❛♠í❧✐❛ ❞❡ ♠♦r✜s♠♦s

{qi}i∈I ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ f :Y →X t❛❧ q✉❡πi◦f =qi ♣❛r❛ t♦❞♦ i∈I✱ ♦✉ s❡❥❛✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

X

πi

Y

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qi //Xi

♣❛r❛ t♦❞♦i∈I✳ ❉❡ ♠❛♥❡✐r❛ ❞✉❛❧✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦♣r♦❞✉t♦ ❣❡♥❡✲

r❛❧✐③❛❞♦✳

❊①❡♠♣❧♦ ✶✳✺✶ ❈♦♥s✐❞❡r❡♠♦s ❛ ❝❛t❡❣♦r✐❛ RM✳ ❙❡❥❛{Mi}i∈I ✉♠❛ ❢❛✲ ♠í❧✐❛ ❞❡ R✲♠ó❞✉❧♦s✱ ❡♥tã♦ Q

i∈I

Mi ♦ ♣r♦❞✉t♦ ❞✐r❡t♦ ❞❡ ♠ó❞✉❧♦s ❥✉♥t❛✲ ♠❡♥t❡ ❝♦♠ ❛s ♣r♦❥❡çõ❡s{πi}i∈I é ♦ ♣r♦❞✉t♦ ❞❡ss❛ ❝❛t❡❣♦r✐❛✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ L

i∈I

Mi ❛ s♦♠❛ ❞✐r❡t❛ ✐♥t❡r♥❛ ❞❡ ♠ó❞✉❧♦s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s ✐♥❝❧✉sõ❡s{ij}j∈I é ♦ ❝♦♣r♦❞✉t♦ ♥❡ss❛ ❝❛t❡❣♦r✐❛✳

Figure

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