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

Livre

0
0
128
1 year ago
Preview
Full text

  

❯♥✐✈❡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

  ❋❧♦r✐❛♥ó♣♦❧✐s ▼❛rç♦ ❞❡ ✷✵✶✺

  

❯♥✐✈❡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❛✳ ▲✉✐s ❆✉❣✉st♦ ❯❧✐❛♥❛

  ❋❧♦r✐❛♥ó♣♦❧✐s ▼❛rç♦ ❞❡ ✷✵✶✺

  

❊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❊❙

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

  ▼❝ ✽✱✸✹

  ❆❣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♦✳

  ❘❡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♥ ▼♦♠❜❡❧❧✐✳

  ❆❜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♥ ▼♦♠❜❡❧❧✐✳

  ❙✉♠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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷

  ■♥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❡s Hom ❡ ⊗✱ 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❞ ❡♠ ✭[✼]✱ ♣✳✶✹✮

  ❚❤❡ ❨♦♥❡❞❛ ▲❡♠♠❛ 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✐♥❣s R✳ ■ ❝♦♥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✲❧✐♥❡❛r C✳ ❆ ❡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❛❞❛ C G ✱ ❝❤❛♠❛❞❛ ❡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 ( A m

  ) GA⊗ k kG m

  ✱ q✉❡ ✐❧✉str❛ ✉♠ ❢❛t♦ ♦❜s❡r✈❛❞♦ ❡♠ [✻]✱ ❞❡ q✉❡ ♣❛r❛ t♦❞❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r ✜♥✐t❛ D✱ ❡①✐st❡ ✉♠❛ á❧❣❡❜r❛ A t❛❧ q✉❡ D é

  ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ❞❡ A✲♠ó❞✉❧♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ ✉♠ ❝♦r♣♦ k✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧✱ ♣r♦✈❛♠♦s q✉❡✱ ❝❛s♦ C s❡❥❛ k✲❧✐♥❡❛r✱ G ❡♥tã♦ C é k✲❧✐♥❡❛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 k → Alg k k →

  ♣❛r❛ q✉❡ ♣✉❞éss❡♠♦s ❢❛❧❛r ❞♦s ❢✉♥t♦r❡s U : Lie ❡ L : Alg Lie k

  ✳ ❯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♥❞✐❝❡ ❈✳

  ❈❛♣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)❀ C

  (U, V ) ✭✐✐✮ ♣❛r❛ t♦❞♦ ♣❛r (U, V ) ❞❡ ♦❜❥❡t♦s ❡♠ C ❤á ✉♠❛ ❝♦❧❡çã♦ Hom ❞❡ ♠♦r✜s♠♦s ❞❡ U ♣❛r❛ V ❀ W ✭✐✐✐✮ ♣❛r❛ q✉❛❧q✉❡r ♦❜❥❡t♦ W ❡♠ Ob(C) ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ I ❡♠ C Hom (W, W )

  ❝❤❛♠❛❞♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡❀ ✭✐✈✮ ♣❛r❛ q✉❛✐sq✉❡r U, V ❡ W ♦❜❥❡t♦s ❡♠ Ob(C) ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ C C → Hom C

  Hom (U, V ) × Hom (V, W ) (U, W ) (f, g) 7→ g ◦ f

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

  U

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

  ❡♠ Hom é t❛❧ q✉❡ f ◦ I ❡ I ✱ ♣❛r❛ q✉❛✐sq✉❡r f ❡♠ C C Hom (U, V ) (V, U )

  ❡ g ❡♠ Hom ❀ C (U, V )

  ✭❜✮ ❞❛❞♦s ♦❜❥❡t♦s U, V, W ❡ Z ❡♠ Ob(C) ❡ ♠♦r✜s♠♦s f ❡♠ Hom ✱ C C g (V, W ) (W, Z) ❡♠ Hom ❡ h ❡♠ Hom ❛ ❝♦♠♣♦s✐çã♦ é ❛ss♦❝✐❛t✐✈❛✱

  ✐✳❡✳✱ h ◦ (g ◦ f) = (h ◦ g) ◦ f✳ C (U, V ) f ❉❡♥♦t❛♠♦s ✉♠ ♠♦r✜s♠♦ f ❡♠ Hom ♣♦r f : U → V ♦✉

  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✐❛ Set é ❛ ❝❛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✳ X

  ❉❡ ❢❛t♦✱ s❡❥❛ X ❡♠ Ob(Set)✳ ❈♦♥s✐❞❡r❛♠♦s ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ I Set (X, X) ❡♠ Hom ❝♦♠♦ s❡♥❞♦ ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡✳ ❆ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s é ❛ss♦❝✐❛t✐✈❛ ❡ ♣♦rt❛♥t♦✱ Set é ✉♠❛ ❝❛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✐❛ Ab é ❛ ❝❛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✐❛ Ring é ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❛♥é✐s ❡ ♦s ♠♦r✜s♠♦s ❡♥tr❡ ♦❜❥❡t♦s sã♦ ♦s ♠♦r✜s♠♦s ❞❡ ❛♥é✐s✳ k ❊①❡♠♣❧♦ ✶✳✻ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r V ect ❛ ❝❛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✳ k

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

  ✲á❧❣❡❜r❛s✳

  R R M ❊①❡♠♣❧♦ ✶✳✽ ❙❡❥❛ R ✉♠ ❛♥❡❧✳ ❉❡♥♦t❛♠♦s ♣♦r ✭M ✮ ❛ ❝❛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❛✮✳ A A M

  ❊①❡♠♣❧♦ ✶✳✾ ❙❡❥❛ A ✉♠❛ k✲á❧❣❡❜r❛✳ ❉❡♥♦t❛♠♦s ♣♦r ✭M ✮ ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s k✲❡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 op é ❛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✳ k ❊①❡♠♣❧♦ ✶✳✶✷ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r Lie ❛ ❝❛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

  2

  = 0 é ✉♠ ❡♥❞♦♠♦r✜s♠♦ ❞❡ ❣r❛✉ ✶ t❛❧ q✉❡ d ✳ ❆ ❝♦♠♣♦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✱ ♣♦✲ N (R) ❞❡♠♦s ❝♦♥s✐❞❡r❛r Ch ♣❛r❛ ✉♠ N ∈ N ❝♦♠ N > 2 ❛ ❝❛t❡❣♦r✐❛ ❞♦s N✲❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛✱ ♦♥❞❡ ♦s ♦❜❥❡t♦s sã♦ ♣❛r❡s (C, d) ❝♦♠♦ ♥❛ N

  = 0 ❝❛t❡❣♦r✐❛ ❛♥t❡r✐♦r ❝♦♠ ❛ r❡ss❛❧✈❛ ❞❡ q✉❡ d ✳

  ❯♠ ❞♦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 ∈ N ✜①♦✳ ❉❡✜♥✐♠♦s ♦ ❣r✉♣♦ ❞❡ n✲tr❛♥ç❛s✱ ❞❡♥♦t❛❞♦ ♣♦r B ✱ ✈✐❛ r❡❧❛çõ❡s s♦❜r❡ ❣❡r❛❞♦r❡s ❝♦♠♦ ❛❜❛✐①♦✳ n n−

  ❉❡✜♥✐çã♦ ✶✳✶✺ ❖ ❣r✉♣♦ B é ♦ ❣r✉♣♦ ❣❡r❛❞♦ ♣♦r σ

  1 ✱ · · · ✱ σ 1 s✉✲

  ❥❡✐t♦s às r❡❧❛çõ❡s✿ ✭✐✮ ❙❡♠♣r❡ q✉❡ 3 ≤ n ❡ 1 ≤ i, j ≤ n − 1 ❝♦♠ | i − j |> 1 t❡♠♦s σ i σ j = σ j σ i .

  ✭✐✐✮ ❚❛♠❜é♠ é ✈á❧✐❞♦ q✉❡ σ i σ i σ i = σ i σ i σ i

  • 1 +1 +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✉♣♦ ❞❡ n ♣❡r♠✉t❛çõ❡s ❞❡ n ❡❧❡♠❡♥t♦s S ✈✐❛ r❡❧❛çõ❡s s♦❜r❡ ❣❡r❛❞♦r❡s ❡♠ q✉❡ σ i = (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♦

  2

  σ = 1 i ♣❛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♦✲

  2

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

  2

  s❡♥t❛r σ i ♣❡❧♦ ❞✐❛❣r❛♠❛ ❖ q✉❡ ♦❝♦rr❡ ♥❡ss❡ ♣r♦❞✉t♦ é q✉❡ ❛♦ ✐♥✈és ❞❡ ✈♦❧t❛r ♣❛r❛ ❛ ❝♦♥✲

  2

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

  3

  ♣♦r ❡①❡♠♣❧♦ σ ✱ ♦❜t❡♠♦s ♠❛✐s ✉♠❛ ✈♦❧t❛✱ ✉♠ ✏♥ó✑ ❡♥tr❡ ❛s ♣♦❧✐❣♦♥❛✐s ❡ i ❛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♦✳

  ❊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 ∈ N t❡♠♦s q✉❡

  ∅, s❡ m 6= n Hom (m, n) =

  B

  B n , s❡ m = n. P❛r❛ ❡ss❡ ❝♦♥❥✉♥t♦ ❞❡ ♠♦r✜s♠♦s ❞❡✜♥✐♠♦s ❛ ❝♦♠♣♦s✐çã♦ ❝♦♠♦ ♦ n

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

  ❡✱ ♣❛r❛ q✉❛✐sq✉❡r U ❡ V ❡♠ 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✉❡ D é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❡ C s❡ D é D C

  (U, V ) = Hom (U, V ) ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ t❛❧ q✉❡ Hom ♣❛r❛ q✉❛✐sq✉❡r U

  ❡ V ❡♠ Ob(D)✳ ❊①❡♠♣❧♦ ✶✳✶✾ ❆ ❝❛t❡❣♦r✐❛ Ab é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❛ ❝❛t❡❣♦r✐❛ Grp k k

  ❡ vect é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❛ ❝❛t❡❣♦r✐❛ V ect ✳ ❊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

  R → M f : 2×2 (R) x 0

  7→ x ❚❛❧ ♠♦r✜s♠♦ ♣❡rt❡♥❝❡ ❛ ❝❛t❡❣♦r✐❛ Ring ♠❛s ♥ã♦ ♣❡rt❡♥❝❡ à ❝❛t❡✲

  ❣♦r✐❛ ring ♣♦✐s ♥ã♦ ♣r❡s❡r✈❛ ✉♥✐❞❛❞❡✳

  ❉❡✜♥✐çã♦ ✶✳✷✶ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❝❛t❡❣♦r✐❛ C é ♣❡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✐❛ C é ❧♦❝❛❧♠❡♥t❡ ♣❡q✉❡♥❛ s❡ C

  (X, Y ) ❞❛❞♦s X ❡ Y ❡♠ C t✐✈❡r♠♦s q✉❡ Hom 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❡✈❡✲ C

  (U, V ) ♠♦s f ∈ Hom ♣❛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❡✳ op

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

  ✳ C op C op (X, Y ) (Y, Z)

  ❆ss✐♠✱ ❞❛❞♦s f ∈ Hom ❡ g ∈ Hom ✱ ❞❡✜♥✐♠♦s ❛ ❝♦♠♣♦s✐çã♦ op g ◦ f = f ◦ g C op

  (Z, W ) q✉❡ é ❛ss♦❝✐❛t✐✈❛✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ f ❡ g ❝♦♠♦ ❛❝✐♠❛ ❡ h ∈ Hom ✳ ❊♥tã♦ op op op h ◦ (g ◦ f ) = (g ◦ f ) ◦ h

  = (f ◦ g) ◦ h = f ◦ (g ◦ h) op = (g ◦ h) ◦ f op op = (h ◦

  g) ◦ f. ❖ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞❛ ❝❛t❡❣♦r✐❛ C é ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ♥❛ op op

  ❝❛t❡❣♦r✐❛ C ✳ ❆ ❝❛t❡❣♦r✐❛ C 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 C C D Hom ×D ((X, Y ), (U, V )) = (Hom (X, U ), Hom (Y, V )).

  ❊♥tã♦ é ♣♦ssí✈❡❧ ❞❡✜♥✐r ❛ s❡❣✉✐♥t❡ ♦♣❡r❛çã♦✿ C D C D (Hom (X, U ), Hom (Y, V )) × (Hom (U, Z), Hom (V, W ))

  ((f, g), (r, s))

  ♣♦r (f, g) ◦ (r, s) = (r ◦ f, s ◦ g) ∈ (Hom ❚❛❧ ♦♣❡r❛çã♦ é ❛ss♦❝✐❛t✐✈❛✳ ❉❡ ❢❛t♦✱

  C D (X, Z), Hom (Y, W )).

  ((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)). X Y

  , I ) ❉❛❞♦ ✉♠ ♦❜❥❡t♦ (X, Y ) ♦ ♠♦r✜s♠♦ (I 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 X : X → Z X : Z → X (X, Z) = {φ C X } ♠♦r✜s♠♦s φ ❡ ψ ✱ ♦✉ s❡❥❛✱ Hom C

  (Z, X) = {ψ X } ❡ Hom ✳

  ❆♥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✳ W : W → Z W : Z → W W ◦ ❊♥tã♦ ❡①✐st❡♠ ú♥✐❝♦s φ ❡ ψ ❡ ♣♦rt❛♥t♦ φ

  ∈ Hom C } ◦ φ ψ W (Z, Z) = {I Z W W =

  ✳ ❆♥❛❧♦❣❛♠❡♥t❡ ❝♦♥❝❧✉í♠♦s q✉❡ ψ

  I W ✳ ▲♦❣♦✱ 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✳

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

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

  X Y >>

  B B }

  B }

  B }

  B } φ ψ B X Y B }

  } !!B

  } Z. ❯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♦s X ❡ Y ❡♠ C✳ ❙❡❥❛♠ Z ❡ Z ❞♦✐s ♦❜❥❡t♦s ③❡r♦s✳ ❙❡❣✉❡ ❞❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛ q✉❡ Z ≃ Z ❡ ❞❡♥♦t❛♠♦s X

  : X → Y ♣♦r φ : Z → Z t❛❧ ✐s♦♠♦r✜s♠♦✳ ❙❡❥❛ 0 Y ♠♦r✜s♠♦ ♥✉❧♦ ♦❜t✐❞♦ X

  = ψ Y ◦ φ X ❡♠ r❡❧❛çã♦ ❛♦ ♦❜❥❡t♦ ③❡r♦ Z✱ ♦✉ s❡❥❛✱ 0 Y ✳ X ′ ′ ′ ′ ′

  ◦ φ → Y = ψ : Z : X → ❱❡r✐✜q✉❡♠♦s q✉❡ 0 Y Y X ❡♠ q✉❡ ψ Y ❡ φ X Z

  ✳ ❉❡ ❢❛t♦✱ ψ Y ◦ φ X = ψ ◦ φ ◦ φ Y ′ ′ X

  1

  = ψ ◦ φ ◦ φ ◦ φ Y ′ ′ X = ψ ◦ φ . Y X Pr♦♣♦s✐çã♦ ✶✳✷✾ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦✱ X, Y, Z, W ∈ C X X Z Y Ob(C) (Y, Z) = 0 ◦ f = 0

  ❡ f ∈ Hom ✳ ❊♥tã♦ f ◦ 0 ❡ 0 ✳ Y Z W W X = f ◦ ψ Y ◦ φ X

  ❉❡♠♦♥str❛çã♦✿ P♦r ❞❡✜♥✐çã♦ t❡♠♦s q✉❡ f ◦ 0 Y ✳ ❆❧é♠ Z = f ◦ ψ Y ❞✐ss♦✱ ψ ✳ P♦rt❛♥t♦✱ X f ◦ 0 = f ◦ ψ Y ◦ φ Y X

  = ψ Z ◦ φ X X = . Z ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ♦❜t❡♠♦s ❛ ♦✉tr❛ ✐❣✉❛❧❞❛❞❡✳ X

  6= I X Pr♦♣♦s✐çã♦ ✶✳✸✵ ❙❡ ❳ ♥ã♦ é ✉♠ ♦❜❥❡t♦ ③❡r♦ ❡♥tã♦ 0 ✳ X X

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

  (X, Y ) = {0 } ♦❜❥❡t♦ q✉❛❧q✉❡r ❡♠ C✳ ▼♦str❡♠♦s q✉❡ Hom Y ❡ q✉❡ C C Y Hom (Y, X) = {0 } (X, Y ) X ✳ ❙❡❥❛ f ∈ Hom ✳ ❊♥tã♦ f = f ◦ I X X

  = f ◦ 0 X X = , Y

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

  6= I X ❞✐③ ❛ ❤✐♣ót❡s❡✳ ▲♦❣♦✱ 0 X ✳ ❉❡✜♥✐çã♦ ✶✳✸✶ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦✱ X, Y ∈ Ob(C) C

  (X, Y ) ❡ f ∈ Hom ✳ ❯♠ ♥ú❝❧❡♦ ❞❡ f é ✉♠ ♣❛r (Ker(f), k) ❡♠ q✉❡ Ker(f ) ∈ Ob(C) Ker ❡ k : Ker(f) → X é ✉♠ ♠♦r✜s♠♦ t❛❧ q✉❡ f ◦ k =

  

(f ) ′ ′ ′ ′

  → X Y ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ♣❛r (K ❝♦♠ k . , k ) : K ′ K ′ = 0 , → Ker(f ) t❛❧ q✉❡ f ◦ k Y ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u : K t❛❧ q✉❡ k ◦ u = k ✱ ✐✳❡✳✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

  K K′ H Y H H H ′ H H k H

  H f $$H u $$

  X // Y ::

  ;;w k w w w w w w w Ker (f ) w Y Ker(f ).

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

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

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

  ◦ v = k k ◦ u = k K K ❡ k ✳ ❙❡❣✉❡ ❞❛ ✉♥✐❝✐❞❛❞❡ q✉❡ u ◦ v = I ❡ v ◦ u = I ✳

  P♦rt❛♥t♦✱ K ≃ K ✳ ❉❡✜♥✐çã♦ ✶✳✸✸ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦✱ X, Y ∈ Ob(C) C

  (X, Y ) ❡ f ∈ Hom ✳ ❯♠ ❝♦♥ú❝❧❡♦ ❞❡ f é ✉♠ ♣❛r (CoKer(f), q) ❡♠ q✉❡ CoKer(f ) ∈ Ob(C) X ❡ q : Y → CoKer(f) é ✉♠ ♠♦r✜s♠♦ t❛❧ q✉❡ q ◦f = ′ ′

  ) : Y → Q CoKer (f ) ✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ♣❛r (Q, q ❝♦♠ q X ◦ f = 0 t❛❧ q✉❡ q ✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u : CoKer(f) → Q t❛❧ Q

  ′

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

  KK eeJJ JJ X JJ JJ Q JJ u J f X X // Y CoKer (f ) tt tt tt tt q zztt ss CoKer(f ).

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

  ✳ ❊♥tã♦✱ ✉♠ ♥ú❝❧❡♦ ♣❛r❛ ♦ ♠♦r✜s♠♦ 0 é ♦ ♣❛r (X, I ❡ ✉♠ X Y ) Y ❝♦♥ú❝❧❡♦ ♣❛r❛ 0 Y é ♦ ♣❛r (Y, I ✳ X X

  ◦ I X = 0 ❉❡ ❢❛t♦✱ é ❝❧❛r♦ q✉❡ 0 Y Y ❡ ❞❛❞♦ q✉❛❧q✉❡r ♦✉tr♦ ♣❛r (K, φ) X K

  ◦ φ = 0 ❡♠ q✉❡ φ : K → X ❡ 0 Y Y ❡♥tã♦ ❝❧❛r❛♠❡♥t❡ φ ❝♦♠✉t❛ ♦ ❞✐❛❣r❛♠❛

  K K A Y A A A φ A A X φ A Y ##

  // Y

  X ;;

  >> I X } } }

  } }

  } X } Y X ◦ φ = φ

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

  ❊①❡♠♣❧♦ ✶✳✸✺ ❊♠ ❝❛t❡❣♦r✐❛s ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛ ♥♦çã♦ ❞❡ ♥ú✲ ❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦ ❛❧❣é❜r✐❝♦ s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❞❛s ❞❡✜♥✐çõ❡s 1.31 ❡

  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♥❝✐❛ [✾]✳

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

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

  ❊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

  (X, Y ) ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ f ∈ Hom ✳

  ❉❡✜♥✐çã♦ ✶✳✸✼ f é ❞✐t♦ ✉♠ ✐s♦♠♦r✜s♠♦ s❡ ❡①✐st❡ g : Y → X ♠♦r✲ Y X ✜s♠♦ t❛❧ q✉❡ f ◦ g = I ❡ g ◦ f = I ✳ ❖s ♦❜❥❡t♦s X, Y sã♦ ❞✐t♦s ✐s♦✲ ♠♦r❢♦s ❡ ❞❡♥♦t❛♠♦s ♣♦r X ≃ 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 = Ker (f ) U k ◦ g ◦ g = 0

  ✳ ❖❜s❡r✈❡♠♦s q✉❡ f ◦ (k ◦ g) = 0 ✳ P♦rt❛♥t♦✱ ❡①✐st❡ Y Y ✉♠ 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 → Q é ✉♠ ❡♣✐♠♦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) z

  = g( ) z

  1

  = g(z)g( ) z

  1 = h(z)g( ). z

  1

  ) ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♣♦r h( z ♦❜t❡♠♦s

  1

  1

  h( ) = h( )h(1) z z

  1

  1

  = h( )h(z)g( ) z z

  1

  = h(1)g( ) z

  1

  = g(1)g( ) z

  1 = g( ). z a

  ❆ss✐♠✱ ❞❛❞♦ q ∈ Q ♣♦❞❡♠♦s ❡s❝r❡✈❡r q = ❝♦♠ a, b ∈ Z✱ b 6= 0✳ a b

  1

  1

  ) = g(a)g( ) = h(a)h( ) = h(q) ∀q ∈ Q ❉❛í g(q) = g( ✳ ▲♦❣♦✱ g = h✳ b b b

  ❊♠ ✭[✾]✱ ❈❤❛♣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 ❝♦✐♥❝✐❞❡♠✳ R M

  Pr♦♣♦s✐çã♦ ✶✳✹✷ ❙❡❥❛ ❘ ✉♠ ❛♥❡❧✳ ❯♠ ♠♦r✜s♠♦ f ❡♠ é ♠♦♥♦✲ ♠♦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♠♦ ♥❛ R M ❝❛t❡❣♦r✐❛ ✳ ❙✉♣♦♥❤❛♠♦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 X t❛❧ q✉❡ k ◦ f = I ✳ ❆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 Y t❛❧ q✉❡ f ◦ k = I ❡ ♣♦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 ❞❡✜♥✐❞♦ ♣♦r h(x) = 0 ♣❛r❛ t♦❞♦ x ∈ Ker(f)✳ ❖❜s❡r✈❛♠♦s q✉❡ f ◦ i = f ◦ h ❡✱ ♥♦ ❡♥t❛♥t♦✱ i 6= 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) ❞❡✜♥✐❞♦ ♣♦r h(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

  ❡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 , π Y ) X : P → X ❞❡ X ❡ Y é ✉♠❛ tr✐♣❧❛ (P, π t❛❧ q✉❡ P ∈ Ob(C)✱ π ❡ π Y : P → Y sã♦ ♠♦r✜s♠♦s ♥❡ss❛ ❝❛t❡❣♦r✐❛✳ ❆❧é♠ ❞✐ss♦✱ s❡ ❡①✐st✐r❡♠ X : Q → X Y : Q → Y

  ♦✉tr♦ ♦❜❥❡t♦ Q ❡ ♠♦r✜s♠♦s q ❡ q ❡♠ C ❡♥tã♦ X ◦φ = q X Y ◦φ = q Y ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ φ : Q → P t❛❧ q✉❡ π ❡ π ✳

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

  33 q ?? X X ~

  ~ ~

  ~ ~ π X

  ~ φ ~ ~

  Q // P @

  @ π @ Y

  @ @

  @ q Y @ @

  • Y.

  Pr♦♣♦s✐çã♦ ✶✳✹✹ ❖ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ♦❜❥❡t♦s X ❡ Y ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ C

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

  ❚❛♠❜é♠✱ ❞♦ ❢❛t♦ ❞❡ (Q, q s❡r ♣r♦❞✉t♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✲ X ◦ ψ = π ◦ ψ = π X Y Y . ✜s♠♦ ψ : P → Q t❛❧ q✉❡ q ❡ q ◆♦t❡♠♦s q✉❡

  ◦ (ψ ◦ φ) = ψ ◦ φ : Q → Q X

  ❡ φ ◦ ψ : P → P sã♦ ♠♦r✜s♠♦s t❛✐s q✉❡ q ◦ φ = q ◦ (ψ ◦ φ) = π ◦ φ = q ◦ (φ ◦ ψ) = q ◦ ψ = π

  π X X Y Y Y X X X ✱ q ✱ π Y Y Y ◦ (φ ◦ ψ) = q ◦ ψ = π

  ❡ π ✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦s ♠♦r✜s♠♦s s❡❣✉❡ q✉❡ ψ ◦ φ = I Q P

  ❡ φ ◦ ψ = I ✳ ❊①❡♠♣❧♦ ✶✳✹✺ ❖ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B ♥❛ ❝❛t❡❣♦r✐❛ Set A , π B ) é ❛ tr✐♣❧❛ (A × B, π ✱ ♦✉ s❡❥❛✱ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❝♦♠ ❛s r❡s✲ A , P B ) ♣❡❝t✐✈❛s ♣r♦❥❡çõ❡s ❝❛♥ô♥✐❝❛s✳ ❉❡ ❢❛t♦✱ s❡❥❛ (D, P ♦✉tr❛ tr✐♣❧❛✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛

  P A B D φ P A × B

  F F π π x A B F xx F F F xx

  F xx F ||xx ##F

  A B A (d), P B (d)) ❡♠ q✉❡ ❞❡✜♥✐♠♦s φ : D → A × B ♣♦r φ(d) = (P ♣❛r❛ t♦❞♦ d ∈ D

  ✳ ◆❡ss❡ ❝❛s♦✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ é ❝❧❛r❛ ❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ φ t❛♠❜é♠✳ ❊①❡♠♣❧♦ ✶✳✹✻ ❉❛❞♦s ❞♦✐s ❣r✉♣♦s G ❡ H ♥❛ ❝❛t❡❣♦r✐❛ Grp✱ ♦ ♣r♦❞✉t♦ G , π H ) ❞❡ss❡s ♦❜❥❡t♦s é ❛ tr✐♣❧❛ (G × H, π 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♦✲ X , i Y ) X : X → Q ❞✉t♦ ❞❡ X ❡ Y é ✉♠❛ tr✐♣❧❛ (Q, i t❛❧ q✉❡ Q ∈ Ob(C)✱ i Y : Y → Q ❡ i sã♦ ♠♦r✜s♠♦s ♥❡ss❛ ❝❛t❡❣♦r✐❛✳ ❆❧é♠ ❞✐ss♦✱ s❡ ❡①✐st✐r❡♠ ′ ′ ′ X : X → Q Y : Y → Q ♦✉tr♦ ♦❜❥❡t♦ Q ❡ ♠♦r✜s♠♦s j ❡ j ❡♠ C ❡♥tã♦ X = j X Y = j Y

  ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ψ : Q → Q t❛❧ q✉❡ ψ◦i ❡ ψ◦i ✳ P♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ❡ss❛ ❞❡✜♥✐çã♦ ✈✐❛ ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ j X i X X {{ ψ ww oo

  Q Q gg cc j Y i Y Y.

  ❖❜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 ♥❛ op ❝❛t❡❣♦r✐❛ C é ♦ ♣r♦❞✉t♦ ❞❡ X ❡ Y ♥❛ ❝❛t❡❣♦r✐❛ C ✳

  ❉❡ ❢❛t♦✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ✐♠♣❧✐❝❛ ♥❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♠♦r✲ ′ ′ C op (Q , Q) X = j X

  ✜s♠♦ ψ : Q → Q ✱ ♦✉ s❡❥❛✱ ψ ∈ Hom t❛❧ q✉❡ ψ ◦ i ❡ op op ψ ◦ i Y = j Y X ◦ ψ = j X Y ◦ ψ = j Y

  ✱ ✐✳ ❡✳✱ i ❡ i ✳ P♦❞❡♠♦s ♣♦rt❛♥t♦

  ❞✐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 ♣♦r A⊔B ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❞❡ A ❡ B✳ ❱❡r✐✜q✉❡✲ A , i B ) A , j B )

  ♠♦s q✉❡ ❛ tr✐♣❧❛ (A ⊔ B, i é ❞❡ ❢❛t♦ ♦ ❝♦♣r♦❞✉t♦✳ ❙❡❥❛ (Q, j A (a) ♦✉tr❛ tr✐♣❧❛✳ ❉❡✜♥✐♠♦s ψ : A ⊔ B → Q ♣♦r ψ(a) = j s❡ a ∈ A ❡ ψ(b) = j B (b) A = j A B = j B s❡ b ∈ B✳ ➱ ✐♠❡❞✐❛t♦ q✉❡ ψ ◦ i ❡ ψ ◦ i ❡ ❝❧❛r❛♠❡♥t❡ ψ é ú♥✐❝❛✳ ❖❜s❡r✈❛çã♦ ✶✳✺✵ ❊ss❛s ♥♦çõ❡s ❞❡ ♣r♦❞✉t♦ ❡ ❝♦♣r♦❞✉t♦ ♣♦❞❡♠ s❡r i i∈I } ❣❡♥❡r❛❧✐③❛❞❛s✳ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ {X ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♦❜❥❡t♦s i i∈I ) i i∈I } } ❡♠ C✳ ❉✐③❡♠♦s q✉❡ (X, {π é ♦ ♣r♦❞✉t♦ ❞❛ ❢❛♠í❧✐❛ {X s❡

  } X ∈ Ob(C) i : X → X i i∈I

  ❡ {π é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♠♦r✜s♠♦s ♥❡ss❛ ❝❛t❡❣♦r✐❛ t❛❧ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ♦❜❥❡t♦ Y ❡ ❢❛♠í❧✐❛ ❞❡ ♠♦r✜s♠♦s {q i } i∈I i ◦ f = q i

  ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ f : Y → X t❛❧ q✉❡ π ♣❛r❛ t♦❞♦ i ∈ I✱ ♦✉ s❡❥❛✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ X f >> π i

  // X i Y q i

  ♣❛r❛ t♦❞♦ i ∈ I✳ ❉❡ ♠❛♥❡✐r❛ ❞✉❛❧✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦♣r♦❞✉t♦ ❣❡♥❡✲ r❛❧✐③❛❞♦✳ R i } i∈I M ❊①❡♠♣❧♦ ✶✳✺✶ ❈♦♥s✐❞❡r❡♠♦s ❛ ❝❛t❡❣♦r✐❛ ✳ ❙❡❥❛ {M ✉♠❛ ❢❛✲

  M i ♠í❧✐❛ ❞❡ R✲♠ó❞✉❧♦s✱ ❡♥tã♦ Q ♦ ♣r♦❞✉t♦ ❞✐r❡t♦ ❞❡ ♠ó❞✉❧♦s ❥✉♥t❛✲ i } i∈I i∈I ♠❡♥t❡ ❝♦♠ ❛s ♣r♦❥❡çõ❡s {π é ♦ ♣r♦❞✉t♦ ❞❡ss❛ ❝❛t❡❣♦r✐❛✳ ❉❛ ♠❡s♠❛

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

  ❈❛♣ít✉❧♦ ✷ ❋✉♥t♦r❡s ✷✳✶ ❋✉♥t♦r❡s ❡ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s

  ◆❡ss❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛s ♥♦çõ❡s ❞❡ ❢✉♥t♦r❡s✱ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s ❡ ❛❞❥✉♥çõ❡s✳ ❍✐st♦r✐❝❛♠❡♥t❡✱ ♦ ❡st✉❞♦ ❞❡ ❝❡rt❛s tr❛♥s❢♦r♠❛✲ çõ❡s ♥❛t✉r❛✐s ❞❡✉ ♦r✐❣❡♠ à ♥♦çã♦ ❞❡ ❢✉♥t♦r q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ❞❡✉ ♦r✐❣❡♠ à ♥♦çã♦ ❞❡ ❝❛t❡❣♦r✐❛✳ ❆s ❛❞❥✉♥çõ❡s sã♦ ✉♠ t✐♣♦ ❡s♣❡❝í✜❝♦ ❞❡ ✐s♦♠♦r✲ ✜s♠♦ ♥❛t✉r❛❧ q✉❡ ❡st❛❜❡❧❡❝❡♠ ✉♠❛ ❝❡rt❛ s✐♠❡tr✐❛ q✉❡ ❡stá ♣r❡s❡♥t❡ ❡♠ ♠✉✐t❛s ❡str✉t✉r❛s ♠❛t❡♠át✐❝❛s✳ ❯♠ r❡s✉❧t❛❞♦ ❢❛♠♦s♦ ❛♣r❡s❡♥t❛❞♦ ♥❡ss❡ ❝❛♣ít✉❧♦ é ♦ ▲❡♠❛ ❞❡ ❨♦♥❡❞❛✳ ❉❡✜♥✐çã♦ ✷✳✶ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s✳ ❯♠ ❢✉♥t♦r ✭❝♦✈❛r✐❛♥t❡✮ F ❝♦♥s✐st❡ ❞❡ ❞✉❛s ❛♣❧✐❝❛çõ❡s✿ ✭✐✮ ✉♠❛ ❛♣❧✐❝❛çã♦ F : Ob(C) → Ob(D) q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♦❜❥❡t♦ X ❡♠ C

  ✉♠ ♦❜❥❡t♦ F (X) ❡♠ D❀ C D (X, Y ) → Hom (F (X), F (Y ))

  ✭✐✐✮ ✉♠❛ ❛♣❧✐❝❛çã♦ F : Hom q✉❡ ❛s✲ C (X, Y ) s♦❝✐❛ ❛ ❝❛❞❛ ♠♦r✜s♠♦ f : X → Y ❡♠ Hom ✉♠ ♠♦r✜s♠♦ D

  F (f ) : F (X) → F (Y ) (F (X), F (Y )) ❡♠ Hom t❛❧ q✉❡ s❡ ✈❡r✐✜❝❛♠ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s

  F (I X ) = I F ❡ F (f ◦ g) = F (f) ◦ F (g)

  (X)

  ❡♠ q✉❡ X é ✉♠ ♦❜❥❡t♦ q✉❛❧q✉❡r ❡♠ C ❡ f, g sã♦ ♠♦r✜s♠♦s ❡♠ C ✭♣♦s✲ sí✈❡✐s ❞❡ s❡ ❝♦♠♣♦r✮✳ ❖❜s❡r✈❛çã♦ ✷✳✷ ❯♠ ❢✉♥t♦r F é ❝❤❛♠❛❞♦ ❝♦♥tr❛✈❛r✐❛♥t❡ s❡ ✐♥✈❡rt❡ ✢❡❝❤❛s✱ ✐✳❡✳✱ ♣❛r❛ t♦❞♦ ♠♦r✜s♠♦ f : X → Y t❡♠♦s F (f) : F (Y ) → F (X)

  ❡ F (f ◦ g) = F (g) ◦ F (f)✳

  ❖❜s❡r✈❛çã♦ ✷✳✸ P❡♥s❛♥❞♦ ❡♠ ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✈♦s✱ ❛ ✐❞❡✐❛ ❞❡ ✉♠ ❢✉♥t♦r é ❧❡✈❛r ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✈♦s ✉♠❛ ❝❛t❡❣♦r✐❛ ❡♠ ❞✐❛❣r❛♠❛s ❝♦✲ C

  (Y, Z) ♠✉t❛t✐✈♦s ❞❡ ♦✉tr❛ ❝❛t❡❣♦r✐❛✳ ❙❡❥❛♠ X, Y, Z ∈ Ob(C)✱ f ∈ Hom C

  (X, Y ) ❡ g ∈ Hom ✳ ❊♥tã♦ ♦ ❞✐❛❣r❛♠❛ f ◦g

  X // Z >>

  ~ g ~ ~ ~

  ~ f ~

  ~ Y

  é ❧❡✈❛❞♦ ♣♦r ✉♠ ❢✉♥t♦r F ❝♦✈❛r✐❛♥t❡ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ❝♦♥tr❛✈❛r✐❛♥t❡✮ ♥♦ ❞✐❛❣r❛♠❛ à ❡sq✉❡r❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡t❛✮ F F

  

(f ◦g) (f ◦g)

  oo F (X) // F (Z) F (X) F (Z)

  OO ;;w v w F w F w vv

  (g) (g)

  w vv w F F

  (f ) (f )

  w vv w w {{vv F (Y ) F (Y ).

  ❖❜s❡r✈❛çã♦ ✷✳✹ ❉❛❞♦ ✉♠ ❢✉♥t♦r ❝♦♥tr❛✈❛r✐❛♥t❡ F ❞❡✜♥✐♠♦s ✉♠ ♥♦✈♦ op op op op → D

  : C (X) = F (X) (f ) = F (f ) ❢✉♥t♦r F ♣♦r F ❡ F ♣❛r❛ t♦❞♦ op X ∈ Ob(C)

  ❡ t♦❞♦ ♠♦r✜s♠♦ f ❡♠ C✳ ◆❡ss❡ ❝❛s♦✱ F é ✉♠ ❢✉♥t♦r op op op (g ◦ f ) = F (f ◦ g) = F (f ◦ g) = F (g) ◦ F (f ) =

  ❝♦✈❛r✐❛♥t❡✱ ♣♦✐s F op op F (g) ◦ F (f )

  ✳ ❉❛q✉✐ ♣♦r ❞✐❛♥t❡✱ ✉t✐❧✐③❛♠♦s ♦ t❡r♠♦ ❢✉♥t♦r ♣❛r❛ ❞❡s✐❣♥❛r ❢✉♥t♦r❡s

  ❝♦✈❛r✐❛♥t❡s✱ ❝❛s♦ ♥❛❞❛ s❡❥❛ ❞✐t♦ ❛♦ ❝♦♥trár✐♦✳ ❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❢✉♥t♦r❡s✳ ❊①❡♠♣❧♦ ✷✳✺ ❉❛❞♦ ✉♠❛ ❝❛t❡❣♦r✐❛ C✱ ❡①✐st❡ ✉♠ ❢✉♥t♦r ❝❤❛♠❛❞♦ ❢✉♥t♦r C C C

  : C → C (X) = X (f ) = f ✐❞❡♥t✐❞❛❞❡ Id ❞❡✜♥✐❞♦ ♣♦r Id ❡ Id ♣❛r❛ t♦❞♦ X ∈ Ob(C) ❡ t♦❞♦ ♠♦r✜s♠♦ f ❡♠ C✳ ❊①❡♠♣❧♦ ✷✳✻ ❙❡❥❛ F : Grp → Set t❛❧ q✉❡ F (G) = G ❡ F (f) = f✳ ❚❛❧ ❢✉♥t♦r é ❝❤❛♠❛❞♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦✱ ♣♦✐s ❡♠ Set é ❡sq✉❡❝✐❞❛ ❛ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ❞♦s ♦❜❥❡t♦s ❞❡ Grp✳ ❆ss✐♠ ❝♦♠♦ ♦s ♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s sã♦ ❝♦♥s✐❞❡r❛❞♦s ❛♣❡♥❛s ❝♦♠♦ ❢✉♥çã♦ ❡♥tr❡ ❝♦♥❥✉♥t♦s✳

  ❖ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ ❛♣❛r❡❝❡ ❡♠ ✈ár✐❛s ♦✉tr❛s ❝❛t❡❣♦r✐❛s ❡ é ♣♦s✲ sí✈❡❧ ❞❛r ♠✉✐t♦s ❡①❡♠♣❧♦s s✐♠✐❧❛r❡s ❛♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✳ ❊①❡♠♣❧♦ ✷✳✼ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❧♦❝❛❧♠❡♥t❡ ♣❡q✉❡♥❛ ❡ X ∈ Ob(C)✳ X : C → Set X : C → Set op ❉❡✜♥✐♠♦s ❢✉♥t♦r❡s F ❡ G t❛✐s q✉❡✱ ♣❛r❛

  X (Y ) = Hom (X, Y ) C X (Y ) =

  q✉❛❧q✉❡r Y ∈ Ob(C)✱ ❞❡✜♥✐♠♦s F ❡ G C Hom (Y, X)

  ❡ s❡ f : Y → Z é ✉♠ ♠♦r✜s♠♦ ❡♠ C✱ ❡♥tã♦ C C F X (f ) : Hom (X, Y ) → Hom (X, Z)

  α 7→ F X (f )(α) = f ◦ αC C

  G X (f ) : Hom (Z, X) → Hom (Y, X) α 7→ G X (f )(α) = α ◦ f sã♦ ❢✉♥t♦r❡s✳ ❉❡ ❢❛t♦✱

  F X (Id Y )(α) = Id Y ◦ α = α = Id F (α). X (Y )

  ❙❡❥❛♠ f : Y → Z ❡ g : Z → W ♠♦r✜s♠♦s ❡♠ C✳ ❊♥tã♦ F X (g ◦ f )(α) = (g ◦ f ) ◦ α

  = g ◦ (f ◦ α) = g ◦ (F X (f )(α)) = F X (g)(F X (f )(α)) X = (F X (g) ◦ F X (f ))(α). ❆ ✈❡r✐✜❝❛çã♦ ❞❡ q✉❡ G é ✉♠ ❢✉♥t♦r é ❛♥á❧♦❣❛✳ ◆❛ ❝♦♥str✉çã♦ ❞❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡

  ▲✐❡ ❡①✐st❡♠ ❛❧❣✉♥s ❢✉♥t♦r❡s q✉❡ ❡stã♦ ✐♠♣❧í❝✐t♦s✱ ❝♦♠♦ ✈❡r❡♠♦s ♥♦s ♣ró①✐♠♦s ❡①❡♠♣❧♦s✳ ❘❡❢❡r✐♠♦s ❛q✉✐ ♦ ❆♣ê♥❞✐❝❡ A✳ k ❊①❡♠♣❧♦ ✷✳✽ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❝❛t❡❣♦r✐❛ Lie ❝✉❥♦s ♦❜❥❡t♦s sã♦ á❧❣❡❜r❛s ❞❡ ▲✐❡ s♦❜r❡ k ❡ ♦s ♠♦r✜s♠♦s sã♦ ♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳ k → Alg k

  ❖ ❢✉♥t♦r U : Lie é ❞❡✜♥✐❞♦ ❝♦♠♦ U(L) ❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ L✳ ′ ′

  ∈ Ob(Lie k ) k ❙❡❥❛♠ L, L ❡ f : L → L ✉♠ ♠♦r✜s♠♦ ❡♠ Lie ✳ ❉❛

  ❞❡✜♥✐çã♦ ❞❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧✱ s❡❣✉❡ q✉❡ ✭✉♠❛ ✈❡③ q✉❡ φ = ι L ◦ f k

  é ✉♠ ♠♦r✜s♠♦ ❡♠ Lie ✮ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ψ : U(L) → U(L ) q✉❡ ✐♥❞✉③ ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ ψ : L

  (U(L)) → L(U(L )) q✉❡ ❝♦♠✉t❛ ♦ ❞✐❛❣r❛♠❛ ι L L

  // (U(L)) f ψ L

  // L(U(L )), L ι L′

  L ◦ f = ψ ◦ ι L

  ✐st♦ é✱ ι ✳ ❉❡✜♥✐♠♦s U(f) = ψ✳ L ) = I U ❆ ✉♥✐❝✐❞❛❞❡ ♥♦s ❣❛r❛♥t❡ q✉❡ ♥♦ ❝❛s♦ ❡♠ q✉❡ L = L ✱ U(I (L) ✳ ′ ′ ′′

  → L ❙❡❥❛♠ f : L → L ❡ g : L ♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ❊♥tã♦ ′ ′ ′′ U

  (f ) : U(L) → U(L ) ) → U(L ) ❡ U(g) : U(L sã♦ ♠♦r✜s♠♦s ❞❡ k✲ L = ι L ◦ f L = ι L ◦ g ′ ′ ′′

  á❧❣❡❜r❛s t❛✐s q✉❡ U(f) ◦ ι ❡ U(g) ◦ ι ✳ ❈♦♠♣♦♥❞♦ f ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ r❡s✉❧t❛ q✉❡

  U ′ ′′ ′′ (g) ◦ ι L ◦ f = ι L ◦ g ◦ f L = ι L ◦ g ◦ f

  ✱ ♦✉ s❡❥❛✱ U(g) ◦ U(f) ◦ ι ✳ ❙❡❣✉❡ ❞❛ ✉♥✐❝✐❞❛❞❡ q✉❡ U(g ◦ f) = U(g) ◦ U(f)✳ k → Lie k ❊①❡♠♣❧♦ ✷✳✾ ❙❡❥❛ L : Alg ♦ ❢✉♥t♦r q✉❡ ❧❡✈❛ t♦❞❛ k✲á❧❣❡❜r❛ ♥❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❝✉❥♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ é ♦ ❝♦♠✉t❛❞♦r✳ ❉❛❞♦ ✉♠ ♠♦r✲ ✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s f : A → B✱ ❡st❡ ✐♥❞✉③ ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛ ❞❡ ▲✐❡ L(f) : L(A) → L(B)✳ ❱❡❥❛ ❊①❡♠♣❧♦ A.3 ❞♦ ❆♣ê♥❞✐❝❡ A✳

  ❊ss❡s ❞♦✐s ú❧t✐♠♦s ❢✉♥t♦r❡s ❡stã♦✱ ❞❡ ❝❡rt❛ ❢♦r♠❛✱ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠♦ ✈❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡✳ k ❊①❡♠♣❧♦ ✷✳✶✵ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ Hopf é ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ ❡ ❝✉❥♦s ♠♦r✜s♠♦s sã♦ ❤♦♠♦♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢✳ k → Lie k

  ❈♦♥s✐❞❡r❡♠♦s P : Hopf ♦ ❢✉♥t♦r q✉❡ t♦♠❛ ♣r✐♠✐t✐✈♦s✳ ❙❡❥❛ H

  ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢✳ ❉❡✜♥✐♠♦s P (H) = {x ∈ H : ∆(x) = x ⊗ 1 + 1 ⊗ x}. ❙❡❣✉❡ ❞♦ ❢❛t♦ ❞❡ q✉❡ ∆ é k✲❧✐♥❡❛r q✉❡ P(H) é ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

  P♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❡♠ P(H) ❝♦♥s✐❞❡r❛♥❞♦ [x, y] = xy − yx

  ❝♦♠♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✱ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ P(H)✳ ❚♦❞♦ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ f : H → W ✐♥❞✉③ ✉♠ ♠♦r✜s♠♦ ❞❡ ′ ′

  : P(H) → P(W ) (x) = f (x) á❧❣❡❜r❛s ❞❡ ▲✐❡ f ❢❛③❡♥❞♦ f ✳ ❉❡ ❢❛t♦✱

  (∗)

  ∆(f (x)) = (f ⊗ f )(∆(x)) = (f ⊗ f )(x ⊗ 1 + 1 ⊗ x)

  (∗∗) = f (x) ⊗ 1 + 1 ⊗ f (x) ∈ P(W ).

  ❆s ✐❣✉❛❧❞❛❞❡s ✭✯✮ ❡ ✭✯✯✮ s❡❣✉❡♠ ❞♦ ❢❛t♦ ❞❡ f s❡r ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ ✭❡ ♣♦rt❛♥t♦✱ ❞❡ ❝♦á❧❣❡❜r❛s ❡ ❞❡ á❧❣❡❜r❛s✮✳ ◆❛t✉r❛❧✲ ♠❡♥t❡✱ ❞♦ ❢❛t♦ ❞❡ f s❡r ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s s❡❣✉❡ q✉❡ f ✐♥❞✉③✐❞❛ é ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳

  ❯♠ ❢✉♥t♦r ❝♦♥❤❡❝✐❞♦ ❞❛ á❧❣❡❜r❛ ❧✐♥❡❛r é ❛q✉❡❧❡ q✉❡ t♦♠❛ ♦ ❞✉♣❧♦ ❞✉❛❧ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ k → V ect k

  ❊①❡♠♣❧♦ ✷✳✶✶ ❈❤❛♠❛♠♦s ❞❡ D : V ect ♦ ❢✉♥t♦r ❞❡✜♥✐❞♦ ∗∗ ♣♦r D(V ) = V ♣❛r❛ t♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V ✳ ❚♦❞❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r f : V → W ✐♥❞✉③ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❝❤❛♠❛❞❛ ❞✉♣❧❛ tr❛♥s♣♦st❛ ∗∗ ∗∗

  D(f ) : V → W ∗ ∗ ∗ T 7→ T ◦ f : W → V

  ❡♠ q✉❡ f é ❛ ❛♣❧✐❝❛çã♦ tr❛♥s♣♦st❛ ❞❡ f✱ ❞❡✜♥✐❞❛ ♣♦r ∗ ∗ f (g) = g ◦ f ✱ ♣❛r❛ t♦❞♦ g ∈ W ✳

  ◆❡♠ s❡♠♣r❡ ♦ q✉❡ ♣❛r❡❝❡ ✐♥t✉✐t✐✈❛♠❡♥t❡ ✉♠ ❢✉♥t♦r é ❞❡ ❢❛t♦ ✉♠ ❢✉♥t♦r✱ ♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦ ♠♦str❛ ✉♠❛ t❡♥t❛t✐✈❛ ❢❛❧❤❛ ❞❡ ❝r✐❛r ✉♠ ❢✉♥t♦r✳ ❊①❡♠♣❧♦ ✷✳✶✷ ▼♦str❡♠♦s q✉❡ ♥ã♦ ❡①✐st❡ ✉♠ ❢✉♥t♦r F : Grp → Ab t❛❧ q✉❡ F (G) = Z(G)✱ ❡♠ q✉❡ Z(G) é ♦ ❝❡♥tr♦ ❞♦ ❣r✉♣♦ G✳

  ❈♦♥s✐❞❡r❡♠♦s ♦s ❣r✉♣♦s ❞❡ ♣❡r♠✉t❛çõ❡s S

  2 ❡ S 3 ✳ ❙❛❜❡♠♦s q✉❡

  Z(S ) = S ) = {e S } → S → S

  2 2 ❡ q✉❡ Z(S

  3 3 ✳ ❙❡❥❛♠ f : S

  2 S 7→ e S 3 ❡ g : S

  3

  2

  ♠♦r✜s♠♦s ❞❡ ❣r✉♣♦✳ ❖ ♠♦r✜s♠♦ f é ❞❡✜♥✐❞♦ ♣♦r e 2 3 ✱ (12) 7→ (12) S , (123), (132) 7→ e S

  ❡ g ♣♦r e 3 2 ✱ (12), (23), (13) 7→ (12)✳ ◆❡ss❡ ❝❛s♦✱ g ◦ f = I S 2 ✳ ❈❛s♦ ❡①✐st✐ss❡ ✉♠ ❢✉♥t♦r F t❛❧ q✉❡ F (G) = Z(G)✱ ♦ ♠❡s♠♦ S ) = I F = I S ❞❡✈❡r✐❛ s❛t✐s❢❛③❡r F (g ◦ f) = F (I 2 (S 2 ) 2 ✳

  → {e S } ❊♥tr❡t❛♥t♦✱ F (f) : S

  2 3 é ♦ ❤♦♠♦♠♦r✜s♠♦ ♥✉❧♦✳ ❆ss✐♠✱

  F (g) ◦ F (f ) : S

  2 → S

  2

  é ♦ ❤♦♠♦♠♦r✜s♠♦ ♥✉❧♦✳ ▲♦❣♦✱ F (g ◦ f) 6= F (g) ◦ F (f )

  ✳ ❉❡✜♥✐çã♦ ✷✳✶✸ ❙❡❥❛♠ F, G : C → D ❢✉♥t♦r❡s✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ X : F (X) → G(X) : ♥❛t✉r❛❧ µ : F → G é ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♠♦r✜s♠♦s {µ X ∈ Ob(C)} t❛❧ q✉❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♦❜❥❡t♦s X, Y ∈ Ob(C) ❡ ♣❛r❛ ❝❛❞❛ ♠♦r✜s♠♦ f : X → Y ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ µ X

  // F G F (X) G(X)

  

(f ) (f )

F (Y ) // G(Y ). µ Y

  ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ µ : F → G é ❞✐t❛ ✉♠ ✐s♦♠♦r✜s♠♦ X : F (X) → G(X) ♥❛t✉r❛❧ s❡✱ ♣❛r❛ t♦❞♦ ♦❜❥❡t♦ X✱ ♦ ♠♦r✜s♠♦ µ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ◆❡ss❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ F é ❡q✉✐✈❛❧❡♥t❡ ❛ G ❡ ❞❡♥♦t❛♠♦s ✐ss♦ ♣♦r F ∼ G✳

  ❯♠❛ ♠❛♥❡✐r❛ ❞❡ ✐♥t❡r♣r❡t❛r ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ é ❝♦♠♦ ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♠♦r✜s♠♦s q✉❡ ❧❡✈❛ ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✈♦s ❞❛ ✏✐♠❛❣❡♠ ❞❡

  ✉♠ ❢✉♥t♦r✑ ❡♠ ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✈♦s ❞❛ ✏✐♠❛❣❡♠ ❞❡ ♦✉tr♦ ❢✉♥t♦r✑✱ ♦✉ s❡❥❛✱ é ❝♦♠♦ s❡ ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ tr❛♥s❧❛❞❛ss❡ ♦s ❞✐❛❣r❛♠❛s ❞❡ ✉♠ ❢✉♥t♦r ♣❛r❛ ♦✉tr♦✳ ■ss♦ ♣♦❞❡ s❡r ✈✐st♦ ♣❡❧♦s ❞✐❛❣r❛♠❛s ❛ s❡❣✉✐r✱ ❝♦♥s✐❞❡r❛♥❞♦ µ : F → G ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✱ ♦✉ s❡❥❛✱ µ X

  // F (X) G(X)

  X >

  G G >

  G G F G > f G (f ) G (f )

  G G >

  G G >

  G G >

  G G G G >

  ##G ##G µ h F (h) G (h) > Y // G(Y )

  F (Y ) Y w w ww ww g ww ww F G

  (g) (g)

  ww ww {{ww {{ww µ Z

  // G(Z) F (Z)

  Z ❡♠ q✉❡ X, Y ❡ Z sã♦ ♦❜❥❡t♦s q✉❛✐sq✉❡r ❡♠ C ❡ f, g ❡ h sã♦ ♠♦r✜s♠♦s q✉❡ ❝♦♠✉t❛♠ ♦ ❞✐❛❣r❛♠❛ à ❡sq✉❡r❞❛✳ ❊①❡♠♣❧♦ ✷✳✶✹ ❈♦♥s✐❞❡r❡♠♦s ♦s ❢✉♥t♦r❡s F : Ab → Grp ❢✉♥t♦r ❡sq✉❡✲ ❝✐♠❡♥t♦ ❡ U : Grp → Ab ♦ ❢✉♥t♦r ❞❡✜♥✐❞♦ ♣♦r U(G) = G/[G, G]✱ ❡♠ q✉❡ [G, G] é ♦ ❝♦♠✉t❛❞♦r ❞♦ ❣r✉♣♦ G✳

  ❖ ❢✉♥t♦r U ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ♣♦✐s [G, G] ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ G/[G, G]

  é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ ❉❛❞♦ ✉♠ ♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s f : G → H

  ✐♥❞✉③✐♠♦s ✉♠ ♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ❝♦♠♦ s❡❣✉❡ →

  U (f ) : G/[G, G] H/[H, H] g[G, G] 7→ f (g)[H, H] G ) = I U ♦ q✉❛❧ ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s U(I (G) ✱ ♣❛r❛ t♦❞♦ ❣r✉♣♦ G✳ ❉❛❞♦s ♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s f : G → H ❡ h : H → W t❡♠♦s q✉❡

  U (h ◦ f )(g[G, G]) = (h ◦ f )(g)[W, W ] = h(f (g))[W, W ] = U (h)(f (g)[H, H]) = U (h)(U (f )(g[G, G])) = (U (h) ◦ U (f ))(g[G, G]),

  ♣❛r❛ t♦❞♦ g ∈ G✳ P♦rt❛♥t♦✱ U(h ◦ f) = U(h) ◦ U(f)✳ ❆ ❝♦❧❡çã♦ ❞❡ ♠♦r✜s♠♦s ♣r♦❥❡çã♦

  P = {P G : G → G/[G, G] : G ∈ Ob(Grp)} Grp é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❡♥tr❡ ♦s ❢✉♥t♦r❡s Id ❡ F ◦ U✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ G, H ∈ Ob(Grp) ❡ f : G → H ✉♠ ♠♦r✜s♠♦ ❡♠ Grp✳ ❊♥tã♦ ♦

  ❞✐❛❣r❛♠❛ s❡❣✉✐♥t❡ ❝♦♠✉t❛ P G // f G G/[G, G]

  (F ◦U )(f )

  // H/[H, H], H P H

  ♣♦✐s (P H ◦ f )(g) = P H (f (g))

  = f (g)[H, H] = U (f )(g[G, G]) = F (U (f ))(g[G, G]) = (F ◦ U )(f )(g[G, G]) = (F ◦ U )(f )(P G (g)) H ◦ f = (F ◦ U )(f ) ◦ P G = ((F ◦ U )(f ) ◦ P G )(g),

  ♣❛r❛ t♦❞♦ g ∈ G✳ ▲♦❣♦✱ P ✳ ❉❡✜♥✐çã♦ ✷✳✶✺ ❉✐③❡♠♦s q✉❡ ❞✉❛s ❝❛t❡❣♦r✐❛s C ❡ D sã♦ ❡q✉✐✈❛❧❡♥t❡s D s❡ ❡①✐st❡♠ ❢✉♥t♦r❡s F : C → D ❡ G : D → C t❛✐s q✉❡ F ◦ G ∼ Id ❡ C

  G ◦ F ∼ Id ✳ ❉❡♥♦t❛♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❝❛t❡❣♦r✐❛s ♣♦r C ∼ D✳

  ❊①✐st❡ ✉♠❛ ♥♦çã♦ ♠❛✐s ❢♦rt❡ q✉❡ é ♦ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❝❛t❡❣♦r✐❛s✱ ❝♦♠♦ é ❞❛❞♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r✳ ❉❡✜♥✐çã♦ ✷✳✶✻ ❉✉❛s ❝❛t❡❣♦r✐❛s C ❡ D sã♦ ❞✐t❛s ✐s♦♠♦r❢❛s s❡ ❡①✐st❡♠ C D ❢✉♥t♦r❡s F : C → D ❡ G : D → C t❛✐s q✉❡ G ◦ F = Id ❡ F ◦ G = Id ✳ ❉❡♥♦t❛♠♦s ♦ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❝❛t❡❣♦r✐❛s ♣♦r C ≃ D✳

  ❚♦❞♦ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❝❛t❡❣♦r✐❛s é ❝❧❛r❛♠❡♥t❡ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❝❛t❡❣♦r✐❛s✱ ♥♦ ❡♥t❛♥t♦✱ ❛ r❡❝í♣r♦❝❛ ♥ã♦ é ✈❡r❞❛❞❡✐r❛✳ ❚r❛❜❛❧❤❛♠♦s ❛♣❡♥❛s ❝♦♠ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❝❛t❡❣♦r✐❛s✱ ❡♥❣❧♦❜❛♥❞♦ ♦s ❝❛s♦s ❡♠ q✉❡ sã♦ ✐s♦♠♦r✜s♠♦s✳ ❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s✳ ❊①❡♠♣❧♦ ✷✳✶✼ ❊ss❡ ❡①❡♠♣❧♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ([✷], p.72 − 77)✳ c ) ∗ M ❉❛❞❛ ✉♠❛ ❝♦á❧❣❡❜r❛ C✱ ❝♦♥s✐❞❡r❛♠♦s ❛ ❝❛t❡❣♦r✐❛ Rat( ❝♦♠♦ ❛q✉❡❧❛ ∗ ∗ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s C ✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ r❛❝✐♦♥❛✐s✱ ♦✉ s❡❥❛✱ C ✲ ♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ❝✉❥❛ ❛çã♦ é ❞❛❞❛ ♣♦r ∗ ∗ X i ) i∈I ⊆ C i ) i∈I ⊆ M c · m = c (c i )m i , i∈I

  ♣❛r❛ ❢❛♠í❧✐❛s ✜♥✐t❛s (c ❡ (m ✭I é ✉♠ ❝♦♥❥✉♥t♦ ∗ ∗ ∗ M ∈ C c )

  ✜♥✐t♦✮✱ ∀m ∈ M ❡ ∀c ✳ ❖s ♠♦r✜s♠♦s ❡♠ Rat( sã♦ ♠♦r✜s♠♦s

  ∗ c c ) ∗ M

  ❞❡ C ✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❊♥tã♦ ❛s ❝❛t❡❣♦r✐❛s M ❡ Rat( sã♦ c ✐s♦♠♦r❢❛s✱ ❡♠ q✉❡ M é ❛ ❝❛t❡❣♦r✐❛ ❞♦s C✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ k → V ect k ❊①❡♠♣❧♦ ✷✳✶✽ ❖ ❢✉♥t♦r D : V ect ❞♦ ❊①❡♠♣❧♦ 2.11 é ❡q✉✐✲ V ect ✈❛❧❡♥t❡ ❛♦ ❢✉♥t♦r Id k ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ ❝❛❞❛ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V ✱ ❞❡✜✲ ♥✐♠♦s ∗∗

  µ V : V → V v 7→ µ V (v) : V → k V f 7→ f (v).

  ❙❛❜❡♠♦s q✉❡ µ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ Pr♦✈❡♠♦s q✉❡ ❞❛❞♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s V ✱ W ❡ ✉♠ ♠♦r✜s♠♦ f : V → W ✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ µ V ∗∗

  // f D

  V V ∗∗ (f ) W // W . µ W ❙❡❥❛♠ v ∈ V ❡ g ∈ W ✳ ❊♥tã♦

  ((D(f ) ◦ µ V )(v))(g) = (D(f )(µ V (v)))(g) = (µ V (v) ◦ f )(g) = µ V (v)(f (g)) = (µ V (v))(g ◦ f ) = (g ◦ f )(v) = (µ W (f (v)))(g)

  ◦ f )(v))(g), ∗ ∗ ∗ = ((µ W : W → V V =

  ❡♠ q✉❡ f é ❛ ❛♣❧✐❝❛çã♦ ❞✉❛❧ ❞❡ f✳ P♦rt❛♥t♦✱ D(f) ◦ µ µ W ◦ f

  ✳ ❉❡✜♥✐çã♦ ✷✳✶✾ ❯♠ ❢✉♥t♦r F : C → D é ❞✐t♦ ✭✐✮ ❋✐❡❧✱ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♦❜❥❡t♦s X, Y ∈ Ob(C) ❛ ❛♣❧✐❝❛çã♦ s✉❜❥❛❝❡♥t❡ C D F : Hom (X, Y ) → Hom (F (X), F (Y ))

  é ✐♥❥❡t♦r❛✳ ✭✐✐✮ P❧❡♥♦✱ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♦❜❥❡t♦s X, Y ∈ Ob(C) ❛ ❛♣❧✐❝❛çã♦ s✉❜❥❛✲ C D

  (X, Y ) → Hom (F (X), F (Y )) ❝❡♥t❡ F : Hom é s♦❜r❡❥❡t♦r❛✳ ✭✐✐✐✮ ❉❡♥s♦✱ s❡ ♣❛r❛ t♦❞♦ ♦❜❥❡t♦ Z ∈ Ob(D) ❡①✐st✐r ✉♠ ♦❜❥❡t♦ X ∈ Ob(C) t❛❧ q✉❡ F (X) ≃ Z✳ ❚❡♦r❡♠❛ ✷✳✷✵ ❉✉❛s ❝❛t❡❣♦r✐❛s C ❡ D sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r ✉♠ ❢✉♥t♦r ♣❧❡♥♦✱ ✜❡❧ ❡ ❞❡♥s♦ F : C → D✳

  ❉❡♠♦♥str❛çã♦✿ ✭⇒✮ ❈♦♠♦ C ❡ D sã♦ ❡q✉✐✈❛❧❡♥t❡s ❡①✐st❡♠ ❢✉♥t♦r❡s D C F : C → D

  ❡ G : D → C t❛✐s q✉❡ F ◦ G ∼ Id ❡ G ◦ F ∼ Id ✳ ❙❡❥❛♠ X, Y ∈ Ob(C)

  ✳ ❊♥tã♦ C D (X, Y ) → Hom (F (X), F (Y ))

  ❆✜r♠❛çã♦ ✶✿ F : Hom é ✐♥❥❡t♦r❛✳ C (X, Y )

  Pr✐♠❡✐r♦ ♦❜s❡r✈❛♠♦s q✉❡ ♣❛r❛ t♦❞♦ h ∈ Hom ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦ µ X

  // (G ◦ F )(X) h (G◦F )(h)

  X // (G ◦ F )(Y ), C → (G ◦ F ) Y µ Y

  ♣♦✐s µ : Id é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❙❡❥❛♠ f, g ∈ C Hom (X, Y ) t❛✐s q✉❡ F (f) = F (g)✳ ❊♥tã♦ G(F (f)) = G(F (g)) ❡ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡

  (G ◦ F )(f ) ◦ µ X = (G ◦ F )(g) ◦ µ X . P❡❧❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ ❛♥t❡r✐♦r ✭❢❛③❡♥❞♦ h = f ❡ h = g✮✱

  ◦ f = (G ◦ F )(f ) ◦ µ ◦ g µ Y X = (G ◦ F )(g) ◦ µ Y X = µ Y

  ❡ ✐ss♦ r❡s✉❧t❛ q✉❡ f = g✱ ♣♦✐s µ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ♠♦str❛♠♦s q✉❡ G é ✜❡❧✳ C D

  (X, Y ) → Hom (F (X), F (Y )) ❆✜r♠❛çã♦ ✷✿ ❆ ❛♣❧✐❝❛çã♦ F : Hom

  é s♦❜r❡❥❡t♦r❛✳ ❙❡❥❛ g : F (X) → F (Y ) ✉♠ ♠♦r✜s♠♦ ❡♠ D✳ ▼♦str❡♠♦s q✉❡ ❡①✐st❡

  ✉♠ ♠♦r✜s♠♦ h : X → Y t❛❧ q✉❡ F (h) = g✳

  1

  ◦ G(g) ◦ µ X : X → Y ❈♦♥s✐❞❡r❛♠♦s h = µ Y ✱ s❡❣✉❡ q✉❡ h é ✉♠ C

  (X, Y ) ♠♦r✜s♠♦ ❡♠ Hom ✳ P❡❧❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ ❛❝✐♠❛ t❡♠♦s q✉❡ C

  (G ◦ F )(h) ◦ µ X = µ Y ◦ h, ∀h ∈ Hom (X, Y ).

  1

  ◦ G(g) ◦ µ X ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ h = µ Y ✱ s❡❣✉❡ q✉❡ − −

  1

  1

  (G ◦ F )(µ ◦ G(g) ◦ µ Y Y − − X ) ◦ µ X = µ Y ◦ µ ◦ G(g) ◦ µ X = G(g) ◦ µ X .

  1

  1

  ◦ G(g) ◦ µ X )) = (G ◦ F )(µ ◦ G(g) ◦ µ X ) = G(g) ▲♦❣♦✱ G(F (µ Y Y ❡

  1

  ◦ G(g) ◦ µ Y ) = g ❝♦♠♦ G é ✜❡❧✱ s❡❣✉❡ q✉❡ F (h) = F (µ Y ✳ P♦rt❛♥t♦✱ F é ♣❧❡♥♦✳

  ❆✜r♠❛çã♦ ✸✿ F é ❞❡♥s♦✳

  ❉❡ ❢❛t♦✱ s❡❥❛ Z ∈ Ob(D)✳ ➱ ❝❧❛r♦ q✉❡ G(Z) ∈ Ob(C)✳ P♦r ❤✐♣ót❡s❡✱ Z : Z → (F ◦ G)(Z) ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ µ ✱ ♦✉ s❡❥❛✱ F (G(Z)) ≃ Z✳

  ✭⇐✮ ❙❛❜❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ❢✉♥t♦r F : C → D ♣❧❡♥♦✱ ✜❡❧ ❡ ❞❡♥s♦✳ D ▼♦str❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ❢✉♥t♦r G : D → C t❛❧ q✉❡ F ◦ G ∼ Id ❡ C G ◦ F ∼ Id

  ✳ P❛r❛ ❝❛❞❛ ♦❜❥❡t♦ Z ∈ Ob(D)✱ ❡①✐st❡ ✉♠ ♦❜❥❡t♦ X ∈ Ob(C) t❛❧ q✉❡

  F (X) ≃ Z ✱ ♣♦✐s F é ❞❡♥s♦✳ Z : Z → (F ◦ G)(Z)

  ❉❡✜♥✐♠♦s G(Z) = X✳ ❉❡❝♦rr❡ ❞✐ss♦ q✉❡ µ ❡ µ W : W → (F ◦G)(W ) sã♦ ✐s♦♠♦r✜s♠♦s✱ ♣❛r❛ q✉❛✐sq✉❡r Z, W ∈ Ob(D) ✭❡♠ q✉❡ F (Y ) ≃ W ✱ ♣❛r❛ ❛❧❣✉♠ Y ∈ Ob(C)✮✳ D

  (Z, W ) ❙❡❥❛ f ∈ Hom ✱ ♣r❡❝✐s❛♠♦s ❞❡✜♥✐r G(f)✳ ❈♦♠♦ F é ✜❡❧ ❡

  ♣❧❡♥♦✱ ❛ ❛♣❧✐❝❛çã♦ C D Hom (G(Z), G(W )) → Hom (F (G(Z)), F (G(W )))

  é ❜✐❥❡t♦r❛✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ❞✐❛❣r❛♠❛ µ Z //

  (F ◦ G)(Z) f Z // (F ◦ G)(W ). W µ W

  ❖ ♠♦r✜s♠♦ ♣♦♥t✐❧❤❛❞♦ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦

1 D

  µ W ◦ f ◦ µ ∈ Hom (F (G(Z)), F (G(W ))) Z C (G(Z), G(W ))

  ❡✱ ♣❡❧♦ ❞✐t♦ ❛❝✐♠❛✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ h ∈ Hom t❛❧ q✉❡

1 F (h) = µ W ◦ f ◦ µ

  Z ✳ ❆ss✐♠✱ ❞❡✜♥✐♠♦s G(f) = h✳ Z ) = ❱❡r✐✜q✉❡♠♦s q✉❡ G é ❞❡ ❢❛t♦ ✉♠ ❢✉♥t♦r✳ ▼♦str❡♠♦s q✉❡ G(I

  I G ♣❛r❛ t♦❞♦ ♦❜❥❡t♦ Z ∈ Ob(D)✳ ❙❡❣✉♥❞♦ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦

  (Z) µ Z

  // (F ◦ G)(Z) I Z Z

  // (F ◦ G)(Z), Z µ Z Z ◦ I Z ◦ µ = I

  1

  ♦ ♠♦r✜s♠♦ ♣♦♥t✐❧❤❛❞♦ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ♣♦r µ F ❡ C Z (G(Z)) (G(Z), G(Z)) s❛❜❡♥❞♦ q✉❡ F é ✜❡❧ ❡ ♣❧❡♥♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ι ∈ Hom t❛❧

  ) = I q✉❡ F (ι) = I F (G(Z)) ✳ ❈♦♠♦ F é ❢✉♥t♦r✱ s❡❣✉❡ q✉❡ F (I G (Z) F (G(Z)) ✳ )

  ❉❛í✱ F (ι) = F (I G (Z) ❡ ❛ss✐♠✱ ι = I G (Z) ✱ ♣♦✐s F é ✜❡❧✳ P♦r ❞❡✜♥✐çã♦✱ G(I Z ) = ι = I G (Z) ✳

  ❆❣♦r❛✱ s❡❥❛♠ f : Z → W ❡ g : W → K ♠♦r✜s♠♦s ❡♠ D✳ ❊♥✲ C (G(Z), G(W )) tã♦ s❛❜❡♠♦s q✉❡ ❡①✐st❡♠ ú♥✐❝♦s h, h ❡♠ Hom ❡ ❡♠ C

  1 Hom (G(W ), G(K)) W ◦ f ◦ µ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❛✐s q✉❡ F (h) = µ ❡ Z

1 F (h ) = µ K ◦ g ◦ µ

  ′ ′ W ✳ P♦rt❛♥t♦✱ F (h ◦ h) = F (h ) ◦ F (h) − −

  1

  1

  = (µ K ◦ g ◦ µ ) ◦ (µ W ◦ f ◦ µ ) W Z

K ◦ (g ◦ f ) ◦ µ

  1

  = µ Z ◦h = G(g)◦G(f )

  ❡✱ ♣♦r ❞❡✜♥✐çã♦✱ G(g ◦f) = h ✳ ❉♦♥❞❡✱ G é ✉♠ ❢✉♥t♦r✳ D C P❛r❛ ✜♥❛❧✐③❛r✱ ❞❡✈❡♠♦s ♣r♦✈❛r q✉❡ F ◦ G ∼ Id ❡ G ◦ F ∼ Id ✳ D Z :

  P❛r❛ ♠♦str❛r♠♦s q✉❡ F ◦ G ∼ Id ✱ ♥♦t❛♠♦s q✉❡ ∀ Z ∈ Ob(D)✱ µ D Z → (F ◦ G)(Z) (Z, W )

  é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❙❡❥❛ f ∈ Hom ✳ ❘❡st❛✲♥♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❝♦♠✉t❛ ♣❛r❛ ❣❛r❛♥t✐r♠♦s q✉❡ µ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ µ Z

  // (F ◦ G)(Z) Id D (f ) (F ◦G)(f ) Z // (F ◦ G)(W ).

  W µ W

  1 W

  ◦ f ◦ µ P❡❧♦ q✉❡ ❞❡s❡♥✈♦❧✈❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ (F ◦ G)(f) = µ Z Z = µ W ◦ f

  ❡ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡ (F ◦ G)(f) ◦ µ ✳ C ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ G◦F ∼ Id ✳ ❙❡❥❛ X ∈ Ob(C)✳ ❊♥tã♦ F (X) ∈ ′ ′

  Ob(D) ∈ Ob(C) ) ≃ F (X) ❡ ❛ss✐♠✱ ❡①✐st❡ X t❛❧ q✉❡ F (X ✭F é ❞❡♥s♦✮✳ ′ ′

  ) = F (G(F (X))) = ❉❛í✱ G(F (X)) = X ❡ ♣♦rt❛♥t♦✱ F (X) ≃ F (X

  (F ◦ G)(F (X)) F : F (X) → (F ◦ G)(F (X)) ✳ ▲♦❣♦✱ µ (X) é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❈♦♠♦ F X : X → (G ◦ F )(X) X ) =

  é ♣❧❡♥♦✱ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ α t❛❧ q✉❡ F (α µ F (X) ✳ X F

  ▼♦str❡♠♦s q✉❡ α é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ µ (X) é ✐s♦✲ : (F ◦ G)(F (X)) → F (X) F ◦

  ♠♦r✜s♠♦✱ ❡①✐st❡ µ t❛❧ q✉❡ µ (X) F ′ ′ (X) ◦ µ

  µ = I F = I F F (F ◦G)(F (X)) ❡ µ F (X) (X) ✳ ◆♦✈❛♠❡♥t❡ ♣♦r s❡r F

  (X) (X)

′ ′ ′

  : (G ◦ F )(X) → X ) = µ ♣❧❡♥♦✱ ❡①✐st❡ α X t❛❧ q✉❡ F (α X ✳ ◆♦t❡♠♦s F (X) q✉❡ ′ ′

  F (α X ◦ α ) = F (α X X ) ◦ F (α ) X = µ ◦ µ F (X) F

  (X)

  =

  I

  (F ◦G)(F (X))

  =

  I F

  (G(F (X)))

  = F (I )

  (G◦F )(X)

  X ◦ α = I

  ❡ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡ α X (G◦F )(X) ✱ ♣♦✐s F é ✜❡❧✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ◦ α X = I X

  é ♣♦ssí✈❡❧ ♠♦str❛r♠♦s q✉❡ α X ✳ ❋✐♥❛❧♠❡♥t❡✱ r❡st❛✲♥♦s ♠♦str❛r q✉❡ ❛ ❝♦❧❡çã♦ ❞❡ ♠♦r✜s♠♦s

  α = {α X : X → (G ◦ F )(X) : X ∈ Ob(C)} C (X, Y )

  é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❙❡❥❛♠ X, Y ∈ Ob(C) ❡ f ∈ Hom ✳ ▼♦str❡♠♦s q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ α X

  // (G ◦ F )(X) Id C (f ) (G◦F )(f )

  X // (G ◦ F )(Y ). Y α Y D

  ❉❡ ❢❛t♦✱ s❛❜❡♠♦s q✉❡ F ◦ G ∼ Id ❡ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡ ♦ ❞✐❛❣r❛♠❛ µ F (X) // F F (X) (F ◦ G)(F (X))

  

(f ) (F ◦G)(F (f ))

  F (Y ) // (F ◦ G)(F (Y )) F ◦ F (f ) = (F ◦ G)(F (f )) ◦ µ F µ F (Y ) ❝♦♠✉t❛✳ P♦rt❛♥t♦✱ µ (Y ) (X) ♦ q✉❡ ✐♠♣❧✐❝❛ F (α Y ) ◦ F (f ) = (F ◦ G)(F (f )) ◦ F (α X )

  ❡ ❛ss✐♠✱ F (α Y ◦ f ) = F ((G ◦ F )(f ) ◦ α Y ◦ f = (G ◦ F )(f ) ◦ α X ). X

  ❈♦♠♦ F é ✜❡❧✱ s❡❣✉❡ q✉❡ α ✳ Pr♦♣♦s✐çã♦ ✷✳✷✶ ❙❡❥❛ F : C → D ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❝❛t❡❣♦r✐❛s✳ ❯♠ ♠♦r✜s♠♦ f : X → Y é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C ✭❡♣✐♠♦r✜s♠♦ ❡♠ C

  ✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ F (f) é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠ D ✭❡♣✐♠♦r✜s♠♦ ❡♠ D

  ✮✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ F é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛✱ ❡①✐st❡ ✉♠ ❢✉♥t♦r G : D D C

  → C t❛❧ q✉❡ F ◦ G ∼ Id ❡ G ◦ F ∼ Id ✳ ▼♦str❡♠♦s ♦ ❝❛s♦ ❞♦ ♠♦♥♦♠♦r✜s♠♦✱ ♣♦✐s ♦ ❞♦ ❡♣✐♠♦r✜s♠♦ é ❛♥á❧♦❣♦✳ C

  → G ◦ F ✭⇒✮ ❙✉♣♦♥❤❛♠♦s µ : Id ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❆ss✐♠✱ X

  ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ Ob(C) ❡ q✉❛❧q✉❡r ♠♦r✜s♠♦ f : X → Y ✱ µ ❡ µ Y sã♦ ✐s♦♠♦r✜s♠♦s ❡ ✈❛❧❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ µ X

  // f X (G ◦ F )(X)

  (G◦F )(f ) // (G ◦ F )(Y ).

  Y µ Y

  ❙❡❥❛♠ i, j : Z → F (X) ♠♦r✜s♠♦s ❡♠ D t❛✐s q✉❡ F (f)◦i = F (f)◦j✳ ❊♥tã♦ G(F (f) ◦ i) = G(F (f) ◦ j)✳ ▼❛s

  G(F (f ) ◦ i) = (G ◦ F )(f ) ◦ G(i)

  1

  = ((G ◦ F )(f ) ◦ µ X ) ◦ µ ◦ G(i) X

  1 ◦ f ◦ µ ◦ G(i).

  = µ Y X Y ◦ f ◦ ❉❡ ♠❛♥❡✐r❛ ✐♥t❡✐r❛♠❡♥t❡ ❛♥á❧♦❣❛✱ s❡❣✉❡ q✉❡ G(F (f) ◦ j) = µ

  1

  µ ◦ G(j) X ✳ ▲♦❣♦✱ − −

  1

  1

  µ Y ◦ f ◦ µ ◦ G(i) = µ Y ◦ f ◦ µ ◦ G(j), − − X X

  1

  

1

  ◦G(i) = f ◦µ ◦G(j) ♦ q✉❡ ✐♠♣❧✐❝❛✱ f ◦µ ❡ ❝♦♠♦ f é ♠♦♥♦♠♦r✜s♠♦✱ − − X X

  1

  1

  ◦ G(i) = µ ◦ G(j) s❡❣✉❡ q✉❡ µ X X ✳ P♦rt❛♥t♦✱ G(i) = G(j)✳ ❈♦♠♦ G é ✜❡❧✱ ♣♦✐s t❡♠♦s ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❝❛t❡❣♦r✐❛s✱ s❡❣✉❡ q✉❡ i = j✳ ▲♦❣♦✱ F (f) é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳

  ✭⇐✮ ❙❡❥❛♠ g, h : W → X ♠♦r✜s♠♦s ❡♠ C t❛✐s q✉❡ f ◦ g = f ◦ h✳ ❊♥tã♦ F (f ◦ g) = F (f ◦ h)✳ P♦rt❛♥t♦✱ F (f) ◦ F (g) = F (f) ◦ F (h) ❡ s❡♥❞♦ F (f) ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ F (g) = F (h)✳ ❈♦♠♦ F é ✜❡❧✱ ♣♦✐s t❡♠♦s ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❝❛t❡❣♦r✐❛s✱ s❡❣✉❡ q✉❡ g = h✳ ▲♦❣♦✱ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳

  ❉❡✜♥✐♠♦s ❛❣♦r❛ ❞♦✐s t✐♣♦s ❞❡ ❝♦♠♣♦s✐çã♦ ❡♥tr❡ tr❛♥s❢♦r♠❛çõ❡s ♥❛✲ t✉r❛✐s✳ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s✱ F, G, H : C → D ❢✉♥t♦r❡s ❡ µ : F → G ❡

  λ : G → H tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s✳ ❆ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ λ ◦ µ : ◦ µ

  F → H X = λ X X ❞❛❞❛ ♣♦r (λ ◦ µ) ♣❛r❛ t♦❞♦ ♦❜❥❡t♦ X ❡♠ C é

  ❝❤❛♠❛❞❛ ❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s µ ❡ λ✳ ❚❛❧ ❝♦♠♣♦s✐çã♦ ♣♦❞❡ s❡r ✈✐s✉❛❧✐③❛❞❛ ❝♦♠♦ F G µ //

  C D H λ // // .

  ❱❡r✐✜q✉❡♠♦s q✉❡ ❡ss❛ ❝♦♠♣♦s✐çã♦ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❙❡❥❛ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ▼♦str❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❝♦♠✉t❛

  F (X) λ X ◦µ X // F

  ❚❛❧ ❝♦♠♣♦s✐çã♦ ♣♦❞❡ s❡r ✈✐s✉❛❧✐③❛❞❛ ❝♦♠♦ F // µ J // λ C D E

  

(λ◦µ) Y

  (J ◦ F )(Y )

  (H◦G)(f )

  (H ◦ G)(X)

  (J◦F )(f )

  //

  

(λ◦µ)

X

  ✳ ❊♥tã♦ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ (J ◦ F )(X)

  // H // . ▼♦str❡♠♦s q✉❡ ❡ss❛ ❝♦♠♣♦s✐çã♦ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❙❡❥❛ f ∈ Hom C (X, Y )

  G

  ◦J(µ X ) ♣❛r❛ t♦❞♦ X ∈ Ob(C)✳

  (f )

  ❡ µ : F → G ❡ λ : J → H tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s✳ ❆ ❝♦♠♣♦s✐çã♦ ❤♦r✐③♦♥t❛❧ λ◦µ : J ◦F → H ◦G é ❞❡✜♥✐❞❛ ♣♦r (λ◦µ) X = λ G (X)

  ❡♠ ✭✯✮ ❡ ❡♠ ✭✯✯✮ ✉t✐❧✐③❛♠♦s ❛s ♥❛t✉r❛❧✐❞❛❞❡s ❞❡ µ ❡ λ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛♠ C✱ D ❡ E ❝❛t❡❣♦r✐❛s✱ F, G : C → D ❡ J, H : D → E ❢✉♥t♦r❡s

  = H(f ) ◦ λ X ◦ µ X = H(f ) ◦ (λ ◦ µ) X ,

  

(∗∗)

  = λ Y ◦ G(f ) ◦ µ X

  

(∗)

  (λ ◦ µ) Y ◦ F (f ) = λ Y ◦ µ Y ◦ F (f )

  F (Y ) λ Y ◦µ Y // H(Y ). ❉❡ ❢❛t♦✱

  (f )

  H(X) H

  // (H ◦ G)(Y )

  é ❝♦♠✉t❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡♠♦s ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✲ ✈♦s µ G (X) X λ

  // // F (X) G(X) (J ◦ G)(X) (H ◦ G)(X) F (f ) G (f ) (J◦G)(f ) (H◦G)(f ) ❡ // G(Y ) // (H ◦ G)(Y ).

  F (Y ) (J ◦ G)(Y ) µ Y λ G (Y ) X = µ Y ◦ F (f ) X ) = ❚❡♠♦s q✉❡ G(f) ◦ µ ❡ ✐ss♦ ✐♠♣❧✐❝❛ J(G(f) ◦ µ

  ◦ F (f )) J(µ Y X ) = J(µ Y ) ◦ J(F (f ))

  ✱ ♦✉ s❡❥❛✱ J(G(f)) ◦ J(µ ✳ ❆ss✐♠✱ (λ ◦ µ) Y ◦ (J ◦ F )(f ) = λ ◦ J(µ Y ) ◦ J(F (f )) G (Y )

  = λ ◦ J(G(f )) ◦ J(µ G (Y ) X )

  (∗)

  ◦ J(µ = H(G(f )) ◦ λ G X )

  (X)

  = (H ◦ G)(f ) ◦ (λ ◦ µ) X . ❆ ✐❣✉❛❧❞❛❞❡ ✭✯✮ s❡❣✉❡ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛✳

  ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡♠♦s ♠❛✐s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❝❛t❡❣♦r✐❛✳ ❊①❡♠♣❧♦ ✷✳✷✷ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s ♣❡q✉❡♥❛s✳ ❉❡♥♦t❛♠♦s ♣♦r F un(C, D)

  ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ❢✉♥t♦r❡s F : C → D✳ ❉❛❞♦s ❢✉♥t♦r❡s F, G : C → D✱ ❝❤❛♠❛♠♦s Nat(F, G) ♦ ❝♦♥❥✉♥t♦ ❞❛s tr❛♥s❢♦r✲ ♠❛çõ❡s ♥❛t✉r❛✐s µ : F → G✳

  Pr♦✈❛♠♦s ❛❣♦r❛ ✉♠ ❞♦s ✐♠♣♦rt❛♥t❡s r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡ ❝❛t❡✲ ❣♦r✐❛s✱ ♦ ▲❡♠❛ ❞❡ ❨♦♥❡❞❛✳ X : C → Set

  ❙❡❥❛ X ✉♠ ♦❜❥❡t♦ ✜①♦ ❡♠ C✳ ❉❡✜♥✐♠♦s ♦ ❢✉♥t♦r L ♣♦r C L X (Y ) = Hom (X, Y )

  ✱ ∀ Y ∈ Ob(C)✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ❝❛❞❛ ♠♦r✜s♠♦ C α : Y → Z X (α)(f ) = α ◦ f (X, Y )

  ❞❡✜♥✐♠♦s L ✱ ♣❛r❛ t♦❞♦ f ∈ Hom ✳ X ❉✐③❡♠♦s q✉❡ ♦ ❢✉♥t♦r L é r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ♦❜❥❡t♦ X✳ ❯♠ ❢✉♥t♦r

  ❞✐③✲s❡ r❡♣r❡s❡♥tá✈❡❧ s❡ ♦ ♠❡s♠♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❢✉♥t♦r r❡♣r❡s❡♥t❛❞♦ ♣♦r ❛❧❣✉♠ ♦❜❥❡t♦✳ ▲❡♠❛ ✷✳✷✸ ✭▲❡♠❛ ❞❡ ❨♦♥❡❞❛✮ ❙❡❥❛♠ F : C → Set ✉♠ ❢✉♥t♦r ❡ X ∈ Ob(C) X , F )

  ✳ ❊♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s Nat(L ❡stá ❡♠ ❜✐❥❡çã♦ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ F (X) ♣❡❧❛ ❢✉♥çã♦

  φ : N at(L X , F ) → F (X) 7→ µ

  µ X (I X ).

  X , F )

  ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ µ ∈ Nat(L ✱ Y ∈ Ob(C) ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❊♥tã♦ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ µ X

  // L F L X (X) F (X) X (f ) (f ) Y ◦ L X (f ) = F (f ) ◦ µ L X (Y ) // F (Y ) X µ Y X (f )(I X ) = ♦✉ s❡❥❛✱ µ ✳ ❖❜s❡r✈❛♠♦s q✉❡ L f ◦ I X = f

  ❡ ♣♦rt❛♥t♦✱ µ Y (f ) = µ Y (L X (f )(I X ))

  ◦ L = (µ Y X (f ))(I X X X ) = (F (f ) ◦ µ )(I ).

  ✭✷✳✶✮ ❉❡✜♥✐♠♦s

  ψ : F (X) → N at(L X , F ) x 7→ ψ(x) : L X → F C (X, Y ) t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ Y ∈ Ob(C) ❡ ❝❛❞❛ f ∈ Hom ✱ Y (f ) = F (f )(x) (ψ(x)) Y : L X (Y ) → F (Y )

  é ❞❡✜♥✐❞❛ ♣♦r (ψ(x)) ✳ ▼♦str❡♠♦s q✉❡ ψ é ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❡ φ✳

  ➱ ♥❡❝❡ssár✐♦ ♣r♦✈❛r♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ x ∈ F (X)✱ ψ(x) : L X → F

  é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❙❡❥❛♠ Y, Z ∈ Ob(C) ❡ f : Y → Z ✉♠ ♠♦r✜s♠♦✳ ❱❡❥❛♠♦s q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

  

(ψ(x))

Y

  // L F X (f ) (f ) L X (Y ) F (Y ) L X (Z) // F (Z). C

(ψ(x))

Z

  (X, Y ) ❙❡❥❛ g ∈ Hom ✳ ❊♥tã♦

  (F (f ) ◦ (ψ(x)) Y )(g) = F (f )((ψ(x)) Y (g)) = F (f )(F (g)(x)) = (F (f ) ◦ F (g))(x) = F (f ◦ g)(x) = (ψ(x)) Z (f ◦ g) = (ψ(x)) Z (L X (f )(g))

  ◦ L = ((ψ(x)) Z X (f ))(g). N at ,F X , F )

  ❋✐♥❛❧♠❡♥t❡✱ ✈❡r✐✜q✉❡♠♦s q✉❡ ψ◦φ = I (L ) ✳ ❙❡❥❛ µ ∈ Nat(L ✳ X ❊♥tã♦✱ ♣❛r❛ t♦❞♦ f : X → Y ✱ t❡♠♦s

  (ψ ◦ φ)(µ) = ψ(φ(µ)) = ψ(µ X (I X ))

  ❡ ❛ss✐♠✱ ((ψ ◦ φ)(µ)) Y (f ) = (ψ(µ X (I X ))) Y (f )

  = F (f )(µ X (I X )) = (F (f ) ◦ µ X )(I X )

  (2.1)

  = µ Y (f ) = (I (µ)) Y (f ). N at ,F

  (L X ) Y = (I N at ,F (µ)) Y

  ❈♦♠♦ f é ❛r❜✐trár✐♦✱ s❡❣✉❡ q✉❡ ((ψ ◦ φ)(µ)) (L X ) ♣❛r❛ t♦❞♦ ♦❜❥❡t♦ Y ❡ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s sã♦ N at ,F (µ) ✐❣✉❛✐s✱ ✐✳❡✳✱ (ψ ◦ φ)(µ) = I (L X ) ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞♦ x ∈ F (X)

  (φ ◦ ψ)(x) = φ(ψ(x)) = (ψ(x)) X (I X ) = F (I X )(x) = I F (x).

  (X)

  Pr♦♣♦s✐çã♦ ✷✳✷✹ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ X, Y ♦❜❥❡t♦s ❡♠ C✳ ❊♥tã♦ ∼ L

  L X Y s❡✱ ❡ s♦♠❡♥t❡ s❡✱ X ≃ Y ✳ X → L Y ❉❡♠♦♥str❛çã♦✿ ✭⇒✮ ❙❡❥❛ µ : L ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ X (I X ) : Y → X Y : L X (Y ) → ❈♦♥s✐❞❡r❡♠♦s ♦ ♠♦r✜s♠♦ µ ✳ ❉♦ ❢❛t♦ ❞❡ µ L Y (Y ) Y (f ) = I Y s❡r ✐s♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ f : X → Y t❛❧ q✉❡ µ ✳ P❡❧❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ µ✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ µ X

  // L (f ) L (f ) X Y L X (X) L Y (X) Y (f ) ◦ µ L X )(I X (Y ) // L Y (Y ). X ) = L Y (f )(µ µ Y X (I X )) = f ◦ µ X (I X ).

  ❚❡♠♦s q✉❡ (L ❉❛í✱

  ♣❡❧❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛✱ s❡❣✉❡ q✉❡ f ◦ µ X (I X ) = (L Y (f ) ◦ µ X )(I X ) ◦ L

  = (µ Y X (f ))(I X ) = µ Y (L X (f )(I X )) = µ Y (f ) = I Y .

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ µ Y // L (µ (I )) L (µ (I )) X X X Y L X (Y ) L Y (Y ) X X

  // L L X (X) Y (X). µ X X (Y )

  ❈♦♠✉t❛♥❞♦ ❡ss❡ ❞✐❛❣r❛♠❛✱ ♣❛r❛ f ∈ L ✱ ♦❜t❡♠♦s q✉❡ µ X (I X ) ◦ f = L X (µ X (I X ))(f )

  1

  = µ ◦ (µ X X ◦ L X (µ X (I X )))(f )

  1

  = µ ◦ (L Y (µ X X (I X )) ◦ µ Y )(f )

  1

  ◦ (µ = µ X X (I X ) ◦ µ Y (f ))

  1

  = µ ◦ (µ X X (I X ) ◦ I Y ) =

  I X . ▲♦❣♦✱ f é ✉♠ ✐s♦♠♦r✜s♠♦ ❡ ♣♦rt❛♥t♦✱ X ≃ Y ✳ ✭⇐✮ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ✉♠ ✐s♦♠♦r✜s♠♦ f : Y → X✳ ❉❡✜♥✐♠♦s

  µ : L X → L Y ❢❛③❡♥❞♦✱ ♣❛r❛ ❝❛❞❛ Z ∈ Ob(C)✱

  µ Z : L X (Z) → L Y (Z) g 7→ g ◦ f. Z ❖ ❢❛t♦ ❞❡ q✉❡✱ ♣❛r❛ t♦❞♦ Z ∈ Ob(C)✱ µ é ✐s♦♠♦r✜s♠♦ é ✐♠❡❞✐❛t♦✱

  ✈✐st♦ q✉❡ f é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❱❡r✐✜q✉❡♠♦s q✉❡ µ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ P❛r❛ ✐ss♦✱ ♠♦str❡♠♦s q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ µ Z

  // L L L X (Z) L Y (Z)

X Y

(g) (g) L X (W ) // L Y (W ), µ W

  ♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦s Z ❡ W ❡ ♠♦r✜s♠♦ g : Z → W ✳ ❙❡❥❛ α : X → Z ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❊♥tã♦

  (µ W ◦ L X (g))(α) = µ W (g ◦ α) = (g ◦ α) ◦ f = g ◦ (α ◦ f ) = L Y (g)((α ◦ f )) = L Y (g)(µ Z (α)) = (L Y (g) ◦ µ Z )(α).

  Pr♦♣♦s✐çã♦ ✷✳✷✺ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ X, Y ∈ Ob(C)✳ ❖ ❝♦♣r♦✲ ❞✉t♦ ❞❡ X ❡ Y ❡①✐st❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ❢✉♥t♦r F : C → Set ❞❡✜♥✐❞♦ C C

  (X, Z) × Hom (Y, Z) ♣♦r F (Z) = Hom ♣❛r❛ t♦❞♦ Z ∈ Ob(C) é r❡♣r❡✲ s❡♥tá✈❡❧✳ ❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❛♠♦s q✉❡✱ ❞❛❞♦ ✉♠ ♠♦r✜s♠♦ f : Z → W ❡♠ C

  ✱ t❡♠♦s C C C C F (f ) : Hom (X, Z) × Hom (Y, Z) → Hom (X, W ) × Hom (Y, W ) (g, h) 7→ (f ◦ g, f ◦ h).

  ✭⇒✮ P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ ♦ ❝♦♣r♦❞✉t♦ ❞❡ X ❡ Y ✱ q✉❡ ❞❡♥♦t❛♠♦s X , i Y ) ♣❡❧❛ tr✐♣❧❛ (X ⊔ Y, i ✳ ❆✜r♠❛♠♦s q✉❡ ♦ ❢✉♥t♦r F é ❡q✉✐✈❛❧❡♥t❡ ❛♦ X⊔Y X⊔Y → F ❢✉♥t♦r L ✳ P❛r❛ ✐ss♦✱ ❞❡✜♥✐♠♦s µ : L t❛❧ q✉❡✱ ♣❛r❛ ❝❛❞❛ A ∈ Ob(C)

  ✱ C C C µ A : Hom (X ⊔ Y, A) → Hom (X, A) × Hom (Y, A)

  φ 7→ (φ ◦ i C C X , φ ◦ i Y ).

  (X, A) × Hom (Y, A) ❙❡❥❛ (g, h) ∈ Hom ✳ ◆❡ss❡ ❝❛s♦✱ g i

  X ww X yy oo

  A X ⊔ Y α ee h gg i Y Y X Y

  ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ α : X ⊔Y → A t❛❧ q✉❡ g = α◦i ❡ h = α◦i ✳ A (α) = (α ◦ i X , α ◦ i Y ) = (g, h) A ❆ss✐♠✱ µ ✱ ♦✉ s❡❥❛✱ µ é s♦❜r❡❥❡t♦r❛✳

  , α : X ⊔ Y → A A (α ) = µ A (α ) ❙❡❥❛♠ ❛❣♦r❛ α

  1 2 t❛✐s q✉❡ µ

  1 2 ✱ ♦✉ s❡❥❛✱

  α ◦ i X = α ◦ i X ◦ i Y = α ◦ i Y = α

  1 2 ❡ α A

  1 2 ❡ s❡❣✉❡ ❞❛ ✉♥✐❝✐❞❛❞❡ q✉❡ α

  1 2 ✳

  P♦rt❛♥t♦✱ µ é ✐♥❥❡t♦r❛✳ ❋❛❧t❛ ✈❡r✐✜❝❛r♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ µ✳ Pr♦✈❡♠♦s q✉❡ ♦ s❡❣✉✐♥t❡

  ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ ♣❛r❛ ✉♠ ♠♦r✜s♠♦ α : A → B✱ ✐st♦ é✱ µ A C C C // Hom (X ⊔ Y, A) Hom (X, A) × Hom (Y, A) L F X⊔Y C // Hom C C (α) (α) Hom (X ⊔ Y, B) (X, B) × Hom (Y, B). µ B

  ❉❡ ❢❛t♦✱ ❞❛❞♦ f : X ⊔ Y → A✱ t❡♠♦s (F (α) ◦ µ A )(f ) = F (α)(µ A (f ))

  = F (α)(f ◦ i X , f ◦ i Y ) = (α ◦ (f ◦ i X ), α ◦ (f ◦ i Y )) = ((α ◦ f ) ◦ i X , (α ◦ f ) ◦ i Y ) = µ B (α ◦ f ) = µ B (L X⊔Y (α)(f )) = (µ B ◦ L X⊔Y (α))(f ).

  ✭⇐✮ P♦r ❤✐♣ót❡s❡✱ F : C → Set é r❡♣r❡s❡♥tá✈❡❧ ❡ ❞❛í✱ ❡①✐st❡ C ∈ Ob(C) C t❛❧ q✉❡ F ∼ L ❛tr❛✈és ❞♦ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ µ : F → L C C X , i Y ) ∈

  ✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ µ é ✉♠ ✐s♦♠♦r✜s♠♦✱ ❡♥tã♦ ❡①✐st❡ (i C C Hom (X, C) × Hom (Y, C) C (i X , i Y ) = I C t❛❧ q✉❡ µ ✳ X , i Y )

  ❆✜r♠❛çã♦✿ ❆ tr✐♣❧❛ (C, i é ✉♠ ❝♦♣r♦❞✉t♦ ❞❡ X ❡ Y ✳ ❉❡ X , j Y ) X ∈ Hom (X, D) Y ∈ C ❢❛t♦✱ s❡❥❛ (D, j ♦✉tr❛ tr✐♣❧❛ ❝♦♠ j ❡ j C C

  Hom (Y, D) D (C, D) D (j ✳ ❈♦♠♦ µ é ✉♠ ✐s♦♠♦r✜s♠♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ φ ∈ Hom X , j Y ) = φ t❛❧ q✉❡ µ ✳ ❆❣♦r❛✱ ✉t✐❧✐③❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ µ ❝♦♠ ♦ ♠♦r✜s♠♦ φ ❡ ♦ ❞✐❛✲

  ❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ µ C C C C // Hom (X, C) × Hom (Y, C) Hom (C, C) F L C C C (φ) (φ) C Hom (X, D) × Hom (Y, D) // Hom (C, D). µ D

  C (φ)(I C ) = φ ◦ I C = φ

  ❖❜s❡r✈❛♠♦s q✉❡ L ✳ ❉❛í✱ φ = L C (φ)(I C )

  = L C (φ)(µ C (i X , i Y )) = (L C (φ) ◦ µ C )(i D ◦ F (φ))(i X Y X , i Y )

  = (µ , i ) = µ D (F (φ)(i X , i Y )) = µ D (φ ◦ i X , φ ◦ i Y ).

1 X , j Y ) = µ

  ◦ φ = (φ ◦ i X , φ ◦ i Y ) X = j X P♦rt❛♥t♦✱ (j D ✳ ❆ss✐♠✱ φ ◦ i Y = j Y ❡ φ ◦ i ✳

  ✷✳✷ ❋✉♥t♦r❡s ❛❞❥✉♥t♦s

  ❉❡✜♥✐çã♦ ✷✳✷✻ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s✳ ❯♠❛ ❛❞❥✉♥çã♦ ❞❡ C ❛ D é ✉♠❛ tr✐♣❧❛ (F, G, φ)✱ ❡♠ q✉❡ F : C → D ❡ G : D → C sã♦ ❢✉♥t♦r❡s X,Y : Hom (F (X), Y ) → Hom (X, G(Y )) : X ∈ Ob(C) D C ❡ {φ

  ❡ Y ∈ Ob(D)}

  é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s✳ ◆❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♦s ❢✉♥t♦r❡s D D op op

  Hom (−, −) ◦ (F × Id ) : C × D → D × D → Set ❡ C C op op op

  Hom (−, −) ◦ (Id × G) : C × D → C × C → Set, C D op (X, U ), Hom (Y, V )) t❛✐s q✉❡✱ ♣❛r❛ ❝❛❞❛ ♠♦r✜s♠♦ (f, g) ∈ (Hom C D

  (U, X), Hom (Y, V )) ✭♦✉ (f, g) ∈ (Hom ✮✱ t❡♠♦s D D D

  (Hom (−, −) ◦ (F × Id ))(f, g) = Hom (F (f ), g), ❡♠ q✉❡ D D → Hom D

  Hom (F (f ), g) : Hom (F (X), Y ) (F (U ), V ) 7→

  α g ◦ α ◦ F (f ) ❡ C C C op

  (Hom (−, −) ◦ (Id × G))(f, g) = Hom (f, G(g)), ❡♠ q✉❡ C C C

  Hom (f, G(g)) : Hom (X, G(Y )) → Hom (U, G(V )) 7→ β G(g) ◦ β ◦ f.

  ❋✐♥❛❧♠❡♥t❡✱ D D C C op φ : Hom (−, −) ◦ (F × Id ) → Hom (−, −) ◦ (Id × G)

  é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❡ s✉❛ ♥❛t✉r❛❧✐❞❛❞❡ é ❡①♣r❡ss❛ ♣❡❧❛ ❝♦♠✉t❛t✐✲ ✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ D // C φ X,Y Hom Hom D (F (f ),g) C (f,G(g)) Hom (F (X), Y ) Hom (X, G(Y )) D C Hom (F (U ), V ) // Hom (U, G(V )). φ U,V

  ◆❛ tr✐♣❧❛ (F, G, φ) ♦ ❢✉♥t♦r F é ❝❤❛♠❛❞♦ ❞❡ ❛❞❥✉♥t♦ à ❡sq✉❡r❞❛ ❞❡ G

  ❡ ♦ ❢✉♥t♦r G é ❝❤❛♠❛❞♦ ❞❡ ❛❞❥✉♥t♦ à ❞✐r❡✐t❛ ❞❡ F ✳ ❖❜s❡r✈❛çã♦ ✷✳✷✼ ❖ t❡r♠♦ ❛❞❥✉♥çã♦ s❡ ❞❡✈❡ ❛ s✐♠❡tr✐❛ ❞♦s sí♠❜♦❧♦s ♥♦ ✐s♦♠♦r✜s♠♦✱ ♣♦✐s ♣❛r❛ ♦❜❥❡t♦s X ∈ Ob(C) ❡ Y ∈ Ob(D) t❡♠♦s ♦ D C

  (F (X), Y ) ≃ Hom (X, G(Y )) ✐s♦♠♦r✜s♠♦ Hom q✉❡ é ❝♦♠♣❛rá✈❡❧ à ❞❡✜♥✐çã♦ ❞❡ ♦♣❡r❛❞♦r❡s ❛❞❥✉♥t♦s ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt (H, h·, ·i)✱ s❡ T : H → H ❡ L : H → H sã♦ ♦♣❡r❛❞♦r❡s ❛❞❥✉♥t♦s ❡♥tã♦ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ hT (x), yi = hx, L(y)i ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ H✳ ❚❡♦r❡♠❛ ✷✳✷✽ ✭❬✶❪✱ Pr♦♣♦s✐t✐♦♥ ✾✳✹✱ ✾✳✺✮ ❙❡❥❛♠ F : C → D ❡ G : D → C

  ❢✉♥t♦r❡s✳ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮ (F, G, φ) é ✉♠❛ ❛❞❥✉♥çã♦✳ D C

  → G ◦ F ✭✐✐✮ ❊①✐st❡♠ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s e : F ◦G → Id ❡ c : Id t❛✐s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r Y ∈ Ob(D) ❡ X ∈ Ob(C)✱ ✈❛❧❡♠ ❛s ✐❣✉❛❧❞❛❞❡s✿

  I G = G(e Y ) ◦ c G

  (Y ) (Y ) ✭✷✳✷✮

  ❡ I = e ◦ F (c F (X) F (X) ✭✷✳✸✮ X ). C

  → G ◦ F ✭✐✐✐✮ ❊①✐st❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ c : Id ❝♦♠ ❛ ♣r♦♣r✐❡✲ ❞❛❞❡ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r X ∈ Ob(C)✱ Y ∈ Ob(D) ❡ q✉❛❧q✉❡r ♠♦r✜s♠♦ f : X → G(Y )

  ❡♠ C ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ g : F (X) → Y ❡♠ D X t❛❧ q✉❡ f = G(g) ◦ c ✱ ♦✉ s❡❥❛✱ t❡♠♦s ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ ❛❜❛✐①♦ c X //

  G(F (X)) F (X)

  X H H H H H G (g) g H f H

  H ##H

  G(Y ) Y.

  D

  ✭✐✈✮ ❊①✐st❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ e : F ◦ G → Id ❝♦♠ ❛ ♣r♦♣r✐❡✲ ❞❛❞❡ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r X ∈ Ob(C)✱ Y ∈ Ob(D) ❡ q✉❛❧q✉❡r ♠♦r✜s♠♦ g : F (X) → Y Y ◦ F (f ) ❡♠ D ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ f : X → G(Y ) ❡♠ C t❛❧ q✉❡ g = e ✳ ❚❛❧ ♣r♦♣r✐❡❞❛❞❡ é r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ ❞✐❛❣r❛♠❛

  ❝♦♠✉t❛t✐✈♦ F (X)

  X g w ww F (f ) f ww ww

  {{ww oo F (G(Y )) G(Y ). Y e Y

  ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ⇒ ✭✐✐✮ ❙❡❥❛♠ X ∈ Ob(C) ❡ Y ∈ Ob(D)✱ ❞❡✜✲ Y = φ (I G )

1 X = φ X,F (I F )

  ♥✐♠♦s e (Y ) ❡ c (X) (X) ✳ ❈♦♥s✐❞❡r❡♠♦s G (Y ),Y g : F (X) → Y ✉♠ ♠♦r✜s♠♦ ❡♠ D✳ ❉❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ φ ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ D // C φ X,F (X) Hom D (F (I ),g) Hom C (I ,G (g)) Hom (F (X), F (X)) Hom (X, G(F (X))) X D // Hom C X Hom (F (X), Y ) (X, G(Y )). φ X,Y

  ❚❡♠♦s q✉❡ C D (Hom (I X , G(g))◦φ )(I ) = (φ X,Y ◦Hom (F (I X,F (X) F (X) F (X) X ), g))(I ) q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ C

  Hom (I X , G(g))(φ X,F (I F )) = φ X,Y (g ◦ I F ◦ F (I X )),

  (X) (X) (X)

  ♦✉ s❡❥❛✱ G(g) ◦ φ X,F (I F ) = φ X,Y (g). Y Y ) ◦ c

(X) (X)

X = ❋❛③❡♥❞♦ g = e s❡❣✉❡✱ ❞❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ q✉❡ G(e

  φ X,Y (e Y ) Y ) ◦ c = φ (e Y ) = I ❡ ♣❛r❛ X = G(Y )✱ G(e G (Y ) G (Y ),Y G (Y ) ✱ Y ) ◦ c = I

  ✐st♦ é✱ G(e G (Y ) G (Y ) ✳ ❆ ♦✉tr❛ ✐❣✉❛❧❞❛❞❡ é ♦❜t✐❞❛ ❞❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r✳

  ❋✐♥❛❧♠❡♥t❡✱ ♣r♦✈❡♠♦s q✉❡ c é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❙❡❥❛♠ X, Y ∈ Ob(C)

  ❡ f : X → Y ✳ ❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ c X //

  (G ◦ F )(X) f (G◦F )(f )

  X // (G ◦ F )(Y ). Y c Y

  P❛r❛ ♣r♦✈❛r♠♦s ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ ❛❝✐♠❛✱ ✉t✐❧✐③❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ φ q✉❡ ❝♦♠✉t❛ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s D // C φ X,F (X) Hom D (F (I ),F (f )) C (I (F (f ))) Hom (F (X), F (X)) Hom (X, G(F (X))) X D C Hom ,G X Hom (F (X), F (Y )) // Hom (X, G(F (Y ))) φ X,F (Y )

  ❡ D // C φ Y,F (Y ) Hom D (F (f ),F (I )) Hom C (f,G(F (I ))) Hom (F (Y ), F (Y )) Hom (Y, G(F (Y ))) D // Hom C Y Y Hom (F (X), F (Y )) (X, G(F (Y ))). φ X,F (Y ) ❉♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ✈❡♠ q✉❡ C D

  (Hom (I X , G(F (f )))◦φ X,F )(I F ) = (φ X,F ◦Hom (F (I X ), F (f )))(I F )

  (X) (X) (Y ) (X)

  ♦✉ s❡❥❛✱ ◦ (F (f ) ◦ I ◦ F (I

  G(F (f )) ◦ φ X,F (I F ) ◦ I X = φ X,F F X ))

  (X) (X) (Y ) (X)

  q✉❡ é G(F (f )) ◦ φ (I ) = φ ◦ F (f ). X,F (X) F (X) X,F (Y ) ❉♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ t❡♠♦s C D

  (Hom (f, G(F (I Y )))◦φ Y,F )(I F ) = (φ X,F ◦Hom (F (f ), F (I Y )))(I F ),

  (Y ) (Y ) (Y ) (Y )

  ♦✉ s❡❥❛✱ G(I ) ◦ φ (I ) ◦ f = φ ◦ (F (I Y ) ◦ I ◦ F (f )) F (Y ) Y,F (Y ) F (Y ) X,F (Y ) F (Y ) q✉❡ é φ Y,F (Y ) (I F (Y ) ) ◦ f = φ X,F (Y ) ◦ F (f ).

  P♦rt❛♥t♦✱ (G ◦ F )(f ) ◦ c X = (G ◦ F )(f ) ◦ φ X,F

  ψ X,Y : Hom C (X, G(Y )) → Hom D (F (X), Y ) β 7→ e Y ◦ F (β). ▼♦str❡♠♦s q✉❡ φ é tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❡♠ ❛♠❜❛s ✈❛r✐á✈❡✐s✳ ❙❡✲

  = G(g ◦ α) ◦ G(F (f )) ◦ c U = G(g ◦ α ◦ F (f )) ◦ c U = φ U,V (g ◦ α ◦ F (f )) = (φ U,V

  

(∗)

  ◦ f = G(g ◦ α) ◦ c X ◦ f

  = G(g) ◦ (G(α) ◦ c X ) ◦ f = G(g) ◦ G(α) ◦ c X

  ❚❡♠♦s✱ ♣❛r❛ α : F (X) → Y ✉♠ ♠♦r✜s♠♦ q✉❛❧q✉❡r ❡♠ D✱ q✉❡ (Hom C (f, G(g)) ◦ φ X,Y )(α) = Hom C (f, G(g))(G(α) ◦ c X )

  // Hom C (U, G(V )). Pr♦✈❡♠♦s q✉❡ ♦ ♠❡s♠♦ ❝♦♠✉t❛ ❡ ❞✐ss♦✱ s❡❣✉❡ ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ φ✳

  Hom C (X, G(Y )) Hom C (f,G(g)) Hom D (F (U ), V ) φ U,V

  Hom D (F (X), Y ) φ X,Y // Hom D (F (f ),g)

  ❥❛♠ f : U → X ❡ g : Y → V ♠♦r✜s♠♦s ❡♠ C ❡ D✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛

  α 7→ G(α) ◦ c X

  (X)

  ❛♣❧✐❝❛çõ❡s✿ φ X,Y : Hom D (F (X), Y ) → Hom C (X, G(Y ))

  ) ◦ f = c Y ◦ f. ❆ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ e s❡ ♠♦str❛ ❞❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r✳ ✭✐✐✮ ⇒ ✭✐✮ P❛r❛ q✉❛✐sq✉❡r X ∈ Ob(C) ❡ Y ∈ Ob(D) ❞❡✜♥✐♠♦s ❛s

  (Y )

  (I F

  (Y )

  (F (f )) = φ Y,F

  (Y )

  ) = φ X,F

  (X)

  (I F

  ◦ Hom D (F (f ), g))(α),

  ❡♠ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ✭✯✮ s❡❣✉❡ ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ c✱ ✐st♦ é✱ ✈❛❧❡ ❛ ❝♦♠✉✲ t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ U c U

  (Hom D (F (f ), g) ◦ ψ X,Y )(β) = Hom D (F (f ), g)(e Y ◦ F (β)) = g ◦ (e Y ◦ F (β)) ◦ F (f ) = g ◦ e Y ◦ F (β) ◦ F (f ) = g ◦ e Y ◦ F (β ◦ f )

  Y g (F ◦ G)(V ) e V // V. ❋✐♥❛❧♠❡♥t❡✱ ♠♦str❡♠♦s q✉❡ ψ é ✐♥✈❡rs❛ ❞❡ φ ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♦❜✲

  (F ◦G)(g)

  //

  ❡♠ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ✭✯✮ s❡❣✉❡ ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ e✱ ✐st♦ é✱ ✈❛❧❡ ❛ ❝♦♠✉✲ t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ (F ◦ G)(Y ) e Y

  = e V ◦ F (G(g) ◦ β ◦ f ) = ψ U,V (G(g) ◦ β ◦ f ) = (ψ U,V ◦ Hom C (f, G(g)))(β),

  = e V ◦ F (G(g)) ◦ F (β ◦ f )

  

(∗)

  Hom C (U, G(V )) ψ U,V // Hom D (F (U ), V ), ❝♦♠✉t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ β : X → G(Y ) ✉♠ ♠♦r✜s♠♦ q✉❛❧q✉❡r ❡♠ C✱ t❡♠♦s

  // f (G ◦ F )(U )

  (F (f ),g)

  Hom D (F (X), Y ) Hom D

  (f,G(g))

  Hom C (X, G(Y )) ψ X,Y // Hom C

  ◆♦t❡♠♦s ❛✐♥❞❛ q✉❡ ψ é t❛♠❜é♠ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❱❡r✐✲ ✜q✉❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛

  X c X // (G ◦ F )(X).

  (G◦F )(f )

  ❥❡t♦s ❡ ❞❛í✱ φ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳

  ❙❡❥❛ g : F (X) → Y ✉♠ ♠♦r✜s♠♦ ❡♠ D✳ ❊♥tã♦ (ψ X,Y

  = G(e Y ) ◦ c G

  → G ◦ F t❛❧ q✉❡ φ X,Y (h) = G(h) ◦ c X ✱ ♣❛r❛ q✉❛❧q✉❡r ♠♦r✜s♠♦ h : F (X) → Y ❡♠ D✳

  ♠♦r✜s♠♦ ❡♠ C✳ ❉❛ ❡q✉✐✈❛❧ê♥❝✐❛ ✭✐✮ ⇔ ✭✐✐✮✱ ❡①✐st❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ c : Id C

  G(Y ) c G (Y ) // (G ◦ F )(G(Y )). ✭✐✮ ⇒ ✭✐✐✐✮ ❙❡❥❛♠ X ∈ Ob(C)✱ Y ∈ Ob(D) ❡ f : X → G(Y ) ✉♠

  (G◦F )(h)

  (G ◦ F )(X)

  X c X // h

  h, ❡♠ ✭✯✯✮ ✉t✐❧✐③❛♠♦s ❛ ✐❣✉❛❧❞❛❞❡ (2.2) ❡ ❡♠ ✭✯✮ ✉t✐❧✐③❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ c ♣❛r❛ ❝♦♠✉t❛r ♦ ❞✐❛❣r❛♠❛

  ◦ h =

  (Y )

  I G

  =

  (∗∗)

  ◦ h

  (Y )

  (∗)

  ◦ φ X,Y )(g) = ψ X,Y (G(g) ◦ c X ) = e Y

  (∗∗)

  ◦ F (G(g) ◦ c X ) = e Y

  ◦ F (G(g)) ◦ F (c X )

  (∗)

  = g ◦ e F

  (X)

  ◦ F (c X )

  = g ◦ I F (X) =

  = G(e Y ) ◦ G(F (h)) ◦ c X

  g, ❡♠ ✭✯✯✮ ✉t✐❧✐③❛♠♦s ❛ ✐❣✉❛❧❞❛❞❡ (2.3) ❡ ❡♠ ✭✯✮ ✉t✐❧✐③❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ e ♣❛r❛ ❝♦♠✉t❛r ♦ ❞✐❛❣r❛♠❛

  (F ◦ G)(F (X)) e F (X) //

  (F ◦G)(g)

  F (X) g (F ◦ G)(Y ) e Y // Y. P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ h : X → G(Y ) ✉♠ ♠♦r✜s♠♦ ❡♠ C✱ t❡♠♦s q✉❡

  (φ X,Y ◦ ψ X,Y )(h) = φ X,Y (e Y ◦ F (h))

  = G(e Y ◦ F (h)) ◦ c X

  ❙❡♥❞♦ φ X,Y ✉♠ ✐s♦♠♦r✜s♠♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ g : F (X) → Y t❛❧ q✉❡ φ X,Y (g) = f ✳ P♦rt❛♥t♦✱ f = φ X,Y (g) = G(g) ◦ c X .

  ✭✐✐✐✮ ⇒ ✭✐✮ ❙❡❥❛♠ X ∈ Ob(C) ❡ Y ∈ Ob(D)✳ ❉❡✜♥✐♠♦s D → Hom C φ X,Y : Hom (F (X), Y ) (X, G(Y ))

  7→ C α G(α) ◦ c X , → G ◦ F

  ❡♠ q✉❡ c : Id é ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❞❛❞❛ ❡♠ ✭✐✐✐✮✳ X,Y ▼♦str❡♠♦s q✉❡ φ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❉❡ ❢❛t♦✱ s❡❥❛ ♦ ♠♦r✜s♠♦ C f ∈ Hom (X, G(Y )) D ✳ P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ g ∈

  Hom (F (X), Y ) X = φ X,Y (g) X,Y t❛❧ q✉❡ f = G(g) ◦ c ✳ P♦rt❛♥t♦✱ φ é s♦❜r❡❥❡t♦r❛✳

  ❆ ✐♥❥❡t✐✈✐❞❛❞❡ s❡❣✉❡ ❞❛ ✉♥✐❝✐❞❛❞❡ ❞❛❞❛ ❡♠ ✭✐✐✐✮✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ X,Y q✉❡ φ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ é ✐♥t❡✐r❛♠❡♥t❡ ❛♥á❧♦❣❛ à ❢❡✐t❛ ❡♠ ✭✐✐✮ ⇒ ✭✐✮✳

  ❆ ❡q✉✐✈❛❧ê♥❝✐❛ ✭✐✮ ⇔ ✭✐✈✮ é ❢❡✐t❛ ❞❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r à ❡q✉✐✈❛❧ê♥❝✐❛ ✭✐✮ ⇔ ✭✐✐✐✮✳

  ❆s tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s c ❡ e sã♦ ❝❤❛♠❛❞❛s ✉♥✐❞❛❞❡ ❡ ❝♦✉♥✐❞❛❞❡ ❞❛ ❛❞❥✉♥çã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❛❞❥✉♥çã♦✳ ❊①❡♠♣❧♦ ✷✳✷✾ ❖s ❢✉♥t♦r❡s U : Grp → Ab ❡ F : Ab → Grp ❞♦ ❊①❡♠✲ ♣❧♦ 2.14 é ✉♠❛ ❛❞❥✉♥çã♦✳ ▼♦str❡♠♦s q✉❡ F é ❛❞❥✉♥t♦ à ❞✐r❡✐t❛ ❞❡ U✳ P❛r❛ ✐ss♦✱ ♣r♦✈❡♠♦s ♦ ✐t❡♠ ✭✐✐✐✮ ❞♦ ❚❡♦r❡♠❛ 2.28✳ ❊st❛♠♦s ❝♦♥s✐❞❡✲ r❛♥❞♦ C = Grp✱ D = Ab✱ F = U ❡ G = F ✳

  ❉❡ ❢❛t♦✱ s❡❥❛♠ G ∈ Ob(Grp)✱ H ∈ Ob(Ab) ❡ f : G → F (H) ✉♠ ♠♦r✜s♠♦ ❡♠ Grp✳ ❉❡✜♥✐♠♦s g : U(G) → H✱ ♦✉ s❡❥❛✱ g : G/[G, G] → H x[G, G] 7→ f (x).

  ❱❡❥❛♠♦s q✉❡ g ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❙❡❥❛♠ x, y ∈ G t❛✐s q✉❡ x[G, G] =

  1

  y[G, G] y ∈ [G, G] ✳ ❊♥tã♦ x ✳ ❉❛í✱ − − − Y n

  1

  1

  

1

  x y = a b a j b j j j j ♣❛r❛ a, b ∈ G.

  =1

  ▲♦❣♦✱ Q n − −

  1

  1

  1

  f (x y) = f ( a b a j b j ) Q n j =1 j j

  1 −

  1

  = f (a j ) f (b j ) f (a j )f (b j ) j

  =1

  = e H , ✐st♦ ♣♦✐s F (H) = H ❡ H é ❛❜❡❧✐❛♥♦✳ P♦rt❛♥t♦✱ f(x) = f(y)✳

  G : G/[G, G] → H

  ❖❜s❡r✈❛♠♦s q✉❡ f = F (g) ◦ P ❡ q✉❡ s❡ g é ′ ′ ) ◦ P G

  ✉♠ ♠♦r✜s♠♦ t❛❧ q✉❡ f = F (g ❡♥tã♦ g = g ✳ ▲♦❣♦✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ P G

  // F (G/[G, G]) = G/[G, G] G/[G, G]

  G P P P P P P P P g F

  P (g) P f P P

  P P ((P

  F (H) H. P♦rt❛♥t♦✱ F é ❛❞❥✉♥t♦ à ❞✐r❡✐t❛ ❞❡ U✳ k → Alg k

  ❊①❡♠♣❧♦ ✷✳✸✵ ❈♦♥s✐❞❡r❡♠♦s ♦s ❢✉♥t♦r❡s U : Lie ❡ L : Alg k → Lie k

  ❞♦s ❊①❡♠♣❧♦s 2.8 ❡ 2.9✳ P❛r❛ ❝❛❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ L✱ ❝♦♥✲ L ) s✐❞❡r❡♠♦s (U(L), ι s✉❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧✱ ❝♦♠♦ ❞❡s❝r✐t❛ ♥♦ ❆♣ê♥❞✐❝❡ ❆✳ ′ ′

  ∈ Ob(Lie k ) ❙❡❥❛♠ L, L ❡ f : L → L ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡

  ▲✐❡✳ P❡❧❛ ❝♦♥str✉çã♦ ❞♦ ❢✉♥t♦r U✱ ❡①✐st❡ ú♥✐❝♦ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s U

  (f ) : U(L) → U(L ) q✉❡ ✐♥❞✉③ ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ U(f) : L

  (U(L)) → L(U(L )) L = i L ◦ f t❛❧ q✉❡ U(f) ◦ i ✱ ♦✉ s❡❥❛✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ ι L

  L //

  (U(L)) f U L (f ) // L(U(L )). L ι L′

  P♦rt❛♥t♦✱ ❛ ❝♦❧❡çã♦ ι = {ι L : L → L(U(L)) : L ∈ Ob(Lie k )} Lie

  é ✉♠❛ tr❛s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❡♥tr❡ Id ❡ L ◦ U✳ Pr♦✈❡♠♦s q✉❡ U é k ❛❞❥✉♥t♦ à ❡sq✉❡r❞❛ ❞❡ L ✉t✐❧✐③❛♥❞♦ ♦ ✐t❡♠ ✭✐✐✐✮ ❞♦ ❚❡♦r❡♠❛ 2.28✳ k k

  ) ) ❙❡❥❛♠ L ∈ Ob(Lie ✱ A ∈ Ob(Alg ❡ f : L → L(A) ✉♠ ♠♦r✜s♠♦ L

  ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ❈♦♠♦ ✭U(L), ι ✮ é ❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ L✱ s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s g : U(L) → A t❛❧ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ ι L

  L U //

  (U(L)) (L) L F F F F F L g

  (g)

  F f F F

  ""F L

  (A) A.

  R

  ❊①❡♠♣❧♦ ✷✳✸✶ ❙❡❥❛ R ✉♠ ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡ 1 ✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❢✉♥✲ R → Ab R M M t♦r ❡sq✉❡❝✐♠❡♥t♦ F : ❡ ♦ ❢✉♥t♦r L : Ab → ❞❡✜♥✐❞♦ ♣♦r L(G) = R ⊗ G

  Z ✱ ♣❛r❛ ❝❛❞❛ ♦❜❥❡t♦ G ∈ Ob(Ab)✳ ❙❡❥❛ f : G → H ✉♠

  ♠♦r✜s♠♦ ❡♠ Ab✳ ❊♥tã♦ L(f ) : R ⊗ G → R ⊗ H

  Z Z r ⊗ g 7→ r ⊗ f (g).

  ❆ ❝♦❧❡çã♦ ❞❡ ♠♦r✜s♠♦s {u G : G → F (R ⊗

  G) : u G (g) = 1 R ⊗ g}

  Z

  é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛✲ ❣r❛♠❛ u G

  // F (R ⊗

  G) h (L(h)) G Z F // R ⊗

  H, H Z u H

  ❡♠ q✉❡ h : G → H é ✉♠ ♠♦r✜s♠♦ ❡♠ Ab✳ ❚❡♠♦s q✉❡ (F (L(h)) ◦ u G )(g) = F (L(h))(u G (g))

  = F (L(h))(1 R ⊗ g) =

  1 R ⊗ h(g) = u H (h(g)) = (u H ◦ h)(g),

  ♣❛r❛ t♦❞♦ g ∈ G✳ R ) M ❋✐♥❛❧♠❡♥t❡✱ s❡❥❛♠ G ∈ Ob(Ab)✱ M ∈ Ob( ❡ f : G → F (M) ✉♠

  G → M ♠♦r✜s♠♦ ❡♠ Ab✳ ❉❡✜♥✐♠♦s t : R ⊗ Z ♣♦r t(r ⊗ g) = rf(g)✱ q✉❡ é ❝❧❛r❛♠❡♥t❡ ✉♠ ♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s✳ G (g) = f (g)

  ❖❜s❡r✈❛♠♦s q✉❡ F (t)◦u ✱ ♣❛r❛ t♦❞♦ g ∈ G✳ ❆ ✉♥✐❝✐❞❛❞❡ : R ⊗ G → M

  ❞❡ t s❡❣✉❡ ❞♦ ❢❛t♦ ❞❡ q✉❡✱ ❞❛❞♦ ♦✉tr♦ ♠♦r✜s♠♦ t Z ❡♠ ′ ′ M R ) ◦ u G (g) = f (g) (1 R ⊗ g) = t❛❧ q✉❡ F (t ✱ ♣❛r❛ t♦❞♦ g ∈ G✱ ❡♥tã♦ t

  ⊗ g) = f (g) = t(1 ⊗ g) F (t )(1 R R ′ ′ ✳ ▲♦❣♦✱ t (r ⊗ g) = t (r(1 R ⊗ g))

  = rt (1 R ⊗ g) = rt(1 R ⊗ g) = t(r ⊗ g)

  = t ❡ ❛ss✐♠✱ t ✳

  ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♥♦s ❞✐③ q✉❡ ♦ ❢✉♥t♦r ❛❞❥✉♥t♦ à ❞✐r❡✐t❛ ❞❡ ✉♠ ❢✉♥t♦r é ú♥✐❝♦✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛✳ Pr♦♣♦s✐çã♦ ✷✳✸✷ ❙❡❥❛ F : C → D ✉♠ ❢✉♥t♦r✳ ❙❡ (F, G, φ) ❡ (F, H, ψ) sã♦ ❛❞❥✉♥çõ❡s ❡♥tã♦ G ∼ H✳ ❉❡♠♦♥str❛çã♦✿ P♦r ❤✐♣ót❡s❡✱ φ ❡ ψ sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r X ∈ Ob(C) ❡ Y ∈ Ob(D)✱ s❡❣✉❡♠ ♦s ✐s♦♠♦r✜s♠♦s D C

  φ X,Y : Hom (F (X), Y ) → Hom (X, G(Y )) ❡ D C ψ X,Y : Hom (F (X), Y ) → Hom (X, H(Y )).

  ❆ss✐♠✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦s ✐s♦♠♦r✜s♠♦s

1 C C α X,Y = ψ X,Y ◦ φ : Hom (X, G(Y )) → Hom (X, H(Y )).

  X,Y ◆♦t❛♠♦s q✉❡ α é tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱ C C op C C op

  α : Hom (−, −) ◦ (Id × G) → Hom (−, −) ◦ (Id × H) ❡♠ q✉❡ ♦s ❢✉♥t♦r❡s C C op op

  Hom (−, −) ◦ (Id × G) : C × D → Set ❡ C C × H) : C × D → Set. op op

  Hom (−, −) ◦ (Id ❈❧❛r❛♠❡♥t❡✱ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ é ❝♦♠✉t❛t✐✈♦ C // D C φ X,Y X,Y 1 ψ Hom C Hom D Hom C Hom (X, G(Y )) Hom (F (X), Y ) // Hom (X, H(Y ))

  (f,G(g)) (F (f ),g) (f,H(g)) C D C

  Hom (U, G(V )) // Hom (F (U ), V ) // Hom (U, H(V )) φ U,V 1 ψ U,V ♣❛r❛ q✉❛✐sq✉❡r ♠♦r✜s♠♦s f : U → X ❡ g : Y → V ❡♠ C ❡ D✱ r❡s✲ X,Y ♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡♥❞♦ α ✐s♦♠♦r✜s♠♦✱ ♣❛r❛ q✉❛✐sq✉❡r X ∈ Ob(C) ❡ Y ∈ Ob(D)

  ✱ s❡❣✉❡ q✉❡ α é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ Y = α G (I G ) : G(Y ) → H(Y ) ❈♦♥s✐❞❡r❡♠♦s λ (Y ),Y (Y ) ♠♦r✜s♠♦ ❡♠

  C ✳ Pr♦✈❡♠♦s q✉❡ λ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ P❛r❛ ✈❡r✐✜❝❛r♠♦s ❛

  ♥❛t✉r❛❧✐❞❛❞❡✱ ❜❛st❛ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❡ λ Y // G H G(Y ) H(Y )

  (g) (g)

  G(W ) // H(W ) λ W

  ♣❛r❛ q✉❛✐sq✉❡r W, Y ∈ Ob(D) ❡ ♠♦r✜s♠♦ g : Y → W ❡♠ D✳ ■ss♦ s❡❣✉❡ ❞♦ ❢❛t♦ ❞❡ α s❡r ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ é ❝♦♠✉t❛t✐✈♦ C // C α G (Y ),Y Hom C (I ,G (g)) Hom C (I ,H (g)) Hom (G(Y ), G(Y )) Hom (G(Y ), H(Y )) G (Y ) C C G (Y ) Hom (G(Y ), G(W )) // Hom (G(Y ), H(W )). α G (Y ),W

  ❆ss✐♠✱ C ◦Hom C (Hom (I G , H(g))◦α G )(I G ) = (α G (I G , G(g)))(I G ),

  (Y ) (Y ),Y (Y ) (Y ),W (Y ) (Y )

  ♦✉ s❡❥❛✱ H(g) ◦ λ Y = α G ◦ G(g).

  (Y ),W

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ♦ ♣ró①✐♠♦ ❞✐❛❣r❛♠❛ t❛♠❜é♠ ❝♦♠✉t❛ α G (W ),W C C // Hom C (G(g),G(I )) C (G(g),H(I )) Hom (G(W ), G(W )) Hom (G(W ), H(W )) C C W Hom W Hom (G(Y ), G(W )) // Hom (G(Y ), H(W )). α G (Y ),W P♦rt❛♥t♦✱ C C

  (Hom (G(g), H(I W ))◦α G )(I G ) = (α G ◦Hom (G(g), G(I W )))(I G

  (W ),W (W ) (Y ),W (W

  q✉❡ ❡q✉✐✈❛❧❡ ❛ α G (I G ) ◦ G(g) = α G ◦ G(g), W ◦ G(g) = α ◦ G(g) = H(g) ◦ λ Y (W ),W (W ) (Y ),W

  ♦✉ s❡❥❛✱ λ G (Y ),W ❡ s❡❣✉❡ ❛ ❝♦♠✉t❛t✐✲ ✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛✳ Y = α (I ) : H(Y ) → G(Y )

  1

  ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡♠♦s σ H (Y ) ♠♦r✲ H

  (Y ),Y Y = λ

  1

  ✜s♠♦ ❡♠ C✳ ▼♦str❡♠♦s q✉❡ σ ✳ ❉❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ α s❡❣✉❡ ❛ Y ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ C // C α G (Y ),Y Hom ,G Hom ,H C (σ Y (I Y )) C (σ Y (I Y )) Hom (G(Y ), G(Y )) Hom (G(Y ), H(Y )) C C Hom (H(Y ), G(Y )) // Hom (H(Y ), H(Y )). α H (Y ),Y

  ❚❡♠♦s (Hom C (σ Y , H(I Y ))◦α G

  (Y ),Y

  )(I G

  (Y )

  ) = (α H

  (Y ),Y

  ◦Hom C (σ Y , G(I Y )))(I G

  (Y )

  ), q✉❡ ❡q✉✐✈❛❧❡ ❛ Hom C (σ Y , H(I Y ))(λ Y ) = α H

  (Y ),Y

  (σ Y ), ♦✉ s❡❥❛✱

  λ Y ◦ σ Y = I H

  (Y ) .

1 X,Y

  1

  ) = (α

  P♦rt❛♥t♦✱ G ∼ H✳

  (Y ) .

  σ Y ◦ λ Y = I G

  (λ Y ), ♦✉ s❡❥❛✱

  1 G (Y ),Y

  ), q✉❡ ❡q✉✐✈❛❧❡ ❛ Hom C (λ Y , G(I Y ))(σ Y ) = α

  (Y )

  ◦Hom C (λ Y , H(I Y )))(I H

  1 G (Y ),Y

  (Y )

  = {α

  )(I H

  1 H (Y ),Y

  ❆ss✐♠✱ (Hom C (λ Y , G(I Y ))◦α

  Hom C (G(Y ), H(Y )) α 1 G (Y ),Y // Hom C (G(Y ), G(Y )).

  (I Y ))

  Hom C (H(Y ), G(Y )) Hom C (λ Y ,G

  ❉❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ α

  // Hom C (λ Y ,H

  ❡ Y ∈ Ob(D)}✱ s❡❣✉❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ Hom C (H(Y ), H(Y )) α 1 H (Y ),Y

  : Hom C (X, H(Y )) → Hom C (X, G(Y )) : X ∈ Ob(C)

  (I Y ))

  ❈❛♣ít✉❧♦ ✸ ❈❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s

  ◆♦ss♦ ♦❜❥❡t✐✈♦ ♥❡ss❡ ❝❛♣ít✉❧♦ é ❡st✉❞❛r ❝❛t❡❣♦r✐❛s ❝♦♠ ✉♠❛ ❡str✉✲ t✉r❛ ❛❞✐❝✐♦♥❛❧ ❞❡ ❛❞✐t✐✈✐❞❛❞❡ ❡♥tr❡ ♠♦r✜s♠♦s✳ ■♥✐❝✐❛♠♦s ❝♦♠ ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s ❡ ❛♦ ✜♥❛❧✱ ❞❡✜♥✐♠♦s ❝❛t❡❣♦r✐❛s k✲❧✐♥❡❛r❡s ✭❛❜❡❧✐❛♥❛s✮ q✉❡ sã♦ ♥❡❝❡ssár✐❛s ♣❛r❛ ❛s ❝♦♥str✉çõ❡s ❞♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✱ q✉❡ tr❛t❛ ❡ss❡♥✲ ❝✐❛❧♠❡♥t❡ ❞❡ ❝❛t❡❣♦r✐❛s k✲❧✐♥❡❛r❡s✳ ❚❛❧ ❡str✉t✉r❛ ❞❡ ❛❞✐t✐✈✐❞❛❞❡ ♥❡ss❛s ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ❝r✐❛ ✉♠❛ ❝❡rt❛ s✐♠❡tr✐❛ ✭❞✉❛❧✐❞❛❞❡✮✱ ❝♦♠♦ é ♦❜s❡r✲ ✈❛❞♦ ♣♦r ▼❛❝▲❛♥❡ ❡♠ [✶✺]✱ ❡♥tr❡ s❡✉s ♦❜❥❡t♦s✳ ❖s r❡s✉❧t❛❞♦s✱ ❞❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ [✶✻]✱ [✶✹]✱ [✺] ❡ [✼]✳ ❉❡✜♥✐çã♦ ✸✳✶ ❯♠❛ ❝❛t❡❣♦r✐❛ C é ❞✐t❛ ♣ré✲❛❞✐t✐✈❛ s❡ ✭✐✮ C ♣♦ss✉✐ ♦❜❥❡t♦ ③❡r♦❀ C

  (X, Y ) ✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ Ob(C)✱ ♦ ❝♦♥❥✉♥t♦ Hom é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦❀ ✭✐✐✐✮ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s é ❜✐❧✐♥❡❛r✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r ♠♦r✲ ′ ′

  : X → Y : Y → Z ✜s♠♦s f, f ❡ g, g ✈❛❧❡♠ ′ ′ g ◦ (f + f ) = g ◦ f + g ◦ f ′ ′ ◦ f.

  (g + g ) ◦ f = g ◦ f + g ❉❡✜♥✐çã♦ ✸✳✷ ❯♠❛ ❝❛t❡❣♦r✐❛ C é ❞✐t❛ ❛❞✐t✐✈❛ s❡ ✭✐✮ C é ♣ré✲❛❞✐t✐✈❛❀

  X , π Y )

  ✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ Ob(C)✱ ❡①✐st❡ ♦ ♣r♦❞✉t♦ (X × Y, π ❞❡

  X ❡ Y ✳

  ❆❧❡rt❛♠♦s ♦ ❧❡✐t♦r q✉❡✱ ♣❛r❛ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s q✉❡ ❢❛r❡♠♦s ❛q✉✐✱ é út✐❧ r❡❧❡♠❜r❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ♦ ♦❜❥❡t♦ ③❡r♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ Pr♦♣♦s✐çã♦ 1.29✳ ▲❡♠❛ ✸✳✸ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ X, Y ∈ Ob(C)✳ ❙❡❥❛ (X × Y, π X , π Y ) X ◦ φ = π ♦ ♣r♦❞✉t♦ ❞❡ X ❡ Y ✳ ❙❡ φ, ψ : Z → X × Y sã♦ ♠♦r✜s♠♦s X ◦ ψ Y ◦ φ = π Y ◦ ψ t❛✐s q✉❡ π ❡ π ✱ ❡♥tã♦ φ = ψ✳ X ◦ φ = π X ◦ ψ Y ◦ φ = π Y ◦ ψ

  ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ π ❡ π ✱ s❡❣✉❡ q✉❡ Z Z Z π X ◦ (φ − ψ) = 0 Y ◦ (φ − ψ) = 0 : Z → X ❡ π Y ✳ ▼❛s ♦ ♠♦r✜s♠♦ 0 X×Y Z Z Z Z X × Y X ◦ 0 = 0 Y ◦ 0 = 0

  é ♦ ú♥✐❝♦ t❛❧ q✉❡ π X×Y Z X ❡ π X×Y Y ✭❞❡✜♥✐çã♦ ❞❡ ♣r♦❞✉t♦✮✳ ▲♦❣♦✱ φ − ψ = 0 X×Y ✱ ♦✉ s❡❥❛✱ φ = ψ✳ R M ❊①❡♠♣❧♦ ✸✳✹ ❆ ❝❛t❡❣♦r✐❛ é ❛❞✐t✐✈❛✳ ❉❡ ❢❛t♦✱ ♦ ♠ó❞✉❧♦ tr✐✈✐❛❧ {e} é ♦ ♦❜❥❡t♦ ③❡r♦ ♥❡ss❛ ❝❛t❡❣♦r✐❛ ❡ ♦ ♣r♦❞✉t♦ ✭❞✐r❡t♦✮ ❞❡ ♠ó❞✉❧♦s ❡①✐st❡ ♣❛r❛ q✉❛✐sq✉❡r ❞♦✐s ♠ó❞✉❧♦s✳ ❊①❡♠♣❧♦ ✸✳✺ ❆ ❝❛t❡❣♦r✐❛ Ab é ❛❞✐t✐✈❛✳ ❖ ♣r♦❞✉t♦ ✭❞✐r❡t♦✮ ♥❡ss❛ ❝❛✲ t❡❣♦r✐❛ é ♦ ❞❡✜♥✐❞♦ ♥♦ ❊①❡♠♣❧♦ 1.46 ♣❛r❛ ❛ ❝❛t❡❣♦r✐❛ Grp✳ ❉❡✜♥✐çã♦ ✸✳✻ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛✳ ❉❛❞♦s ♦❜❥❡t♦s X, Y ∈ Ob(C) X , π Y , i X , i Y )

  ✱ ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ X ❡ Y é ✉♠❛ q✉í♥t✉♣❧❛ (X⊕Y, π ✱ X : X ⊕ Y → X Y : X ⊕ Y → Y X : X → X ⊕ Y ❡♠ q✉❡ π ✱ π ✱ i ❡ i Y : Y → X ⊕ Y sã♦ ♠♦r✜s♠♦s q✉❡ s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s X ◦ i ◦ i X = I X Y Y = I Y

  ✭✐✮ π ❡ π ❀ X ◦ π X + i Y ◦ π Y = I X⊕Y ✭✐✐✮ i ✳ ❖❜s❡r✈❛çã♦ ✸✳✼ ❙❡ C é ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❡♥tã♦ s❡❣✉❡✱ ❞❛ ♣r♦✲ Y X X Y = 0 Y ◦ i ◦ i X = 0

  ♣r✐❡❞❛❞❡ ✭✐✐✮ ❛❝✐♠❛✱ q✉❡ π X ❡ π Y ✳ ❉❡ ❢❛t♦✱ π X = π X ◦ I X⊕Y

  = π X ◦ (i X ◦ π X + i Y ◦ π Y ) = π X ◦ (i X ◦ π X ) + π X ◦ (i Y ◦ π Y )

  ◦ i ◦ i = (π X X ) ◦ π X + (π X Y ) ◦ π Y

  ◦ i X⊕Y X⊕Y = π X + (π X Y ) ◦ π Y . ◦ i ◦ i

  = (π X Y ) ◦ π Y Y = ❆ss✐♠✱ 0 X ❡ ✐st♦ ✐♠♣❧✐❝❛ q✉❡ 0 Y X X

  ◦i ◦i ◦i ◦i (π X Y ) ◦ π Y Y , = π X Y Y X = 0

  ♦✉ s❡❥❛✱ 0 X ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ π Y ✳

  X , π Y , i X , i Y )

  Pr♦♣♦s✐çã♦ ✸✳✽ ❙❡ (X ⊕ Y, π é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ X ❡ Y X , π Y ) X , i Y )

  ✱ ❡♥tã♦ (X ⊕Y, π é ✉♠ ♣r♦❞✉t♦ ❡ (X ⊕Y, i é ✉♠ ❝♦♣r♦❞✉t♦ ❞❡ X ❡ Y ✳ X , π Y ) ❉❡♠♦♥str❛çã♦✿ ❱❡r✐✜q✉❡♠♦s q✉❡ (X ⊕ Y, π é ✉♠ ♣r♦❞✉t♦ ❞❡

  X X , i Y ) ❡ Y ✳ ❆ ✈❡r✐✜❝❛çã♦ ❞❡ q✉❡ (X ⊕ Y, i é ✉♠ ❝♦♣r♦❞✉t♦ ❞❡ X ❡ Y

  é s✐♠✐❧❛r✳ X , p Y ) X : Z → X Y : Z → Y ❙❡❥❛ (Z, p ✉♠❛ tr✐♣❧❛ ❡♠ q✉❡ p ❡ p sã♦

  ♠♦r✜s♠♦s ❡♠ C✳ ❉❡✜♥✐♠♦s φ = i X ◦ p X + i Y ◦ p Y : Z → X ⊕ Y. ◆♦t❡♠♦s q✉❡

  π X ◦ φ = π X ◦ (i X ◦ p X + i Y ◦ p Y ) = π X ◦ (i X ◦ p X ) + π X ◦ (i Y ◦ p Y ) = (π X ◦ i X ) ◦ p X + (π X ◦ i Y ) ◦ p Y Y ◦ φ = p Y = p X .

  ❆♥❛❧♦❣❛♠❡♥t❡✱ π ✳ P❛r❛ ✈❡r✐✜❝❛r♠♦s ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ φ✱ X ◦ ψ = s✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ✉♠ ♠♦r✜s♠♦ ψ : Z → X ⊕ Y t❛❧ q✉❡ π p X Y ◦ ψ = p Y X Y

  ❡ π ✳ ❆♣❧✐❝❛♥❞♦ i ❡ i ♥❛s r❡s♣❡❝t✐✈❛s ✐❣✉❛❧❞❛❞❡s✱ X ◦ (π X ◦ ψ) = i X ◦ p X Y ◦ (π Y ◦ ψ) = i Y ◦ p Y ♦❜t❡♠♦s i ❡ i ✳ ❙♦♠❛♥❞♦ ❛s X ◦ (π X ◦ ψ) + i Y ◦ (π Y ◦ ψ) = ✐❣✉❛❧❞❛❞❡s ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ s❡❣✉❡ q✉❡ i i X ◦ p X + i Y ◦ p Y

  ✳ P♦rt❛♥t♦✱ φ = i X ◦ p X + i Y ◦ p Y

  = i X ◦ (π X ◦ ψ) + i Y ◦ (π Y ◦ ψ) = (i X ◦ π X ) ◦ ψ + (i Y ◦ π Y ) ◦ ψ = (i X ◦ π X + i Y ◦ π Y ) ◦ ψ = ψ.

  ❊♠ ✈✐st❛ ❞❡ss❛ ♣r♦♣♦s✐çã♦✱ ❛ s♦♠❛ ❞✐r❡t❛ é t❛♠❜é♠ ❝❤❛♠❛❞❛ ❜✐♣r♦✲ ❞✉t♦✳ ❙❡❣✉❡ ❞❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♣r♦❞✉t♦ ❡ ❝♦♣r♦❞✉t♦ q✉❡ ❛ s♦♠❛ ❞✐r❡t❛ é ú♥✐❝❛✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ R M ❊①❡♠♣❧♦ ✸✳✾ ❈♦♥s✐❞❡r❡♠♦s ❛ ❝❛t❡❣♦r✐❛ ✳ ❙❡❥❛♠ M ❡ N R✲♠ó❞✉❧♦s✳ M , π N , i M , i N ) M N ❊♥tã♦ ❛ q✉í♥t✉♣❧❛ (M ⊕ N, π ✱ ❡♠ q✉❡ π ❡ π sã♦ ❛s M N ♣r♦❥❡çõ❡s ❝❛♥ô♥✐❝❛s ❡ i ❡ i sã♦ ❛s ✐♥❝❧✉sõ❡s ❝❛♥ô♥✐❝❛s✱ é ❛ s♦♠❛ ❞✐r❡t❛ ❞❡ M ❡ N✳ Pr♦♣♦s✐çã♦ ✸✳✶✵ ✭❬✶✹❪✱ ❚❤❡♦r❡♠ ✷✱ ♣✳ ✶✾✵✮ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❡ X, Y ∈ Ob(C)✳ ❊♥tã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

  ✭✐✮ ❡①✐st❡ ♣r♦❞✉t♦ ❞❡ X ❡ Y ❀ ✭✐✐✮ ❡①✐st❡ ❝♦♣r♦❞✉t♦ ❞❡ X ❡ Y ❀ ✭✐✐✐✮ ❡①✐st❡ s♦♠❛ ❞✐r❡t❛ ❞❡ X ❡ Y ✳ ❉❡♠♦♥str❛çã♦✿ ❆s ✐♠♣❧✐❝❛çõ❡s (iii) ⇒ (ii) ❡ (iii) ⇒ (i) s❡❣✉❡♠ ❞✐r❡✲ t❛♠❡♥t❡ ❞❛ Pr♦♣♦s✐çã♦ 3.8✳

  (i) ⇒ (iii) X , π Y ) P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ ♦ ♣r♦❞✉t♦ (X × Y, π ❞❡ X ❡ Y

  ❡ ❛ss✐♠ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✈♦s

  22

  22 I X X X Y

  X ;;w ;;w w w w w w w w w π π w X w X w w i i w w X Y w w

  X // X × Y Y // X × Y G G G G π π

  G Y G Y G G G G G G G G X G G I Y Y ##G ##G ,, ,,

  Y Y X ◦ i ◦ i ◦ X = I X Y Y = I Y Y q✉❡✱ ✐♠❡❞✐❛t❛♠❡♥t❡✱ ♥♦s ❞ã♦ q✉❡ π ✱ π ✱ π X Y ◦ i i X = 0 Y ❡ π X Y = 0 X ✳ ❇❛st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ ❛ q✉í♥t✉♣❧❛ (X ×

  Y, π X , π Y , i X , i Y ) s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭✐✐✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ s♦♠❛ ❞✐r❡t❛✳ ❖❜s❡r✈❛♠♦s q✉❡

  π X ◦ (i X ◦ π X + i Y ◦ π Y ) = π X ◦ (i X ◦ π X ) + π X ◦ (i Y ◦ π Y ) = (π X ◦ i X ) ◦ π X + (π X ◦ i Y ) ◦ π Y Y ◦ (i X ◦ π = π X Y ◦ π Y Y X .

  • i ) = π ❆♥❛❧♦❣❛♠❡♥t❡✱ π ✳ P♦r ♦✉tr♦ ❧❛❞♦✱

  π X ◦ I X×Y = π X Y ◦ I X×Y = π Y X ◦ ❡ π ✳ ❙❡❣✉❡✱ ❞❛ ✉♥✐❝✐❞❛❞❡✱ q✉❡ i

  π X + i Y ◦ π Y = I X×Y ✳ ❆ ✐♠♣❧✐❝❛çã♦ (ii) ⇒ (iii) é s✐♠✐❧❛r✳

  ❉❡✈✐❞♦ à ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ ❡♠ s❡ tr❛t❛♥❞♦ ❞❡ s♦♠❛ ❞✐r❡t❛✱ ♣r♦❞✉t♦ ♦✉ ❝♦♣r♦❞✉t♦ ❞❡ q✉❛✐sq✉❡r ♦❜❥❡t♦s X ❡ Y ♥✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛✱ ✉s❛✲ ♠♦s ❛ ♥♦t❛çã♦ X ⊕Y ♣❛r❛ ❞❡s✐❣♥❛r q✉❛❧q✉❡r ✉♠ ❞♦s três ♠❡♥❝✐♦♥❛❞♦s✳

  ❆❣♦r❛✱ s❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ ⊕ : C × C → C ♦ ❢✉♥t♦r t❛❧ q✉❡ ⊕(X, Y ) = X ⊕ Y ♣❛r❛ ♦❜❥❡t♦s X, Y ∈ Ob(C)✳ ❉❛❞♦s ♠♦r✜s♠♦s C C ′ ′ f ∈ Hom (X, Y ) (X , Y ) ′ ′ ′ ′ ′ ′ ❡ g ∈ Hom ✱ ❝♦♥s✐❞❡r❡♠♦s s♦♠❛s ❞✐r❡t❛s (X ⊕ X , π X , π X , i X , i X ) , p Y , p Y , j Y , j Y )

  ❡ (Y ⊕ Y ✳

  ′ ′

  → Y ⊕ Y ❉❡✜♥✐çã♦ ✸✳✶✶ ✭❬✶✹❪✱ ♣✳ ✶✾✶✮ ❉❡✜♥❡✲s❡ f ⊕ g : X ⊕ X ❝♦♠♦ s❡♥❞♦ ♦ ú♥✐❝♦ ♠♦r✜s♠♦ q✉❡ ❝♦♠✉t❛ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛

  X ⊕ X H π H X X′ w H π

  H ww H H ww

  H H ww

  ##H {{ww f ⊕g

  X f

  X g Y ⊕ Y

  H H p Y H w p Y ′ ww H H H ww

  H H ww

  ##H {{ww

  Y Y ′ ′ Y ◦ (f ⊕ g) = f ◦ π ◦ (f ⊕ g) = g ◦ π X Y X ♦✉ s❡❥❛✱ p ❡ p ✳

  ❆♥t❡s ❞❡ ♣❛ss❛r♠♦s à ♣ró①✐♠❛ ♦❜s❡r✈❛çã♦✱ ♥♦t❡♠♦s q✉❡ ♦ ♠♦r✜s♠♦ f ⊕ g ❡①✐st❡ ❡ é ú♥✐❝♦✳ ❈♦♠♦ t❡♠♦s ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ Y ❡ Y ✱ ❛ tr✐♣❧❛

  (Y ⊕ Y , p Y , p Y ) é ✉♠ ♣r♦❞✉t♦ ❡ ❛ss✐♠✱ ♣♦r ❞❡✜♥✐çã♦ ✭❞❡ ♣r♦❞✉t♦✮✱

  ❡①✐st❡ ❡ é ú♥✐❝♦ t❛❧ ♠♦r✜s♠♦✳ ❖❜s❡r✈❛çã♦ ✸✳✶✷ ◆♦t❡♠♦s q✉❡✱ r❡❧❛t✐✈♦ à ❡str✉t✉r❛ ❞❡ ❝♦♣r♦❞✉t♦ ❡♠✲ ❜✉t✐❞❛ ♥❛ ❡str✉t✉r❛ ❞❡ s♦♠❛ ❞✐r❡t❛✱ t❡♠♦s ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛

  X X G G v G G vv

  G G vv i i X G

  G vv X′ f ##G zzvv g X ⊕ X f ⊕g

  Y Y G G v

  G G vv

  G G vv j j Y G vv G Y ′

  ##G {{vv X = j Y ◦ f Y ⊕ Y , X = j Y ◦ g ′ ′ ♦✉ s❡❥❛✱ (f ⊕ g) ◦ i ❡ (f ⊕ g) ◦ i ✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ p Y ◦ (f ⊕ g) = f ◦ π X

  ✱ s❡❣✉❡ q✉❡ p Y ◦ ((f ⊕ g) ◦ i X ) = (p Y ◦ (f ⊕ g)) ◦ i X = (f ◦ π X ) ◦ i X

  = f ◦ (π X ◦ i X ) = f = (p Y ◦ j Y ) ◦ f ◦ (j ◦ f ). = p Y Y

  Y ◦ (f ⊕ g) = g ◦ π ′ ′ X

  ❆✐♥❞❛✱ p ✱ s❡❣✉❡ q✉❡ ′ ′ p Y ◦ ((f ⊕ g) ◦ i X ) = (p Y ◦ (f ⊕ g)) ◦ i X = (g ◦ π X ) ◦ i X

  = g ◦ (π X X ◦ i X ) = Y = (p Y ◦ j Y ) ◦ f = p Y ◦ (j Y ◦ f ). X Y ◦ f

  P♦rt❛♥t♦✱ ♣❡❧♦ ▲❡♠❛ ✸✳✸ ✭❢❛③❡♥❞♦ φ = (f ⊕ g) ◦ i ❡ ψ = j ✮✱ X = j Y ◦ f X = j Y ◦ g ′ ′ r❡s✉❧t❛ q✉❡ (f ⊕ g) ◦ i ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ (f ⊕ g) ◦ i ✳ ❉❡✜♥✐çã♦ ✸✳✶✸ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ X ∈ Ob(C)✳ ❉❡✜♥✐✲ ♠♦s ♦s ♠♦r✜s♠♦s X : X → X ⊕ X ✭✐✮ ❞✐❛❣♦♥❛❧ ❞❡ X✱ ❞❡♥♦t❛❞♦ ♣♦r ∆ ✱ q✉❡ ❝♦♠✉t❛ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦

  22 I X X ;;v v v v v v 1 v π X v v

  ∆ X X // X ⊕ X

  G 2 G π G X G G G G I X ##G G

  ,, X : X ⊕ X → X X, ✭✐✐✮ ❝♦❞✐❛❣♦♥❛❧ ❞❡ X✱ ❞❡♥♦t❛❞♦ ♣♦r δ ✱ q✉❡ ❝♦♠✉t❛ ♦ ❞✐❛❣r❛♠❛ I X

  X v vv vv i X δ {{vv X vv yy oo

  X ⊕ X

  X ee ccHH i X HH HH HH I X H X.

  ❉❡❝♦rr❡ ❞❛s ❞❡✜♥✐çõ❡s ❞❡ ♣r♦❞✉t♦ ❡ ❝♦♣r♦❞✉t♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ X X q✉❡ ∆ ❡ δ ✱ ♣❛r❛ q✉❛❧q✉❡r X ∈ Ob(C)✱ sã♦ ♦s ú♥✐❝♦s ♠♦r✜s♠♦s q✉❡ ❝♦♠✉t❛♠ ♦s r❡s♣❡❝t✐✈♦s ❞✐❛❣r❛♠❛s✳ ❈♦♠ ❛ ♥♦t❛çã♦ ❛❝✐♠❛✱ ❡♥✉♥❝✐❛♠♦s ♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✳

  ▲❡♠❛ ✸✳✶✹ ✭❬✶✹❪✱ Pr♦♣♦s✐t✐♦♥ ✸✱ ♣✳✶✾✷✮ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ′ ′ ′ : X → Y = δ Y ◦ (f ⊕ f ) ◦ ∆ X

  ❡ ♠♦r✜s♠♦s f, f ✳ ❊♥tã♦ f + f ✱

  1

  2

  1

  2

  , π , i , i ) ❡♠ q✉❡ sã♦ ❝♦♥s✐❞❡r❛❞❛s ❛s s♦♠❛s ❞✐r❡t❛s (X ⊕ X, π X X X X ❡

  1

  2

  1

  2

  (Y ⊕ Y, p , p , j , j ) Y Y y Y ✳ ❉❡♠♦♥str❛çã♦✿ ◆♦t❡♠♦s q✉❡ ′ ′

  1

  1

  2

  2

  ◦ (f ⊕ f ◦ (f ⊕ f ◦ π ◦ π δ Y ) ◦ ∆ X = δ Y ) ◦ (i + i ) ◦ ∆ X X X X X

  1

  1

  2

  2

  = δ Y ◦ (f ⊕ f ) ◦ (i ◦ π ◦ ∆ X X X + i ◦ π ◦ ∆ X X X )

  1

  2

  = δ Y ◦ (f ⊕ f ) ◦ (i + i ) ′ ′ X X

  1

  2

  = δ Y ◦ ((f ⊕ f ) ◦ i + (f ⊕ f ) ◦ i ) Y Y

  1

  2

  = δ Y ◦ (j ◦ f + j ◦ f ) Y Y

  1

  2

  = δ Y ◦ j ◦ f + δ Y ◦ j ◦ f Y Y = f + f .

  ▲❡♠❛ ✸✳✶✺ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ Z ∈ Ob(C)✳ ❊♥tã♦ Z é Z : Z → Z ⊕ Z ✉♠ ♦❜❥❡t♦ ③❡r♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ∆ é ✉♠ ✐s♦♠♦r✜s♠♦✱

  1

  2

  , π ) ❡♠ q✉❡ (Z ⊕ Z, π Z Z é ✉♠ ♣r♦❞✉t♦ ❞❡ Z✳ i Z Z = I Z ◦ ∆ ❉❡♠♦♥str❛çã♦✿ (⇐) ❈♦♠♦ ∆ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡ π Z i Z = I Z⊕Z ◦ π ♣❛r❛ i = 1, 2✱ ❡♥tã♦ ∆ Z ✱ ♣❛r❛ i = 1, 2✱ ❡ ✐ss♦ ♥♦s ❞✐③ q✉❡

  1

  1

  2

  π = π = ∆ Z Z Z ✳ ❙❡❥❛♠ Y ∈ Ob(C) ❡ ♠♦r✜s♠♦s f, g : Y → Z✳ ❊♥tã♦ ✭♣❡❧❛ ❞❡✜♥✐çã♦

  ❞❡ ♣r♦❞✉t♦✮ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ φ : Y → Z × Z t❛❧ q✉❡ g =

  1

  2

  π ◦ φ = π ◦ φ = f Z Z ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r Y = Z✱ Z Z f = I Z Z = 0 ❡ g = 0 Z ❡ ❛ss✐♠✱ I Z ✳ ▲♦❣♦✱ Z é ✉♠ ♦❜❥❡t♦ ③❡r♦✱ ✈❡❥❛

  Pr♦♣♦s✐çã♦ ✶✳✸✵✳

  1

  2

  (⇒) = π Z ❈♦♠♦ Z é ✉♠ ♦❜❥❡t♦ ③❡r♦✱ s❡❣✉❡ q✉❡ π Z Z ✳ ❙❡♥❞♦ ∆ ♦

  1

  ◦ ∆ Z = I Z ♠♦r✜s♠♦ ❞✐❛❣♦♥❛❧✱ ❡♥tã♦ π Z ✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛

  Z ⊕ Z π Z 1 π π Z Z 1 Z 1

  ∆ Z

  Z ⊕ Z 1 F 1 π π Z F Z x F F xx

  F xx F F F xx

  ##F ||xx Z Z.

  ❚❛❧ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛✱ ♣♦✐s

  1

  1

  1

  1

  π ◦ (∆ Z ◦ π ) = (π ◦ ∆ Z ) ◦ π Z Z Z Z

  1

  =

  I Z ◦ π Z

  1 = π . Z

  1

  1

  1

  ◦ I Z⊕Z = π Z ◦ π = I Z⊕Z ▼❛s π Z Z ✳ P♦rt❛♥t♦✱ ∆ Z ❡ ✐ss♦ s❡❣✉❡ ❞♦ Z ◦ π Z⊕Z = π

  1

  1

  2

  ▲❡♠❛ ✸✳✸ ✭❢❛③❡♥❞♦ φ = ∆ Z ❡ ψ = I ✱ ✉♠❛ ✈❡③ q✉❡ π Z Z ✮✳ ❖ ♦❜❥❡t✐✈♦ ❞❛ ♣ró①✐♠❛ ❞❡✜♥✐çã♦ é✱ s♦❜r❡t✉❞♦✱ s✐♠♣❧✐✜❝❛r ♦s ❡♥✉♥✲

  ❝✐❛❞♦s ❞♦s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s✳ ❉❡✜♥✐çã♦ ✸✳✶✻ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s ❡ F : C → D ✉♠ ❢✉♥t♦r✳ ❙❡❥❛♠ X, Y ∈ Ob(C)✳ ❉✐③❡♠♦s q✉❡ X , π Y , i X , i Y ) ✭✐✮ F ♣r❡s❡r✈❛ s♦♠❛ ❞✐r❡t❛ s❡ (X ⊕Y, π é ✉♠❛ s♦♠❛ ❞✐r❡t❛ X ), F (π Y ), F (i X ), F (i Y )) ❞❡ X ❡ Y ✱ ❡♥tã♦ (F (X ⊕Y ), F (π é ✉♠❛ s♦♠❛ ❞✐r❡t❛ F (X) ❡ F (Y )❀ X , π Y ) ✭✐✐✮ F ♣r❡s❡r✈❛ ♣r♦❞✉t♦ s❡ (X ⊕ Y, π é ✉♠ ♣r♦❞✉t♦ ❞❡ X ❡ Y ✱ X ), F (π Y )) ❡♥tã♦ (F (X ⊕ Y ), F (π é ✉♠ ♣r♦❞✉t♦ ❞❡ F (X) ❡ F (Y )❀ X , i Y ) ✭✐✐✐✮ F ♣r❡s❡r✈❛ ❝♦♣r♦❞✉t♦ s❡ (X ⊕ Y, i é ✉♠ ❝♦♣r♦❞✉t♦ ❞❡ X ❡ Y ✱ X ), F (i Y )) ❡♥tã♦ (F (X ⊕ Y ), F (i é ✉♠ ❝♦♣r♦❞✉t♦ ❞❡ F (X) ❡ F (Y )✳ ▲❡♠❛ ✸✳✶✼ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s ❡ F : C → D ✉♠ ❢✉♥t♦r q✉❡ ♣r❡s❡r✈❛ ♣r♦❞✉t♦s✳ ❊♥tã♦ sã♦ ✈á❧✐❞❛s ❛s ❛✜r♠❛çõ❡s s❡❣✉✐♥t❡s✳ ✭✐✮ ❙❡ Z é ✉♠ ♦❜❥❡t♦ ③❡r♦ ❡♠ C ❡♥tã♦ F (Z) é ✉♠ ♦❜❥❡t♦ ③❡r♦ ❡♠ D✳ F X X (X)

  : X → Y ) = 0 ✭✐✐✮ ❙❡ 0 Y é ✉♠ ♠♦r✜s♠♦ ♥✉❧♦ ❡♠ C ❡♥tã♦ F (0 Y é F (Y ) ✉♠ ♠♦r✜s♠♦ ♥✉❧♦ ❡♠ D✳

  1

  2

  , π ) ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙❡❥❛ (F (Z) ⊕ F (Z), π ✉♠ ♣r♦❞✉t♦ ❞❡ F F

  (Z) (Z)

  1

  2 F (Z) ), F (π ))

  ✳ P♦r ❤✐♣ót❡s❡✱ ❛ tr✐♣❧❛ (F (Z ⊕ Z), F (π Z Z é ✉♠ ♣r♦❞✉t♦ ❞❡ F (Z)✳ Z :

  ❙❡❣✉❡✱ ❞❛ ❞❡✜♥✐çã♦ ❞♦ ♣r♦❞✉t♦✱ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ f

  1

  1

  2 F (Z ⊕ Z) → F (Z) ⊕ F (Z) ◦ f Z = F (π ) ◦ f Z =

  t❛❧ q✉❡ π ❡ π F Z F

  (Z) (Z)

2 F (π ) Z

  Z ✳ ❖ ♠♦r✜s♠♦ f é ✉♠ ✐s♦♠♦r✜s♠♦ ❡ ✐ss♦ s❡❣✉❡ ❞❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♣r♦❞✉t♦✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛

  F F (Z) I F (Z) F (Z) (∆ Z ) I F (Z ⊕ Z) f Z π R π 1 R F (Z) ⊕ F (Z) 2 F (Z) lll R F (Z) R R R R lll

  R R R lll

  R R lll R

  R ))R uulll

  F (Z) F (Z). ❚❛❧ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛✱ ♣♦✐s

  1

  1

  π ◦ (f Z ◦ F (∆ Z )) = (π ◦ f Z ) ◦ F (∆ Z ) F (Z) F (Z)

  1

  = F (π ) ◦ F (∆ Z ) Z

  1

  = F (π ◦ ∆ Z ) Z = F (I Z ) = I F .

  (Z)

  2

  ◦ (f Z ◦ F (∆ Z )) = I F ❆♥❛❧♦❣❛♠❡♥t❡✱ π (Z) ✳ ❙❡❣✉❡✱ ❞❛ ✉♥✐❝✐❞❛❞❡ F (Z)

  = f Z ◦ F (∆ Z ) ❞♦ ♠♦r✜s♠♦ ❞✐❛❣♦♥❛❧✱ q✉❡ ∆ F (Z) ✳ ❈♦♠♦ Z é ✉♠ ♦❜❥❡t♦ Z Z ) ③❡r♦✱ r❡s✉❧t❛ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r q✉❡ ∆ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡ ❛ss✐♠✱ F (∆ t❛♠❜é♠ ♦ é✳ P♦rt❛♥t♦✱ ∆ F (Z) é ✉♠ ✐s♦♠♦r✜s♠♦✳ ◆♦✈❛♠❡♥t❡✱ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ F (Z) é ✉♠ ♦❜❥❡t♦ ③❡r♦ ❡♠ D✳ X

  : X → Y ✭✐✐✮ ❉❛❞♦s X, Y ∈ Ob(C)✱ s❡ 0 é ✉♠ ♠♦r✜s♠♦ ♥✉❧♦ ❡♠ C X (X) F Y

  ) = 0 ❡♥tã♦ F (0 ❡ ✐ss♦ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞♦ ✐t❡♠ ✭✐✮✳ Y F

  (Y )

  ❉❡✜♥✐çã♦ ✸✳✶✽ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s ♣ré✲❛❞✐t✐✈❛s✳ ❯♠ ❢✉♥t♦r F : C

  → D é ❞✐t♦ ❛❞✐t✐✈♦ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r f, g : X → Y ♠♦r✜s♠♦s ❡♠ C✱

  ❡♥tã♦ F (f + g) = F (f ) + F (g). Pr♦♣♦s✐çã♦ ✸✳✶✾ ✭❬✶✹❪✱ Pr♦♣♦s✐t✐♦♥ ✹✱ ♣✳ ✶✾✸✮ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s ❡ F : C → D ✉♠ ❢✉♥t♦r✳ ❙❡❥❛♠ X, Y ∈ Ob(C)✳ ❊♥tã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s

  ✭✐✮ F é ❛❞✐t✐✈♦❀ ✭✐✐✮ F ♣r❡s❡r✈❛ s♦♠❛ ❞✐r❡t❛❀ ✭✐✐✐✮ F ♣r❡s❡r✈❛ ♣r♦❞✉t♦❀ ✭✐✈✮ F ♣r❡s❡r✈❛ ❝♦♣r♦❞✉t♦✳ X , π Y , i X , i Y ) ❉❡♠♦♥str❛çã♦✿ (i) ⇒ (ii) ❙❡❥❛ (X ⊕ Y, π ✉♠❛ s♦♠❛ ❞✐✲ X ), F (π Y ), F (i X ), F (i Y )) r❡t❛ ❞❡ X ❡ Y ✳ ▼♦str❡♠♦s q✉❡ (F (X⊕Y ), F (π X ◦i X = I X

  é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ F (X) ❡ F (Y )✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ π ✱ s❡❣✉❡ q✉❡

  I F = F (I X )

  (X)

  ◦ i = F (π X X )

  = F (π X ) ◦ F (i X ). Y ) ◦ F (i Y ) = I F X ◦ π ◦ π X + i Y Y = ❆♥❛❧♦❣❛♠❡♥t❡✱ F (π (Y ) ❡ s❡♥❞♦ i

  I X⊕Y ✱ s❡❣✉❡ q✉❡

  I = F (I X⊕Y ) F (X⊕Y ) = F (i X ◦ π X + i Y ◦ π Y ) = F (i X ◦ π X ) + F (i Y ◦ π Y )

  ❋ é ❛❞✐t✐✈♦ = F (i X ) ◦ F (π X ) + F (i Y ) ◦ F (π Y ). X ), F (π Y ), F (i X ), F (i Y ))

  P♦rt❛♥t♦✱ ❛ q✉í♥t✉♣❧❛ (F (X ⊕ Y ), F (π é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ F (X) ❡ F (Y )✳

  1

  2

  1

  2

  (ii) ⇒ (i) , π , i , i ) ❙❡❥❛♠ X ∈ Ob(C) ❡ (X ⊕ X, π X X X X ✉♠❛ s♦♠❛

  1

  2

  1

  2

  ), F (π ), F (i ), F (i )) ❞✐r❡t❛ ❞❡ X✳ P♦r ✭✐✐✮✱ (F (X ⊕ X), F (π X X X X é ✉♠❛

  1

  2

  ), F (π )) s♦♠❛ ❞✐r❡t❛ ❞❡ F (X) ❡ ✐ss♦ ♥♦s ❞á (F (X ⊕ X), F (π X X é ✉♠ ♣r♦❞✉t♦ ❞❡ F (X)✳

  

1

  2

  1

  2

  , π , i , i ) P♦r ♦✉tr♦ ❧❛❞♦✱ (F (X)⊕F (X), π é t❛♠❜é♠ F F F F

  (X) (X) (X) (X)

  1

  2

  , π ) ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ F (X) ❡ ♣♦rt❛♥t♦✱ (F (X) ⊕ F (X), π é F F

  (X) (X)

  t❛♠❜é♠ ✉♠ ♣r♦❞✉t♦ ❞❡ F (X)✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦

  1

  1

  f X : F (X ⊕ X) → F (X) ⊕ F (X) ◦ f X = F (π ) t❛❧ q✉❡ π F (X) X ❡

  2

  2

  π ◦ f F (X) X = F (π ) X ✳ ◆♦t❡♠♦s q✉❡

  1

  1

  1

  1

  π ◦ (f F F X ◦ F (i )) = (π ◦ f X X ) ◦ F (i ) X

  (X) (X)

  1

  1

  = F (π ) ◦ F (i ) X X

  1

  1

  = F (π ◦ i ) X X = F (I X ) =

  I F (X)

  1

  1

  = π ◦ i F F

  (X) (X)

  ❡ π

1 X

1 X

  ◦ (f X ◦ F (∆ X )) = (π

  2 X

  ) = i

  2 F (X)

  ✳ ❊ss❛s ✐❣✉❛❧❞❛❞❡s sã♦ ✉s❛❞❛s ♠❛✐s ❛❜❛✐①♦✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛ F (X) F (∆ X

  ) I F (X) I F (X)

  F (X ⊕ X) f X F (X) ⊕ F (X) π 1 F (X) uukkk kkk kkk kkk kkk π 2 F (X)

  ))S S S S S S S S S S S S S S S F (X) F (X).

  ◆♦t❡♠♦s q✉❡ π

  1 F (X)

  ◦ f X ) ◦ F (∆ X ) = F (π

  1 F (X)

  1 F (X)

  1 X

  ) ◦ F (∆ X ) = F (π

  1 X

  ◦ ∆ X ) = F (I X ) =

  I F (X) . ❆♥❛❧♦❣❛♠❡♥t❡✱ π

  ◦ (f X ◦ F (∆ X )) = I F

  (X) ✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦

  ♠♦r✜s♠♦ ❞✐❛❣♦♥❛❧✱ t❡♠♦s q✉❡ ∆ F (X) = f X ◦ F (∆ X )

  ✳ ❆❣♦r❛✱ ♦✉tr♦

  ❡ f X ◦ F (i

  ) = i

  2 F (X)

  2 X

  ◦ (f X ◦ F (i

  )) = (π

  2 F (X)

  ◦ f X ) ◦ F (i

  1 X

  ) = F (π

  2 X

  ) ◦ F (i

  1 X

  ) = F (π

  ◦ i

  ◦ F (i

  1 X

  ) = F (0 X X )

  

(∗)

  = F

  (X) F (X)

  = π

  2 F (X)

  ◦ i

  1 F (X)

  . ❈♦♠♦ F ♣r❡s❡r✈❛ s♦♠❛ ❞✐r❡t❛✱ F ♣r❡s❡r✈❛ ♣r♦❞✉t♦ ✭♣❛r❛ ❝♦♠♦❞✐✲

  ❞❛❞❡ ❞♦ ❧❡✐t♦r✱ ✈❡❥❛ ✐♠♣❧✐❝❛çã♦ ✭✐✐✮ ⇒ ✭✐✐✐✮ ♠❛✐s ❛ ❢r❡♥t❡✮ ❡ ❛ss✐♠✱ ❛♣❧✐❝❛✲ ♠♦s ♦ ▲❡♠❛ 3.17 ✭✐✐✮ ♣❛r❛ ♦❜t❡r♠♦s ❛ ✐❣✉❛❧❞❛❞❡ ✭✯✮✳ ❋✐♥❛❧♠❡♥t❡✱ ♣❡❧♦ ▲❡♠❛ ✸✳✸✱ f X

2 F (X)

  ❞✐❛❣r❛♠❛ F (X)

1 X

  1 Y

  2 F (X)

  = I F

  (X) ✳ ❉❛ ✉♥✐❝✐❞❛❞❡ ❞♦

  ♠♦r✜s♠♦ ❝♦❞✐❛❣♦♥❛❧✱ s❡❣✉❡ q✉❡ δ F (X) = F (δ X ) ◦ f

  1 X

  ✳ ❋✐♥❛❧♠❡♥t❡✱ s❡❥❛ Y ∈ Ob(C) ♦✉tr♦ ♦❜❥❡t♦✳ ❈♦♥s✐❞❡r❡♠♦s ❛ s♦♠❛

  ❞✐r❡t❛ (Y ⊕ Y, π

  , π

  ❆♥❛❧♦❣❛♠❡♥t❡✱ (F (δ X ) ◦ f

  2 Y

  , i

  1 Y

  , i

  2 Y

  ) ✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♣❛r❛ ♦ ♦❜❥❡t♦ X✱

  ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ f Y : F (Y ⊕ Y ) → F (Y ) ⊕ F (Y ) ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡s❝r✐t❛s ❛❝✐♠❛✳

  ) ◦ i

  (X) .

  F (X ⊕ X) F

  1 F (X)

  (δ X )

  OO F (X) ⊕ F (X) f 1 X OO

  F (X) i 2 F (X) 55k k k k k k k k k k k k k k k I F (X)

  AA F (X). i 1 F (X) iiSSS SSSS SSSS SSSS I F (X)

  ]] ◆♦t❡♠♦s q✉❡

  (F (δ X ) ◦ f

  ) ◦ i

  = F (δ X ) ◦ (f

  I F

  1 X

  ◦ i

  1 F (X)

  ) = F (δ X ) ◦ F (i

  1 X

  ) = F (δ X ◦ i

  1 X

  ) = F (I X ) =

1 X

  ❙❡❥❛♠ f, f : X → Y

1 X

  ✳ ❋✐♥❛❧♠❡♥t❡✱

  = (δ F

  (∗∗)

  ) ◦ ∆ X ) = F (δ Y ) ◦ F (f ⊕ f ) ◦ F (∆ X )

  = F (δ Y ◦ (f ⊕ f

  (∗)

  F (f + f )

  ) = f Y ◦ F (f ⊕ f ) ◦ f

  1 X

  ◦ f Y ) ◦ (f

  ) ✱ s❡❣✉❡ q✉❡ F (f) ⊕ F (f

  ✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♠♦r✜s♠♦ F (f) ⊕ F (f

  2 F (X)

  ) = F (f ) ◦ π

  ◦ (f Y ◦ F (f ⊕ f ) ◦ f

  2 F (Y )

  (Y )

  ◦ (F (f ) ⊕ F (f )) ◦ f X ) ◦ (f

  

1

Y

  (X)

  ) = F (f ) + F (f ) ❡ F é ❛❞✐t✐✈♦✳

  ✳ P♦rt❛♥t♦✱ F (f + f

  1 X

  ❡ F (f)⊕F (f ) = f Y ◦F (f ⊕f )◦f

  1 Y

  = F (δ Y )◦f

  = f X ◦F (∆ X ) ✱ δ F (Y )

  = F (f ) + F (f ), ❡♠ ✭✯✮ ❡ ✭✯✯✯✮ ✉t✐❧✐③❛♠♦s ♦ ▲❡♠❛ 3.14 ❡ ❡♠ ✭✯✯✮ ✉t✐❧✐③❛♠♦s q✉❡ ∆ F

  . ❆ ✐❣✉❛❧❞❛❞❡ (△) r❡s✉❧t❛ ❞❡ ✭✶✮ ❞❛ ❉❡✜♥✐çã♦ ✸✳✶✶✳ ❆♥❛❧♦❣❛♠❡♥t❡✱

  (X) (∗∗∗)

  ◦ (F (f ) ⊕ F (f )) ◦ ∆ F

  (Y )

  ) = δ F

  (X)

  ◦ ∆ F

  1 X

  π

  1 F (X)

  ♠♦r✜s♠♦s✳ ❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ F (X) ⊕ F (X) π 1 F (X) uukkk kkk kkk kkk kkk π 2 F (X)

  ))S S S S S S S S S S S S S S S F (Y ) F (Y ).

  1 X

  ◦ f Y ) ◦ F (f ⊕ f ) ◦ f

  1 F (Y )

  ) = (π

  ◦ (f Y ◦ F (f ⊕ f ) ◦ f

  1 F (Y )

  ◆♦t❡♠♦s q✉❡ π

  F (Y ⊕ Y ) f Y F (Y ) ⊕ F (Y ) π 1 F (Y ) uukkk kkk kkk kkk kkk π 2 F (Y )

  1 Y

  F (f )

  F (X)

  (f ⊕f )

  F (X ⊕ X) F

  F (f )

  F (X)

  ))S S S S S S S S S S S S S S S f 1 X

  = F (π

  ) ◦ F (f ⊕ f ) ◦ f

  = F (f ) ◦ π

  = F (f ) ◦ F (π

  1 X

  ◦ f X ◦ f

  1 F (X)

  = F (f ) ◦ π

  1 X

  ) ◦ f

  1 X

  1 X

  1 X

  ) ◦ f

  1 X

  = F (f ◦ π

  1 X (△)

  ◦ (f ⊕ f )) ◦ f

  1 Y

  = F (π

1 X

  (ii) ⇒ (iii) X , π Y ) ❙❡❥❛ (X ⊕ Y, π ✉♠ ♣r♦❞✉t♦ ❞❡ X ❡ Y ✳ ❊♥tã♦✱ X , π Y , i X , i Y )

  ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✶✵✱ (X ⊕ Y, π é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ X X Y ❡ Y ✭i ❡ i sã♦ ♠♦r✜s♠♦s ❝♦♠♦ ♥❛ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✶✵✮✳ P♦r X ), F (π Y ), F (i X ), F (i Y )) ✭✐✐✮✱ (F (X ⊕Y ), F (π é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❡✱ ♣❡❧❛ X ), F (π Y )) Pr♦♣♦s✐çã♦ ✸✳✽✱ (F (X ⊕ Y ), F (π é ✉♠ ♣r♦❞✉t♦ ❞❡ F (X) ❡ F (Y )

  ✳ ▲♦❣♦✱ F ♣r❡s❡r✈❛ ♣r♦❞✉t♦✳ ❆ ✐♠♣❧✐❝❛çã♦ (ii) ⇒ (iv) é ❛♥á❧♦❣❛✳ ❆❣♦r❛ ♣r♦✈❡♠♦s ❛ ✐♠♣❧✐❝❛çã♦ (iii) ⇒ (ii)✱ ❛ ✐♠♣❧✐❝❛çã♦ (iv) ⇒ (ii) é

  ❢❡✐t❛ ❞❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r✳ (iii) ⇒ (ii) X , π Y , i X , i Y )

  ❙❡❥❛ (X ⊕ Y, π ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ X ❡ Y ✳ X , π Y ) P❡❧❛ Pr♦♣♦s✐çã♦ 3.8✱ (X ⊕ Y, π é ✉♠ ♣r♦❞✉t♦ ❞❡ X ❡ Y ✳ X ), F (π Y ))

  P♦r ✭✐✐✐✮✱ (F (X ⊕Y ), F (π é ✉♠ ♣r♦❞✉t♦ ❞❡ F (X) ❡ F (Y )✳ , π )

  P♦r ♦✉tr♦ ❧❛❞♦✱ (F (X)⊕F (Y ), π F (X) F (Y ) é t❛♠❜é♠ ✉♠ ♣r♦❞✉t♦ ❞❡ F (X)

  ❡ F (Y )✳ X,Y : F (X ⊕ Y ) → F (X) ⊕ ❆ss✐♠✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ f

  ◦ f ◦ f F (Y ) F X,Y = F (π X ) F X,Y = F (π Y ) t❛❧ q✉❡ π (X) ❡ π (Y ) ✳ X ), F (π Y ), F (i X ), F (i Y ))

  ▼♦str❡♠♦s q✉❡ (F (X⊕Y ), F (π é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ F (X) ❡ F (Y )✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛ F (i ) F (X) X I F (X) F (Y ) F (X)

  F (X ⊕ Y ) f X,Y F (X) ⊕ F (Y )

  R π R π F (X) kkk R F (Y ) R R R kkk R

  R R kkk

  R R R kkk R

  R uukkk ))R F (X) F (Y )

  ◆♦t❡♠♦s q✉❡ π F ◦ f X,Y ◦ F (i X ) = F (π X ) ◦ F (i X )

  (X)

  = F (π X ◦ i X ) = F (I X ) = I F .

  (X)

  ❆❧é♠ ❞✐ss♦✱ π F ◦ f X,Y ◦ F (i X ) = F (π Y ) ◦ F (i X )

  (Y )

  = F (π Y ◦ i X X ) = F (0 ) Y

  (∗) F (X)

  = , F

  (Y )

  ❡♠ ✭✯✮ ✉t✐❧✐③❛♠♦s ♦ ▲❡♠❛ 3.17✳ ❙❛❜❡♠♦s✱ ❞❛ Pr♦♣♦s✐çã♦ 3.10✱ q✉❡ ♦ F : F (X) → F (X) ⊕ F (Y ) ♠♦r✜s♠♦ i (X) é ♦ ú♥✐❝♦ q✉❡ ❝♦♠✉t❛ t❛❧ F = f X,Y ◦ F (i X ) ❞✐❛❣r❛♠❛✳ ❆ss✐♠✱ i (X) ✳ ❈♦♠ r❛❝✐♦❝í♥✐♦ ✐♥t❡✐r❛♠❡♥t❡ F = f X,Y ◦ F (i Y ) ❛♥á❧♦❣♦✱ r❡s✉❧t❛ q✉❡ i (Y ) ✳ F ◦ i F = π F ◦ f X,Y ◦ F (i X ) = I F

  P♦rt❛♥t♦✱ π (X) (X) (X) (X) ❡ ❞❛ ♠❡s♠❛ F ◦ i F = I F ❢♦r♠❛ π (Y ) (Y ) (Y ) ✳ F , π F )

  ❆❣♦r❛ ♦❜s❡r✈❛♠♦s q✉❡ ❝♦♠♦ (F (X) ⊕ F (Y ), π (X) (Y ) é ✉♠ ♣r♦❞✉t♦ ❞❡ F (X) ❡ F (Y ) s❡❣✉❡✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✶✵✱ q✉❡ (F (X) ⊕ F (Y ), π F , π F , i F , i F ) F = i F ◦ π F + i F ◦ π F .

  (X) (Y ) (X) (Y ) é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ F (X) ❡ F (Y )✳

  ❉❛í✱ I (X)⊕F (Y ) (X) (X) (Y ) (Y ) ❆ss✐♠✱

1 I = f ◦ I ◦ f X,Y

  F F

  (X⊕Y ) X,Y (X)⊕F (Y )

  1

  ◦ (i ◦ π ◦ π = f F F + i F F ) ◦ f X,Y X,Y (X) (X) (Y ) (Y ) − −

  1

  1

  ◦ i ◦ π ◦ f ◦ i ◦ π ◦ f = f F F X,Y + f F F X,Y X,Y (X) (X) X,Y (Y ) (Y ) = F (i X ) ◦ F (π X ) + F (i Y ) ◦ F (π Y ).

  ❚❡♦r❡♠❛ ✸✳✷✵ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✱ F : C → D ❡ G : D

  → C ❢✉♥t♦r❡s t❛✐s q✉❡ (F, G, α) s❡❥❛ ✉♠❛ ❛❞❥✉♥çã♦ ❞❡ C ❛ D✳ ❊♥tã♦

  F ❡ G sã♦ ❢✉♥t♦r❡s ❛❞✐t✐✈♦s✳

  ❉❡♠♦♥str❛çã♦✿ Pr♦✈❡♠♦s q✉❡ G é ❛❞✐t✐✈♦✳ P❛r❛ ✐ss♦✱ ✉t✐❧✐③❛♠♦s ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ♠♦str❛♥❞♦ q✉❡ G ♣r❡s❡r✈❛ ♣r♦❞✉t♦s✳ ❙❡❥❛ (X ⊕ Y, π X , π Y )

  ✉♠ ♣r♦❞✉t♦ ❞❡ X ❡ Y ♦❜❥❡t♦s ❡♠ D✳ Pr❡❝✐s❛♠♦s ♠♦str❛r X ), G(π Y )) q✉❡ (G(X ⊕ Y ), G(π é ✉♠ ♣r♦❞✉t♦ ❞❡ G(X) ❡ G(Y )✳ G , p G ) G : W → G(X) ❙❡❥❛ (W, p (X) (Y ) ✉♠❛ tr✐♣❧❛ t❛❧ q✉❡ p (X) ❡ p G : W → G(Y )

  

(Y ) s❡❥❛♠ ♠♦r✜s♠♦s ❡♠ C✳ ❈♦♥s✐❞❡r❡♠♦s ♦s ✐s♦♠♦r✜s✲

  ♠♦s ♥❛t✉r❛✐s D C α W,X : Hom (F (W ), X) → Hom (W, G(X)) D C

  α W,Y : Hom (F (W ), Y ) → Hom (W, G(Y )) X : F (W ) → X Y : F (W ) → ❡ ♣♦rt❛♥t♦✱ ❡①✐st❡♠ ú♥✐❝♦s ♠♦r✜s♠♦s q ❡ q Y t❛✐s q✉❡

  α W,X (q X ) = p W,Y (q Y ) = p . G (X) ❡ α G (Y ) X , π Y )

  ❈♦♠♦ (X ⊕ Y, π é ✉♠ ♣r♦❞✉t♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ φ : F (W ) → X ⊕ Y X = π X ◦ φ Y = π Y ◦ φ t❛❧ q✉❡ q ❡ q ✳ ❈♦♥s✐❞❡r❡♠♦s t❛♠❜é♠ ♦ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ D C α W,X⊕Y : Hom (F (W ), X ⊕ Y ) → Hom (W, G(X ⊕ Y )). C

  (W, G(X⊕ ❘❡❧❛t✐✈♦ ❛♦ ♠♦r✜s♠♦ φ✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ψ ∈ Hom

  Y )) W,X⊕Y (φ) t❛❧ q✉❡ ψ = α ✳ ❈♦♠♦ α é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✱ ✈❛❧❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ α W,X⊕Y D C // Hom D (F (I ),π ) Hom C (I ,G (π )) Hom (F (W ), X ⊕ Y ) Hom (W, G(X ⊕ Y )) W D // Hom C X W X Hom (F (W ), X) (W, G(X)). α W,X

  ◆♦t❡♠♦s q✉❡ D (α W,X ◦ Hom (F (I W ), π X ))(φ) = α W,X (π X ◦ φ) = α W,X (q X ) = p G

  (X)

  ❡ C C (Hom (I W , G(π X ))◦α W,X⊕Y )(φ) = Hom (I W , G(π G = G(π X ))(ψ) = G(π X ) ◦ ψ X )◦ψ.

  P❡❧❛ ❝♦♠✉t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛✱ s❡❣✉❡ q✉❡ p (X) ✳ ❉❡ G = G(π Y ) ◦ ψ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ♦❜t❡♠♦s p (Y ) ✳

  : W → G(X ⊕ Pr♦✈❡♠♦s q✉❡ ψ é ú♥✐❝❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ψ ′ ′

  Y ) G = G(π X ) ◦ ψ G = G(π Y ) ◦ ψ W,X⊕Y t❛❧ q✉❡ p (X) ❡ p (Y ) ✳ ❈♦♠♦ α : F (W ) → X ⊕ Y

  é ✐s♦♠♦r✜s♠♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ φ t❛❧ q✉❡ ′ ′ α W,X⊕Y (φ ) = ψ C // D ✳ ❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ α W,X⊕Y 1 Hom C ,G Hom D Hom (W, G(X ⊕ Y )) Hom (F (W ), X ⊕ Y )

  (I W (π X )) (F (I W ),π C D X ) Hom (W, G(X)) // Hom (F (W ), X). α W,X 1

  ❆ss✐♠✱ − − D D 1 ′ 1 ′ (Hom (F (I W ), π X ) ◦ α )(ψ ) = Hom (F (I W ), π W,X⊕Y W,X⊕Y D X )(α (ψ ))

  = Hom (F (I W ), π X )(φ ) ◦ φ

  = π X ,

  ❡ − −

  1 ′ C C 1 ′

  (α ◦ Hom (I W , G(π W,X W,X X )))(ψ ) = α (Hom (I W , G(π X ))(ψ ))

  1 ′

  = α (G(π W,X X ) ◦ ψ )

  1

  = α (p ) W,X G (X) X ◦ φ = q ′ ′ X Y ◦ φ = q Y = q X .

  ▲♦❣♦✱ π ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ π ✳ P♦rt❛♥t♦✱ ❞❛ ′ ′ = φ = ψ

  ✉♥✐❝✐❞❛❞❡ ❞❡ φ s❡❣✉❡ q✉❡ φ ✳ ❆ss✐♠✱ ψ ✳ ▲♦❣♦✱ (G(X ⊕ Y ), G(π X ), G(π Y ))

  é ✉♠ ♣r♦❞✉t♦ ❞❡ G(X) ❡ G(Y )✳ P❛r❛ ♠♦str❛r q✉❡ F

  é ❛❞✐t✐✈♦✱ ❛ ♣r♦✈❛ é s✐♠✐❧❛r✳ ❉❡✜♥✐çã♦ ✸✳✷✶ ❯♠❛ ❝❛t❡❣♦r✐❛ C é ❞✐t❛ ❛❜❡❧✐❛♥❛ s❡ ✭✐✮ C é ❛❞✐t✐✈❛❀ ✭✐✐✮ t♦❞♦ ♠♦r✜s♠♦ ❡♠ C ♣♦ss✉✐ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦❀ ✭✐✐✐✮ t♦❞♦ ♠♦♥♦♠♦r✜s♠♦ é ✉♠ ♥ú❝❧❡♦ ❡ t♦❞♦ ❡♣✐♠♦r✜s♠♦ é ✉♠ ❝♦♥ú❝❧❡♦✳ R M ❊①❡♠♣❧♦ ✸✳✷✷ ❙❡❥❛ R ✉♠ ❛♥❡❧✳ ❆ ❝❛t❡❣♦r✐❛ é ❛❜❡❧✐❛♥❛✳ ❉❡ ❢❛t♦✱ ♣♦✐s ❛ ♠❡s♠❛ é ❝❧❛r❛♠❡♥t❡ ❛❞✐t✐✈❛✳ ❉❛❞♦ ✉♠ ♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s f : X → Y

  ✱ s❡✉ ♥ú❝❧❡♦ é ♦ ♣❛r (Ker(f), i)✱ ❡♠ q✉❡ Ker(f) = {m ∈ X : f (m) = 0} ❡ i : Ker(f) → X é ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✳

  ❖ ❝♦♥ú❝❧❡♦ ❞❡ f é ♦ ♣❛r (Y/Im(f), π)✱ ❡♠ q✉❡ Y/Im(f) é ♦ q✉♦✲ ❝✐❡♥t❡ ❞♦ R✲♠ó❞✉❧♦ Y ♣❡❧❛ Im(f) ❡ π : Y → Y/Im(f) é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳

  ❉❛❞♦ ✉♠ ♠♦♥♦♠♦r✜s♠♦ f : Z → W ✱ ❡st❡ é ♥ú❝❧❡♦ ❞♦ ♠♦r✜s♠♦ p : W → W/f (Z)

  ✭♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✮ ❡✱ ♣❛r❛ ✉♠ ❡♣✐♠♦r✜s♠♦ q✉❛❧q✉❡r✱ g : W → Z ✱ ❡st❡ é ❝♦♥ú❝❧❡♦ ❞♦ ♠♦r✜s♠♦ j : Ker(g) → W ✭✐♥❝❧✉sã♦

  ❝❛♥ô♥✐❝❛✮✳ ▲❡♠❛ ✸✳✷✸ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✳ ❊♥tã♦ t♦❞♦ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C é ✉♠ ♥ú❝❧❡♦ ❞❡ s❡✉ ❝♦♥ú❝❧❡♦✳ ❉✉❛❧♠❡♥t❡✱ t♦❞♦ ❡♣✐♠♦r✜s♠♦ ❡♠ C

  é ✉♠ ❝♦♥ú❝❧❡♦ ❞❡ s❡✉ ♥ú❝❧❡♦✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ f : X → Y ✉♠ ❡♣✐♠♦r✜s♠♦ ❡♠ C ❡ (Ker(f), k) ✉♠ ♥ú❝❧❡♦ ❞❡ f✳ ❈♦♠♦ C é ❛❜❡❧✐❛♥❛✱ ♣♦r ✭✐✐✐✮ ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ g : Z → X t❛❧ q✉❡ ♦ ♣❛r (Y, f) é ✉♠ ❝♦♥ú❝❧❡♦ ❞❡ g✳ ❆ss✐♠✱ Z f ◦ g = 0 Y ✳

  ❈♦♠♦ (Ker(f), k) é ✉♠ ♥ú❝❧❡♦ ❞❡ f✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u : Z → Ker(f ) t❛❧ q✉❡ k ◦ u = g✳ ▼♦str❡♠♦s q✉❡ ♦ ♣❛r (Y, f) é ✉♠

  Ker (f )

  ❝♦♥ú❝❧❡♦ ❞❡ k✳ ❙❛❜❡♠♦s q✉❡ f ◦ k = 0 Y ✳ ❙❡❥❛ (W, h)✱ h : X → W ✱ Ker

  

(f )

  ✉♠ ♦✉tr♦ ♣❛r t❛❧ q✉❡ h ◦ k = 0 W ✳ ❊♥tã♦ h ◦ g = h ◦ (k ◦ u) = (h ◦ k) ◦ u Ker

  (f )

  = ◦ u Z W = . W : Y →

  ❈♦♠♦ (Y, f) é ✉♠ ❝♦♥ú❝❧❡♦ ❞❡ g✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u W ◦ f = h t❛❧ q✉❡ u ❡ s❡❣✉❡ ♦ ❞❡s❡❥❛❞♦✳ ❖ ♦✉tr♦ ❝❛s♦ é s✐♠✐❧❛r✳ ❉❡✜♥✐çã♦ ✸✳✷✹ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❯♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛ C é ❞✐t❛ k✲ C

  (X, Y ) ❧✐♥❡❛r s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦s X, Y ∈ Ob(C)✱ ♦ ❝♦♥❥✉♥t♦ Hom é ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s é k✲❜✐❧✐♥❡❛r✳ k ❊①❡♠♣❧♦ ✸✳✷✺ ❆ ❝❛t❡❣♦r✐❛ V ect é ❝❧❛r❛♠❡♥t❡ ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r✱ ❡♠ q✉❡ ♦ ♣r♦❞✉t♦ ♣♦r ❡s❝❛❧❛r ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♠♦r✜s♠♦s é ❞❡✜♥✐❞♦ ♣♦♥t♦ ❛ ♣♦♥t♦✱ ♣❛r❛ ❝❛❞❛ ♠♦r✜s♠♦✳ ❉❡✜♥✐çã♦ ✸✳✷✻ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s k✲❧✐♥❡❛r❡s✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❢✉♥t♦r F : C → D é k✲❧✐♥❡❛r s❡ F é ❛❞✐t✐✈♦ ❡ F (rf) = rF (f)✱ ♣❛r❛ t♦❞♦ r ∈ k

  ❡ t♦❞♦ ♠♦r✜s♠♦ f ❡♠ C✳ k k → V ect ❊①❡♠♣❧♦ ✸✳✷✼ ❖ ❢✉♥t♦r D : V ect ❞♦ ❊①❡♠♣❧♦ 2.11 é k✲ ❧✐♥❡❛r✳

  ❙❡❥❛♠ V ❡ W k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❖❜s❡r✈❛♠♦s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r

  f, g ∈ Hom V ect (V, W ) k ❡ r ∈ k✱ t❡♠♦s ((f + rg) (h))(v) = (h ◦ (f + rg))(v)

  = h((f + rg)(v)) = h(f (v) + rg(v)) = h(f (v)) + rh(g(v)) ∗ ∗ = f (h)(v) + rg (h)(v) ∗ ∗ ∗ ∗ ∗ ∗ = (f + rg )(h)(v),

  = f +rg ♣❛r❛ q✉❛✐sq✉❡r h ∈ W ❡ v ∈ V ✳ P♦rt❛♥t♦✱ (f +rg) ✳ ❆ss✐♠✱ ∗∗

  D(f + rg) = (f + rg) ∗ ∗ ∗ = (f + rg ) ∗∗ ∗∗ = f + rg = D(f ) + rD(g).

  ❈❛♣ít✉❧♦ ✹ ❊q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ❝❛t❡❣♦r✐❛s k✲❧✐♥❡❛r❡s

  ❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ ❝❛♣ít✉❧♦ é ❞❡✜♥✐r ✉♠❛ ♥♦✈❛ ❝❛t❡❣♦r✐❛✱ ❝❤❛♠❛❞❛ ❡q✉✐✈❛r✐❛♥t✐③❛çã♦✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r ❞❛❞❛✳ P❛r❛ ✐ss♦✱ ♥❡❝❡ss✐t❛♠♦s ❞❛ ♥♦çã♦ ❞❡ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦ ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r✳ ❈♦♠♦ ♠♦t✐✈❛çã♦ ♣❛r❛ ❡ss❛ ❝♦♥str✉çã♦✱ ♦❜s❡r✈❛♠♦s q✉❡ ♣❛r❛ ✉♠❛ ❝❛✲ t❡❣♦r✐❛ k✲❧✐♥❡❛r ✜♥✐t❛ ✭q✉❡ ♥ã♦ ♥♦s ❝♦♥✈é♠ ❞❡✜♥✐r ❛❣♦r❛✮ ❡①✐st❡ ✉♠ r❡s✉❧t❛❞♦✱ q✉❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ [✻]✱ q✉❡ ♥♦s ❣❛r❛♥t❡ q✉❡ ❡❧❛ s❡rá ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ❞❡ A✲♠ó❞✉❧♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ ✉♠ ❝♦r♣♦ k✱ s❡♥❞♦ A ✉♠❛ á❧❣❡❜r❛ ✜♥✐t♦ ❞✐♠❡♥s✐♦♥❛❧✳ ■ss♦ ♦r✐❣✐♥❛ ✉♠❛ r❡✲ ❧❛çã♦ ❝♦♠ ❛ t❡♦r✐❛ ❞❡ r❡♣r❡s❡♥t❛çõ❡s✱ q✉❡ ❞❡✉ ♦r✐❣❡♠ à ❝♦♥str✉çã♦ ❞❛ ❡q✉✐✈❛r✐❛♥t✐③❛çã♦✳ G

  ❙❡❥❛ G ✉♠ ❣r✉♣♦✱ ❞❡♥♦t❛♠♦s ♣♦r 1 ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ G✳ ❉❡✜♥✐çã♦ ✹✳✶ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ C ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r✳ ❯♠❛ g : C → C} g∈G ❛çã♦ ❞❡ G ❡♠ C é ✉♠❛ ❝♦❧❡çã♦ ❞❡ ❢✉♥t♦r❡s k✲❧✐♥❡❛r❡s {F ♠✉♥✐❞❛ ❞❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s C

  γ g,h : F g ◦ F h → F gh : Id → F , ❡ γ

  1 G

  t❛✐s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r f, g, h ∈ G ❡ X ∈ Ob(C)✱ ✈❛❧❡♠ ❛s ✐❣✉❛❧❞❛❞❡s (γ gh,f ) X ◦ (γ g,h ) = (γ g,hf ) F (X) ✭✹✳✶✮ f X ◦ F g ((γ h,f ) X )

  (γ g, ) X ◦ F g ((γ ) X ) = (γ ,g ) X ◦ (γ )

1 G G g

  1 F (X), ✭✹✳✷✮

  ♦✉ s❡❥❛✱ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠ F

g h,f

((γ ) ) X //

  (F g ◦ F h ◦ F f )(X) (F g ◦ F hf )(X)

  (γ g,h ) Ff (X) (γ g,hf ) X

  (F gh ◦ F f )(X) // F ghf (X)

  

(γ )

gh,f

X (γ )

Fg (X)

  // F g (X) (F ◦ F g )(X) F

  1 G g

((γ ) ) (γ )

X 1G,g X (F g ◦ F )(X) // F g (X).

  1 G

g, ) 1G X

  ■♥t✉✐t✐✈❛♠❡♥t❡✱ ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ♥♦s ❞✐③ q✉❡ ❡ss❛ ❛çã♦ é ✏❛ss♦❝✐✲

  1

  ❛t✐✈❛ ♥♦s ❢✉♥t♦r❡s✑ ❡ ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ ♥♦s ❞✐③ q✉❡ F G é ❝♦♠♦ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ✏✉♥✐❞❛❞❡✑✳ ▲❡♠❛ ✹✳✷ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ C ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r t❛❧ q✉❡ G g : C → C ❛❣❡ ❡♠ C✳ P❛r❛ ❝❛❞❛ g ∈ G✱ ♦ ❢✉♥t♦r F é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛✳ g : C → C ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ g ∈ G✳ ❈♦♥s✐❞❡r❡♠♦s ♦s ❢✉♥t♦r❡s F ❡ − − 1 1 C

  F g : C → C g ◦ F g ∼ Id ✳ ▼♦str❡♠♦s q✉❡ F ✳ P♦r ❤✐♣ót❡s❡ ❡①✐st❡♠

  ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s − − − 1 1 1 C γ : F g ◦ F → F = F : Id → F . g,g g gg

  1 ❡ γ G G

  1

  ❆ ❝♦♠♣♦s✐çã♦

  1 1 − 1 C

  µ = (γ ) ◦ γ g,g : F g ◦ F g → Id X é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✱ ♣♦✐s µ é ❝❧❛r❛♠❡♥t❡ ✉♠ ✐s♦♠♦r✜s♠♦ ♣❛r❛ t♦❞♦ X ∈ Ob(C) ❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ ✭❛♠❜♦s ❞✐❛❣r❛♠❛s ♠❡✲ ♥♦r❡s ❝♦♠✉t❛♠✮ 1 1 // (γ ) g,g−1 ((γ ) ) X X

  (F g ◦ F )(X) g F (X) // X ◦F F

  1 G (F )(f ) (f ) f 1 g g−1 1G

  (F g ◦ F g )(Y ) // F (Y ) // Y,

  1 G − 1 (γ ) g,g−1 ((γ ) ) Y Y

  ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ Ob(C) ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❆♥❛❧♦✲ 1 C ◦ F g ∼ Id

  ❣❛♠❡♥t❡✱ F g ✳

  ❊①❡♠♣❧♦ ✹✳✸ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ A ✉♠❛ k✲á❧❣❡❜r❛ ❞❡ ❞✐♠❡♥sã♦ ✜✲ ♥✐t❛✳ ❈♦♥s✐❞❡r❡♠♦s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G✱ ❡①✐st❛♠ ♠♦r✜s♠♦s ❞❡ ∗ : A → A g,h ∈ U (A) á❧❣❡❜r❛s g ❡ ❡❧❡♠❡♥t♦s θ ✭❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ✐♥✈❡rtí✈❡✐s ❞❡ A✮ t❛✐s q✉❡✱ ♣❛r❛ t♦❞♦ a ∈ A ❡ ♣❛r❛ q✉❛✐sq✉❡r g, h, f ∈ G✱ t❡♥❤❛♠♦s

  (1 G ) = I A , ✭✹✳✸✮ θ g,h h ∗ (g ∗ (a)) = (gh) ∗ (a)θ g,h ,

  ✭✹✳✹✮ θ gh,f f ∗ (θ g,h ) = θ g,hf θ h,f . G ✭✹✳✺✮ ❖❜s❡r✈❛♠♦s q✉❡ ❞❡ (4.3) ❡ (4.4)✱ ❢❛③❡♥❞♦ h = g = 1 ✱ ♦❜t❡♠♦s q✉❡

  θ , a = aθ , ,

1 G G G G

  1

  1 1 ✭✹✳✻✮ 1 ,

  1

  ♣❛r❛ t♦❞♦ a ∈ A✱ ✐ss♦ ♥♦s ❞✐③ q✉❡ θ G G ❡stá ♥♦ ❝❡♥tr♦ ❞❛ á❧❣❡❜r❛ A G g,

  1 G∗ (θ g, ) =

  1

  1

  ✳ ❯t✐❧✐③❛♥❞♦ (4.5)✱ ❝♦♠ h = f = 1 ✱ ♦❜t❡♠♦s θ G G θ g, θ , g, θ g, = θ g, θ ,

1 G G G G G G G G

  1 1 ✱ ♦✉ s❡❥❛✱ θ

  1

  1

  

1

  1 1 ❡ ♣♦rt❛♥t♦✱ ♣❛r❛ q✉❛❧q✉❡r

  g ∈ G ✱

  θ g,

  1 = θ G G G ✭✹✳✼✮

1 ,

1 . G 1 ,f f ∗ (θ 1 , 1 ) =

  ❯t✐❧✐③❛♥❞♦ (4.5)✱ ❝♦♠ h = g = 1 ✱ ♦❜t❡♠♦s θ G G G θ

  1 ,f θ G G ✱ ♦✉ s❡❥❛✱ 1 ,f

  f ∗ (θ , ) = θ ,f ,

  1 G

  1 G

  1 G ✭✹✳✽✮

  ♣❛r❛ t♦❞♦ f ∈ G✳ A m ❈♦♥s✐❞❡r❡♠♦s ❛ ❝❛t❡❣♦r✐❛ ❞♦s A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ❞❡ ❞✐♠❡♥✲ sã♦ ✜♥✐t❛ s♦❜r❡ k✱ ❝✉❥♦s ♠♦r✜s♠♦s sã♦ ♠♦r✜s♠♦s ❞❡ k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s A m

  ❡ ❞❡ A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ➱ ❝❧❛r♦ q✉❡ é ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r✳ ❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ ❡①❡♠♣❧♦ é ♠♦str❛r♠♦s q✉❡ t❡♠♦s ✉♠❛ ❛çã♦ ❞❡ G ❡♠ A m

  ✳ g : A → A m m ❙❡❥❛ g ∈ G✳ ❉❡✜♥✐♠♦s ♦ ❢✉♥t♦r F ✱ ❢❛③❡♥❞♦ ♣❛r❛ ❝❛❞❛ m

  M ∈ Ob( A ) g (M ) = M g m = g ∗ (a)m ✱ F ✭✐❣✉❛❧❞❛❞❡ ❝♦♠♦ ❝♦♥❥✉♥t♦s✮ ❝♦♠ ❛ ❛çã♦ ❞❛❞❛ ♣♦r a · ✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ A ❡ m ∈ M✳ ❉❛❞♦ ✉♠ A g (f ) = f m ♠♦r✜s♠♦ f : M → N ❡♠ ✱ ❞❡✜♥✐♠♦s F ✳ ❉✐ss♦ s❡❣✉❡ q✉❡ F g

  é ✉♠ ❢✉♥t♦r k✲❧✐♥❡❛r✱ ♣❛r❛ t♦❞♦ g ∈ G✳ A g,h g ◦ F h → m ) : F

  ❙❡❥❛♠ g, h ∈ G✱ M ∈ Ob( ❡ m ∈ M✳ ❉❡✜♥✐♠♦s γ F gh g,h ) M (m) = θ g,h m

  ♣♦r (γ ✳ g ◦ F h )(M ) = F g (F h (M )) ❆❣♦r❛ ❡s❝r❡✈❡♠♦s ❛ ❛çã♦ ❞❡ A ❡♠ (F ✳ h (M ) = (M, · h ) h m = h ∗ (a)m

  Pr✐♠❡✐r❛♠❡♥t❡✱ F ✱ ✐st♦ é✱ a · ❡ ♣♦rt❛♥t♦✱ F g (F h (M )) = F g (M, · h ) g (M, · h ) m =

  ❡ ❛ ❛çã♦ ❞❡ A ❡♠ F é ❞❛❞❛ ♣♦r a· g g ∗ (a) · h m = h ∗ (g ∗ (a))m ✳

  ❱❡r✐✜q✉❡♠♦s q✉❡ (γ g,h ) M : (F g ◦ F h )(M ) → F gh (M ) é ✉♠ ✐s♦♠♦r✲ ✜s♠♦ ❞❡ A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱

1 M

  m ✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ g, h ∈ G✳

  ✳ ❙❡❥❛♠ g, h, f ∈ G✱ M ∈ Ob( A m )

  ❋✐♥❛❧♠❡♥t❡✱ ✈❡r✐✜q✉❡♠♦s ❛s ✐❣✉❛❧❞❛❞❡s ✭✹✳✶✮ ❡ ✭✹✳✷✮ ♣❛r❛ ❝♦♥❝❧✉✐r✲ ♠♦s q✉❡ t❡♠♦s ✉♠❛ ❛çã♦ ❞❡ G ❡♠ A m

  A ✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ▲♦❣♦✱ γ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳

  ) M é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡

  m ♣❛r❛ t♦❞♦ m ∈ M✱ s❡❣✉❡ q✉❡ (γ

  1 G ,

  1

  θ

  ♣♦r (γ ) M (m) =

  1 G

  ✳ ❉❡✜♥✐♥❞♦ γ : Id A m → F

  ❙❡❥❛ M ∈ Ob( A m )

  1 g,h

  (γ g,h ) M (a · g m) = θ g,h (a · g m) = θ g,h (g ∗ (a) · h m) = θ g,h ((h ∗ (g ∗ (a))m) = (θ g,h (h ∗ (g ∗ (a))))m

  (m) = θ

  ✱ (γ g,h ) M é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❇❛st❛ ❞❡✜♥✐r✲ ♠♦s (γ g,h )

  P❛r❛ ❝❛❞❛ M ∈ Ob( A m )

  ◦ F h )(f )(m)) = ((γ g,h ) N ◦ (F g ◦ F h )(f ))(m).

  = f ((γ g,h ) M (m)) = f (θ g,h m) = θ g,h f (m) = (γ g,h ) N (f (m)) = (γ g,h ) N ((F g

  ❉❡ ❢❛t♦✱ s❡❥❛ m ∈ (F g ◦ F h )(M ) ✳ ❊♥tã♦ (F gh (f ) ◦ (γ g,h ) M )(m) = (f ◦ (γ g,h ) M )(m)

  

g,h ) N // F gh (M ).

  // F gh (M ) F gh (f ) (F g ◦ F h )(N )

  (F g ◦F h )(f )

g,h ) M

  ✳ ❱❡r✐✜q✉❡♠♦s q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ (F g ◦ F h )(M )

  ) ❡ f : M → N ✉♠ ♠♦r✜s♠♦ ❡♠ A m

  ❆❧é♠ ❞✐ss♦✱ γ g,h é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❙❡❥❛♠ M, N ∈ Ob( A m

  = ((gh) ∗ (a)θ g,h )m = (gh) ∗ (a)(θ g,h m) = a · gh (θ g,h m) = a · gh (γ g,h ) M (m).

  (4.4)

1 G

  ❡ m ∈ M✳ ❊♥tã♦ ((γ g,hf ) M ◦ F g ((γ h,f ) M ))(m) = (γ g,hf ) M (F g ((γ h,f ) M )(m))

  m = (γ g,

  m) = (γ g,

  1 G

  1 G ,

  1

  ) M (θ

  1 G

  1 G

  ) M ((γ ) M (m)) = (γ g,

  1 G ,

  1

  θ

  1 G

  = θ g,

  

(4.7)

  1 G

  1 G

  = (γ g,hf ) M ((γ h,f ) M (m)) = (γ g,hf ) M (θ h,f m) = (θ g,hf θ h,f )m

  (γ ) X

  ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G✳

  // F g (X) s g F gh (X) s gh // X,

  (γ g,h )

X

F g (s h )

  X (F g ◦ F h )(X)

  (X) s 1G

  // F

  G G G G G G G G G

  ) M (F g ((γ ) M )(m)) = ((γ g,

  X I X ##G

  ❞✐❛❣r❛♠❛s sã♦ ❝♦♠✉t❛t✐✈♦s

  ◦ (γ ) X = I X ❡ s g ◦ F g (s h ) = s gh ◦ (γ g,h ) X ✱ ✐✳❡✳✱ ♦s s❡❣✉✐♥t❡s

  ❛❣❡ ❡♠ C✳ ❯♠ ♦❜❥❡t♦ X ∈ Ob(C) é ❞✐t♦ ❡q✉✐✈❛r✐❛♥t❡✱ s❡ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ s = {s g : F g (X) → X} g∈G ❞❡ ✐s♦♠♦r✜s♠♦s ❡♠ C t❛✐s q✉❡ s

  ❉❡✜♥✐çã♦ ✹✳✹ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ C ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r t❛❧ q✉❡ G

  1 G ) M ◦ F g ((γ ) M ))(m).

  m = m

  1

  1 G ,g )

  1 G ,g ) M ((γ ) F g (M )

  · g m) = (γ

  1 G

  1 G ,

  1

  1 G ,g ) M (θ

  (m)) = (γ

  )(m) = (γ

  1 G ,g (θ

  1 G ,g ) M ◦ (γ ) F g (M )

  ❙❡❥❛ m ∈ F g (M ) ✳ ❊♥tã♦ ((γ

  (M ) )(m).

  · f m) = (γ gh,f ) M ((γ g,h ) F f (M ) (m)) = ((γ gh,f ) M ◦ (γ g,h ) F f

  = θ gh,f f ∗ (θ g,h )m = (γ gh,f ) M (f ∗ (θ g,h )m) = (γ gh,f ) M (θ g,h

  

(4.5)

  1 G ,g ) M (g ∗ (θ

  1

  1 G ,

  1 G

  = θ

  

(4.8)

  m

  1

  ))

  1 G

  1 G ,

  1 G ,g (g ∗ (θ

  )m = θ

  1 G

  1 G ,

  1

  1 G ,g g ∗ (θ

  )m) = θ

1 G

1 G

  ❉❡✜♥✐çã♦ ✹✳✺ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ C ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r t❛❧ q✉❡ G G

  ❛❣❡ ❡♠ C✳ ❆ ❝❛t❡❣♦r✐❛ C ✱ ❝❤❛♠❛❞❛ ❡q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ C ♣♦r G✱ é ❞❡✜♥✐❞❛ ♣♦r G

  ) ✭✐✮ Ob(C é ❛ ❝♦❧❡çã♦ ❞♦s ♣❛r❡s (X, s)✱ ❡♠ q✉❡ X é ✉♠ ♦❜❥❡t♦ ❡q✉✐✈❛✲ r✐❛♥t❡ ❞❡ C ❡ s ❛ ❢❛♠í❧✐❛ ❞❡ ✐s♦♠♦r✜s♠♦s ❛ss♦❝✐❛❞❛✳ G

  ) ✭✐✐✮ ❉❛❞♦s (X, s), (Y, r) ∈ Ob(C ✱ ✉♠ ♠♦r✜s♠♦ f : (X, s) → (Y, r) ❡♠ G C

  ✭♦✉ ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✮ é ✉♠ ♠♦r✜s♠♦ f : X → Y ❡♠ C g = r g g (f ) ◦ F t❛❧ q✉❡ f ◦ s ✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛ s❡❣✉✐♥t❡ ❝♦♠✉t❛ g // F g F (f ) g F (X) (Y ) s r g g

  // Y,

  X f ♣❛r❛ t♦❞♦ g ∈ G✳

  ❈♦♥s✐❞❡r❡♠♦s ♦ ❊①❡♠♣❧♦ 4.3✳ ■♥tr♦❞✉③✐♠♦s ♥♦ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ A ⊗ k kG

  ♦ ♣r♦❞✉t♦ (a ⊗ g)(b ⊗ h) = θ g,h h ∗ (a)b ⊗ gh,

  ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G ❡ a, b ∈ A✳ k kG ❈♦♠ ❡ss❡ ♣r♦❞✉t♦✱ A ⊗ é ✉♠❛ k✲á❧❣❡❜r❛✳ ❉❡ ❢❛t♦✱ ♦ ❡❧❡♠❡♥t♦

  1

  ⊗ 1 ∈ A ⊗ θ G k kG

  1 , G G 1 é s✉❛ ✉♥✐❞❛❞❡✱ ♣♦✐s − −

  1

  1

  (θ ⊗ 1 G )(a ⊗ g) = θ ,g g ∗ (θ )a ⊗ 1 G g ,

  1 G ,

1 G G G G

  1

  1

  1

  1

  = θ ,g (g ∗ (θ , )) a ⊗ g

  1 G G G

  1

  1 (4.8)

  1

  = θ

  1 ,g θ a ⊗ g G ,g

  1 G

  = a ⊗ g ❡

  

1 −

  1

  (a ⊗ g)(θ ⊗ 1 G ) = θ , g, (1 G ) ∗ (a)θ ⊗ g1 G

  1 ,

  1 G G G G

  1 G

  1

  1 (4.3)

  1

  = θ g, aθ ⊗ g

  1 G ,

  1 G G

  1 (4.7)

  1

  = θ , aθ ⊗ g

1 G

  1 G ,

  1 G

  1 G (4.6)

  1

  ⊗ g = aθ

  1 , G G 1 θ 1 , G G

  1

  = a ⊗ g, ♣❛r❛ q✉❛✐sq✉❡r g ∈ G ❡ a ∈ A✳

  ❖ ♣r♦❞✉t♦ é ❛ss♦❝✐❛t✐✈♦✳ ❙❡❥❛♠ g, h, f ∈ G ❡ a, b, c ∈ A✳ ❊♥tã♦ (a ⊗ g)((b ⊗ h)(c ⊗ f )) = (a ⊗ g)(θ h,f f ∗ (b)c ⊗ hf )

  = θ g,hf (hf ) ∗ (a)θ h,f f ∗ (b)c ⊗ g(hf ) = θ g,hf ((hf ) ∗ (a)θ h,f )f ∗ (b)c ⊗ g(hf ) = θ g,hf (θ h,f f ∗ (h ∗ (a)))f ∗ (b)c ⊗ (gh)f = (θ g,hf θ h,f )f ∗ (h ∗ (a))f ∗ (b)c ⊗ (gh)f = (θ gh,f f ∗ (θ g,h ))f ∗ (h ∗ (a))f ∗ (b)c ⊗ (gh)f = θ gh,f (f ∗ (θ g,h )f ∗ (h ∗ (a))f ∗ (b))c ⊗ (gh)f = θ gh,f f ∗ (θ g,h h ∗ (a)b)c ⊗ (gh)f = (θ g,h h ∗ (a)b ⊗ gh)(c ⊗ f ) k kG = ((a ⊗ g)(b ⊗ h))(c ⊗ f ).

  ❯♠❛ ✈❡③ q✉❡ A ⊗ é ✉♠❛ k✲á❧❣❡❜r❛✱ ❢❛③ s❡♥t✐❞♦ ❡♥✉♥❝✐❛r♠♦s ♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ q✉❡ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛♦ q✉❡ ❢♦✐ ❞✐t♦ ♥❛ ✐♥tr♦❞✉✲ çã♦ ❞❡ss❡ ❝❛♣ít✉❧♦✳ ❆❧é♠ ❞✐ss♦✱ ♦ ♠❡s♠♦ ♥♦s tr❛③ ✉♠ ❡①❛♠♣❧♦ ✏♠❛✐s ❝♦♥❝r❡t♦✑ ❞❡ ✉♠❛ ❡q✉✐✈❛r✐❛♥t✐③❛çã♦✳ ❚❡♦r❡♠❛ ✹✳✻ ◆❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✱ ❡①✐st❡ ♦ s❡❣✉✐♥t❡ ✐s♦♠♦r✜s♠♦ ❡♥✲ tr❡ ❝❛t❡❣♦r✐❛s G m m

  ( A ) ≃ kG

  (A⊗ ) k G A ) → m m

  ❉❡♠♦♥str❛çã♦✿ ❉❡✜♥✐♠♦s ♦ ❢✉♥t♦r H : ( (A⊗ kG ) t❛❧ q✉❡✱ G k A ) ) m ♣❛r❛ t♦❞♦ (M, s) ∈ Ob(( ✱ H((M, s)) = M✱ ❝♦♠ ❛ ❛çã♦ ❞❛❞❛ ♣♦r (a ⊗ g)m = s g (am)

  ✳ ❉❡ ❢❛t♦✱

  1

  1 A⊗ kG · m = (θ ⊗ 1 G ) · m k ,

  1 G G

  1

  1

  = s (θ m)

  1 G ,

  1 G G

  1

  = s ((γ ) M (m))

  1 G

  = (s ◦ (γ ) M )(m)

  1 G

  =

  I M (m) = m, ♣❛r❛ t♦❞♦ m ∈ M✳ ❙❡❥❛♠ a, b ∈ A✱ g, h ∈ H ❡ m ∈ M✳ ❊♥tã♦

  (a ⊗ g)((b ⊗ h)m) = (a ⊗ g)(s h (bm)) = s g (as h (bm)) = s g (s h (a · h bm)) = s g (s h (h ∗ (a)bm)) = s g (F g (s h )(h ∗ (a)bm)) = (s g ◦ F g (s h ))(h ∗ (a)bm) = (s gh ◦ (γ g,h ) M )(h ∗ (a)bm) = s gh ((γ g,h ) M (h ∗ (a)bm)) = s gh (θ g,h h ∗ (a)bm) = (θ g,h h ∗ (a)b ⊗ gh)m = ((a ⊗ g)(b ⊗ h))m.

  ❉❛❞♦ f : (M, s) → (N, r) ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✱ t❡♠♦s H(f) = f : M → N ✉♠ ♠♦r✜s♠♦ ❞❡ A ⊗ k kG ✲♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ f é

  1

  1

  g ∗ (a)θ

  1 G

  ⊗ g)m = (θ g,

  1 G

  1 G ,

  1

  (1 G ) ∗ (1)g ∗ (a)θ

  1 G

  ⊗ 1 G ))m = (θ g,

  1 G

  1 G ,

  ⊗ 1 G )m) = ((1 ⊗ g)(g ∗ (a)θ

  1 G

  1 G

  1 G ,

  1

  ⊗ 1 G )m) = (1 ⊗ g)((g ∗ (a)θ

  1 G

  1 G ,

  1

  = s g ((g ∗ (a)θ

  ⊗1 G )m = m ✱ ✉♠❛ ✈❡③ q✉❡ M é ✉♠ A ⊗ k kG ✲♠ó❞✉❧♦✳ ◆♦t❡♠♦s t❛♠❜é♠ q✉❡ s g (a · g m) = s g (g ∗ (a) · m)

  1 G

  1 G ,

  1

  ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ A✳ ◆❛ ✐❣✉❛❧❞❛❞❡ ✭✯✮ ✉s❛♠♦s q✉❡ M é ✉♠ A⊗ k kG ✲ ♠ó❞✉❧♦ ❡ ❡♠ ✭✯✯✮ ✉s❛♠♦s ✭✹✳✻✮✳ ❈❧❛r❛♠❡♥t❡✱ 1·m = (1θ

  1 G ,

  ⊗ g)m

  1 G

  1 G ,

  ⊗ 1 G )(1 ⊗ g))m = a · ((1 ⊗ g)m) = a · s g (m).

  1 G

,

  

1

  ) ⊗ g)m = ((aθ

  1 G

  1 G ,

  1

  1 G

,g g ∗ (aθ

  = (θ

  (4.8)

  ) ⊗ g)m

  1 G

  1

  (∗)

  )g ∗ (aθ

  1 G

  1 G

,

  = (g ∗ (θ

  (4.6)

  ) ⊗ g)m

  1 G

  1 G ,

  1

  θ

  1 G

  

1 G

,

  = (g ∗ (a) ⊗ g)m = (g ∗ (aθ

  ⊗ 1 G )m = (ab) · m,

  1 G ,

  ❡q✉✐✈❛r✐❛♥t❡✱ t❡♠♦s f ◦ s g = r g ◦ F g (f ) ♣❛r❛ t♦❞♦ g ∈ G ❡ ❛ss✐♠✱ f ((a ⊗ g)m) = f (s g (am)) = r g (F g (f )(am)) = r g (f (am)) = r g (af (m)) = (a ⊗ g)f (m),

  

1 G

  1

  = ((aθ

  (∗)

  ⊗ 1 G )m)

  1 G

  1 G ,

  1

  ⊗ 1 G )((bθ

  1 G

  1 G ,

  1

  ⊗ 1 G )m) = (aθ

  1 G

,

  1 G

  1

  ⊗ 1 G )m ♣❛r❛ t♦❞♦ a ∈ A✳ ❉❡ ❢❛t♦✱ a · (b · m) = a · ((bθ

  1 G

  1 G ,

  1

  ♣❛r❛ t♦❞♦ g ∈ G ❡ ❛ ❡str✉t✉r❛ ❞❡ A✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❞❡ M é ❞❛❞❛ ♣♦r a · m = (aθ

  ❡♠ q✉❡ s = {s g : F g (M ) = M → M } é ❛ ❢❛♠í❧✐❛ ❞❡✜♥✐❞❛ ♣♦r s g (m) = (1 ⊗ g)m

  (M ) = (M, s) ✱

  ) G ♣♦r H

  m → ( A m

  (A⊗ k kG

)

  :

  ♣❛r❛ q✉❛✐sq✉❡r a ∈ A✱ g ∈ G ❡ m ∈ M✳ ❉❡✜♥✐♠♦s ♦ ❢✉♥t♦r H

  1 G ,

  ⊗ 1 G )(bθ

  1

  1 G ,

  = (abθ

  (∗∗)

  ⊗ 1 G )m

  1 G

  1 G ,

  1

  bθ

  1 G

  

1 G

,

  

1

  aθ

  1 G

  ⊗ 1 G )m = (θ

  1

  1 G

  1 G ,

  1

  )bθ

  1 G

  1 G ,

  1

  (1 G ) ∗ (aθ

  1 G

  1 G ,

  ⊗ 1 G ))m = (θ

  1 G

  1 G ,

1 G

  ❯s❛♠♦s ❡♠ ✭✯✮ ❛s ✐❣✉❛❧❞❛❞❡s ✭✹✳✼✮ ❡ ✭✹✳✻✮✱ ♥❡ss❛ ♦r❞❡♠✳ P❛r❛ ♠♦s✲ tr❛r♠♦s q✉❡ s g é ✉♠ ✐s♦♠♦r✜s♠♦✱ ❜❛st❛ ❡♥❝♦♥tr❛r ✉♠ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ♣❛r❛ 1 ⊗ g ❡♠ A ⊗ k kG ✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❡❧❡♠❡♥t♦ θ

1 G

1 G

  θ

  ⊗ 1 G =

  1 A⊗ k kG . ❊♠ ✭✯✮ ✉t✐❧✐③❛♠♦s ❛ ✐❣✉❛❧❞❛❞❡ (4.5) ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡ θ

  1 G ,g g ∗ (θ g,g 1

  ) = θ g,

  θ g 1 ,g ❡ ❛ss✐♠✱ (g ∗ (θ g,g 1 ))

  1

  = θ

   1 g 1 ,g

  1 G

  1 g,

  1 G ,

  θ

  1 G ,g

  ✳ ❋✐♥❛❧♠❡♥t❡✱ ♠♦str❡♠♦s q✉❡ ❛ ❢❛♠í❧✐❛ s = {s g : F g (M ) = M →

  M } ❝♦♠✉t❛ ♦s ❞✐❛❣r❛♠❛s ❞❛ ❉❡✜♥✐çã♦ 4.4✳ ❙❡❥❛♠ g, h ∈ G✱ M ∈

  Ob(

  (A⊗ k kG )

  m )

  ❡ m ∈ M✳ ❊♥tã♦ (s

  1 G

  ⊗ 1 G = θ

  1

  1

  = θ

  1

  1 G ,

  1 G

  θ

  1 G ,g (g ∗ (θ

  1 G ,

  1 G

  ))

  ⊗ 1 G

  1 G

  (4.8)

  = θ

  1

  1 G ,

  1 G

  θ

  1 G ,g (θ

  1 G ,g )

  1

  ◦ (γ ) M )(m) = s

  1

  (θ

  1 G ,

  θ

  1

  1 G ,

  1 G

  ⊗ 1 G ))m = (θ

  1 G

,

  

1

G

  θ

  1

  1 G

  1 G ,

  θ

  1

  1 G ,

  1 G

  ⊗ 1 G )m = (θ

  1

  1 G

,

  

1

G

  ⊗ 1 G )m = m

  1 G

  1

  1 g,g 1 θ

  1

  

1

G ,

  1 G

  · m) = s

  1 G

  ((θ

  1

  1 G ,

  1 G

  θ

  1 G ,

  ⊗ 1 G )m) = ((1 ⊗ 1 G )(θ

  1 G

  ⊗ 1 G )m) = (1 ⊗ 1 G )((θ

  1

  1 G ,

  1 G

  θ

  1

  1 G ,

  1 G

  (4.7)

  ⊗ 1 G

  1

  1 g,g 1

  1 G ,

  1 G

  ⊗ 1 G = θ

  1

  

1

G ,

  1 G

  ⊗ 1 G =

  1 A⊗ k kG

  (θ

  θ

  1 g,g 1 θ

  1

  1 G ,

  ⊗ g

  1

  )(1 ⊗ g) = θ g 1 ,g g ∗ (θ

  1 g,g 1

  θ

  1

  1 G ,

  1

  = θ g,g 1 θ

  ) ⊗ g

  1 G ,

  1

  1 G ,

  1 G

  ⊗ g

  1

  ✳ ❚❡♠♦s

  (1 ⊗ g)(θ

  1 g,g 1 θ

  1

  ⊗ g

  1

  1

  ) = θ g,g 1 (g

  1

  ) ∗ (1)θ

  1 g,g 1 θ

  1

  1 G ,

  1 G

  ⊗ gg

  1 G

  1

  ))

  θ

  1 G ,g g ∗ (θ

  1 g,

  1 G

  ) ⊗ 1 G

  (4.7)

  = θ

  1

  1 G ,

  1 G

  1 G ,g g ∗ (θ

  1 G

  1 g,

  1 G

  ) ⊗ 1 G = θ

  1

  1 G ,

  1 G

  θ

  1 G ,g (g ∗ (θ g,

  1 G

  θ

  1 g,

  g = θ g 1 ,g g ∗ (θ

  1 G

  1 g,g 1 θ

  1

  1 G ,

  1 G

  ) ⊗ 1 G = θ g 1 ,g g ∗ (θ

  1 g,g 1

  )g ∗ (θ

  1

  1 G ,

  ) ⊗ 1 G

  ) ⊗ 1 G = θ

  (∗)

  = θ g 1 ,g θ

   1 g 1 ,g

  θ

  1 g,

  1 G

  θ

  1 G ,g g ∗ (θ

  1 g,

  1 G

1 G

1 G

  ❡ (s g ◦ F g (s h ))(m) = s g (F g (s h )(m))

  1

  1 G

  ⊗ 1 G )m) = s ghg,h · m) = s gh ((γ g,h ) M (m)) = (s gh ◦ (γ g,h ) M )(m).

  ◆❛ ✐❣✉❛❧❞❛❞❡ ✭✯✮ ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ θ

  1 G ,

  1 G

  s❡♥❞♦ ❝❡♥tr❛❧ ✐♠♣❧✐❝❛ q✉❡ θ

  1

  1 G ,

  t❛♠❜é♠ ♦ s❡❥❛✳ ❙❡❥❛ f : M → N ✉♠ ♠♦r✜s♠♦ ❡♠ (A⊗ k kG ) m

  ✳ ❉❡✜♥✐♠♦s H (f ) :

  (M, s) → (N, r) ♣♦r H

  (f ) = f ✳ ▼♦str❡♠♦s q✉❡ f é ✉♠ ♠♦r✜s♠♦

  ❡q✉✐✈❛r✐❛♥t❡✳ ❉❡ ❢❛t♦✱ f (a · m) = f ((aθ

  1 G ,

  1

  1 G

  ⊗ 1 G )m) = (aθ

  1

  

1

G ,

  1 G

  ⊗ 1 G )f (m) = a · f (m),

  ♣❛r❛ q✉❛✐sq✉❡r a ∈ A ❡ m ∈ M✳ ❚❛♠❜é♠ (f ◦ s g )(m) = f (s g (m))

  = f ((1 ⊗ g)m) = (1 ⊗ g)f (m) = r g (f (m)) = r g (F (f )(m)) = (r g

  ◦ F (f ))(m), ♣❛r❛ t♦❞♦ m ∈ F g (M )

  ❙❡❥❛ M ∈ Ob( (A⊗ k kG ) m )

  ✳ ❈♦♥s✐❞❡r❡♠♦s ♦ (A ⊗ k kG) ✲ ♠ó❞✉❧♦ à ❡sq✉❡r❞❛ (H ◦ H

  )(M ) ❝♦♠ ❛ ❛çã♦ ❞❡♥♦t❛❞❛ ♣♦r ∗✱ ❡♠ q✉❡ H

  (M ) =

  1 G ,

  ⊗ 1 G )m) = s gh ((θ g,h θ

  = s g ((1 ⊗ h)m) = (1 ⊗ g)((1 ⊗ h)m) = ((1 ⊗ g)(1 ⊗ h))m = (θ g,h h ∗ (1) ⊗ gh)m = (θ g,h ⊗ gh)m

  1 G ,

  (4.7)

  = ((θ gh,

  1 G

  θ

  1

  1 G ,

  1 G

  )θ g,h ⊗ gh)m

  (∗)

  = (θ gh,

  1 G

  θ g,h θ

  1

  1 G

  1 G

  ⊗ gh)m = (θ gh,

  1 G

  1 G∗ (1)θ g,h θ

  1

  1 G ,

  1 G

  ⊗ gh)m = ((1 ⊗ gh)(θ g,h θ

  1

  1 G ,

  1 G

  ⊗ 1 G ))m = (1 ⊗ gh)((θ g,h θ

  1

  1 G ,

1 G

  (M, s) ✳ ❊♥tã♦

  (a ⊗ g) ∗ m = s g (a · m)

  1

  = s g ((aθ ⊗ 1 G )m)

  1 , G G

  1

  1

  = (1 ⊗ g)((aθ ⊗ 1 G )m) ,

  1 G G

  1

  1

  = (θ g, (1 G ) ∗ (1)aθ ⊗ g)m

  1 G ,

  1 G

  1 G

  1

  = (θ g,

  1 θ a ⊗ g)m G ,

  1 G

  1 G

  = (a ⊗ g)m, )(M ) = M =

  ♣❛r❛ q✉❛✐sq✉❡r a ∈ A✱ g ∈ G ❡ m ∈ M✳ ▲♦❣♦✱ (H ◦ H m m Id (M ) = Id

  ✱ ♦✉ s❡❥❛✱ H ◦ H ✳

  (A⊗kkG) (A⊗kkG) G ′ A ) ◦H)((M, s)), r) m

  ❙❡❥❛ (M, s) ∈ ( ✳ ❈♦♥s✐❞❡r❡♠♦s ((H ✉♠ ♦❜❥❡t♦ ❡q✉✐✈❛r✐❛♥t❡ ❝✉❥❛ ❛çã♦ ❡♠ M é ❞❡♥♦t❛❞❛ ♣♦r ∗✳ ❊♥tã♦

  1

  ⊗ 1 a ∗ m = (aθ G )m ,

  

1 G

  1 G

  1

  = s (aθ m)

  1 G 1 , G G

  1

  1

  = s (θ am)

  1 ,

G

  1 G G

  1

  = s

  1 ((γ ) M (am))

G

  ◦ (γ = (s

  1 ) M )(am)

G

  = am, ♣❛r❛ q✉❛✐sq✉❡r a ∈ A ❡ m ∈ M✳ ❆❧é♠ ❞✐ss♦✱ r g (m) = (1 ⊗ g)m = s g (m), G

  ◦ H = Id m ♣❛r❛ t♦❞♦ m ∈ M✳ P♦rt❛♥t♦✱ H ( ) ✳ A ❚❡♦r❡♠❛ ✹✳✼ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ C ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r t❛❧ q✉❡ G G

  ❛❣❡ ❡♠ C✳ ❊♥tã♦ ❛ ❝❛t❡❣♦r✐❛ C é t❛♠❜é♠ k✲❧✐♥❡❛r✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ ❛ ❝❛t❡❣♦r✐❛ C é ❛❞✐t✐✈❛✱ ❡①✐st❡ Z ∈ Ob(C) ♦❜✲ g : C → C ❥❡t♦ ③❡r♦✳ P❛r❛ ❝❛❞❛ g ∈ G✱ F é ✉♠ ❢✉♥t♦r ❛❞✐t✐✈♦ ✭♣♦rt❛♥t♦✱ g (Z) ∈ Ob(C) ♣r❡s❡r✈❛ ♣r♦❞✉t♦✮ ❡ ❛ss✐♠✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✼ ✭✐✮✱ F é ✉♠ g : F g (Z) → Z ♦❜❥❡t♦ ③❡r♦✳ ▲♦❣♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ s ✳ ❈♦♥✲ g : F g (Z) → Z} s✐❞❡r❡♠♦s ❛ ❝♦❧❡çã♦ s = {s ✳ G

  ❆✜r♠❛çã♦ ✭✐✮ ✿ (Z, s) é ✉♠ ♦❜❥❡t♦ ③❡r♦ ❡♠ C ✳ C (Z, Z) = {I Z }

  ❉❡ ❢❛t♦✱ s❛❜❡♠♦s q✉❡ Hom ✳ ❆ss✐♠✱ ♣❛r❛ t♦❞♦ g ∈ G✱ s ◦ (γ ) Z : Z → Z ◦ (γ ) Z = I Z

  1 é ✉♠ ♠♦r✜s♠♦ ❡♠ C✱ ♦✉ s❡❥❛✱ s G g ◦ F h )(Z) 1 ✳ G

  ❙❡❥❛♠ g, h ∈ G✳ ❊♥tã♦ (F é t❛♠❜é♠ ✉♠ ♦❜❥❡t♦ ③❡r♦ ❡♠ C g ◦F h )(Z) ✭▲❡♠❛ ✸✳✶✼ ✭✐✮✮ ❡ ❛ss✐♠✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ (F ❡ Z✳ g ◦ F g (s h ) : (F g ◦ F h )(Z) → Z gh ◦ (γ g,h ) Z : (F g ◦

  P♦r ♦✉tr♦ ❧❛❞♦✱ s ❡ s F h )(Z) → Z g ◦ F g (s h ) = s gh ◦ (γ g,h ) Z sã♦ ✐s♦♠♦r✜s♠♦s ❡ ♣♦rt❛♥t♦✱ s ✳ G

  ▼♦str❛♠♦s ❡♥tã♦ q✉❡ (Z, s) é ✉♠ ♦❜❥❡t♦ ❡♠ C ✳ Pr♦✈❡♠♦s ❛ ♦✉tr❛ ♣❛rt❡ ❞❛ ❛✜r♠❛çã♦✱ ✐st♦ é✱ q✉❡ (Z, s) é ✉♠ ♦❜❥❡t♦ G G

  ) ③❡r♦ ♥❛ ❝❛t❡❣♦r✐❛ C ✳ ❙❡❥❛ (X, r) ∈ Ob(C ✳ C C

  (Z, X) = {ψ X } (X, Z) = {φ X } ❙❛❜❡♠♦s q✉❡ Hom ❡ Hom ✳ ❖❜✲ g ◦ F g (φ X ) X ◦ r g (F g (X), Z) C s❡r✈❡♠♦s q✉❡ s ❡ φ sã♦ ♠♦r✜♠♦s Hom q✉❡ g ◦ F g (φ X ) = φ X ◦ r g

  é ✉♠ ❝♦♥❥✉♥t♦ ✉♥✐tár✐♦✱ ♣❛r❛ ❝❛❞❛ g ∈ G✳ ▲♦❣♦✱ s ✳ g ◦F g (ψ X ) = ψ X ◦r g X X ❆♥❛❧♦❣❛♠❡♥t❡✱ s ✳ P♦rt❛♥t♦✱ φ ❡ ψ sã♦ ♠♦r✜s♠♦s ❡q✉✐✈❛r✐❛♥t❡s✳

  ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ✉♠ ♦✉tr♦ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡ ξ : (Z, s) → (X, r) X

  ✳ P♦r ❞❡✜♥✐çã♦✱ ξ é ✉♠ ♠♦r✜s♠♦ ❡♠ C ❡ ♣♦rt❛♥t♦✱ ξ = φ ✳ X P♦❞❡♠♦s ❞✐③❡r ♦ ♠❡s♠♦ ♣❛r❛ ♦ ♠♦r✜s♠♦ ψ ✳ ▲♦❣♦✱ (Z, s) é ✉♠ ♦❜❥❡t♦ G ③❡r♦ ❡♠ C ✳ G

  ) P❛r❛ ♦ q✉❡ s❡❣✉❡✱ ❝♦♥s✐❞❡r❡♠♦s q✉❛✐sq✉❡r (X, s), (Y, r) ∈ Ob(C ✳ C ((X, s), (Y, r)) G ❆✜r♠❛çã♦ ✭✐✐✮✿ Hom é ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❛ s♦♠❛ ❞❡ ♠♦r✜s♠♦s ❡ ♣r♦❞✉t♦ ♣♦r ❡s❝❛❧❛r ✐♥❞✉③✐❞♦s ♣❡❧❛ ❝❛t❡❣♦r✐❛ C✳ G

  ❉❡ ❢❛t♦✱ s❡❥❛♠ f, l : (X, s) → (Y, r) ♠♦r✜s♠♦s ❡♠ C ✳ ❊♠ ♣❛rt✐❝✉✲ ❧❛r✱ f, l : X → Y sã♦ ♠♦r✜s♠♦s ❡♠ C ♦♥❞❡ t❡♠♦s ❞❡✜♥✐❞❛ ✉♠❛ s♦♠❛ f + l : X → Y

  ✳ ◆♦t❡♠♦s q✉❡ (f + l) ◦ s g = f ◦ s g + l ◦ s g

  = r g ◦ F g (f ) + r g ◦ F g (l) = r g ◦ (F g (f ) + F g (l)) = r g ◦ F g (f + l),

  ♣❛r❛ t♦❞♦ g ∈ G✳ ▲♦❣♦✱ f + l é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✳ X : X → Y X ❖❜s❡r✈❛♠♦s q✉❡ 0 é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✱ ♣♦✐s F F g (X) g (X) Y X Y Y Y ✱ ♣❛r❛ t♦❞♦ g ∈ G✱ ❛ ú❧t✐♠❛ ◦ s g = 0 = r g ◦ 0 = r g ◦ F g (0 ) F (Y ) g

  ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞♦ ▲❡♠❛ ✸✳✶✼ ✭✐✐✮✳ ❉❛❞♦ ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡ q✉❛❧q✉❡r f : X → Y ✱ ♦ ♠♦r✜s♠♦

  ♦♣♦st♦ −f : X → Y ❡♠ r❡❧❛çã♦ à s♦♠❛ ❡♠ C é t❛♠❜é♠ ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✳ ❉❡ ❢❛t♦✱ f ◦ s g + (−f ) ◦ s g = (f + (−f )) ◦ s g X

  ◦ s = g Y g ◦ F g X

  = r (0 ) Y = r g ◦ F g (f + (−f )) = r g ◦ (F g (f ) + F g (−f )) = r g ◦ F g (f ) + r g ◦ F g ((−f )), g = r g ◦F g ((−f )) g = r g ◦F g (f )

  ♣❛r❛ t♦❞♦ g ∈ G✳ ▲♦❣♦✱ (−f)◦s ✱ ♣♦✐s f◦s ✳

  C ((X, s), (Y, r)) G

  P♦rt❛♥t♦✱ s❡❣✉❡ ❞❛ s♦♠❛ ✐♥❞✉③✐❞❛ q✉❡ Hom é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ ❈♦♠ ♦ ♠❡s♠♦ ♣r♦❞✉t♦ ♣♦r ❡s❝❛❧❛r ❞❡✜♥✐❞♦ ♥❛ ❝❛✲ C ((X, s), (Y, r)) G t❡❣♦r✐❛ C✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ Hom é ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ G

  ) ❆✜r♠❛çã♦ ✭✐✐✐✮✿ ❙❡❥❛♠ (X, s), (Y, r) ∈ Ob(C ✳ ❊♥tã♦ ❡①✐st❡ ♦ ♣r♦✲ G

  ❞✉t♦ ❞❡ (X, s) ❡ (Y, r) ❡♠ C ✳ X , π Y ) ❙❛❜❡♠♦s q✉❡ ❡①✐st❡ ♦ ♣r♦❞✉t♦ (X ⊕Y, π ❞❡ X ❡ Y ❡♠ C✳ P❛r❛ g ⊕ r g : F g (X) ⊕ F g (Y ) →

  ❝❛❞❛ g ∈ G✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ♠♦r✜s♠♦ s X ⊕ Y q✉❡ ❝♦♠✉t❛ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ✭✈❡❥❛ ❉❡✜♥✐çã♦ ✸✳✶✶✮

  F g (X) ⊕ F g (Y ) π N π N Fg (X) Fg (Y ) pp N N pp N

  N pp N N pp N

  N wwppp ''N s ⊕r g g F g (X) F g (Y ) s r g g

  X ⊕ Y O O π π oo O X Y O oo O

  O oo O O oo O

  O oo O O wwooo ''O

  Y,

  X ♦✉ s❡❥❛✱

  π X ◦ (s g ⊕ r g ) = s g ◦ π F Y ◦ (s g ⊕ r g ) = r g ◦ π F . g g (X) ❡ π (Y ) ❊ss❛s ✐❣✉❛❧❞❛❞❡s ❞❡st❛❝❛❞❛s ❛❝✐♠❛ sã♦ ♠✉✐t♦ ✉s❛❞❛s ♥♦ q✉❡ s❡❣✉❡ ❡

  ♥ã♦ ❢❛③❡♠♦s ♥❡♥❤✉♠❛ ♠❡♥çã♦ ❛ ❡❧❛s ♣❛r❛ ❡✈✐t❛r ❝❛rr❡❣❛r ♦ t❡①t♦ ❝♦♠ ♠❛✐s r❡❢❡rê♥❝✐❛s✳ g ⊕ r g

  Pr♦✈❡♠♦s q✉❡ s é ✉♠ ✐s♦♠♦r✜s♠♦✳ P❛r❛ ✐ss♦✱ é s✉✜❝✐❡♥t❡ g (X) ⊕ F g (Y ), s g ◦ π F , r g ◦ π F ) ♠♦str❛r♠♦s q✉❡ ❛ tr✐♣❧❛ (F (X) (Y ) é ✉♠ g g ♣r♦❞✉t♦ ❞❡ X ❡ Y ❡♠ C✳ F : F g (X) → F g (X) ⊕ F g (Y ) F : F g (Y ) → F g (X) ⊕ F g (Y ) g (X) ⊕ ❙❛❜❡♠♦s q✉❡ ❡①✐st❡♠ ♠♦r✜s♠♦s i (X) g ❡ i (Y ) t❛✐s q✉❡ ❛ q✉í♥t✉♣❧❛ (F g F g (Y ), π F , π F , i F , i F ) g (X) g g g g (X) (Y ) (X) (Y ) é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ F ❡

  F g (Y ) ✱ ✐ss♦ s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✶✵✳ X : C → X, c Y : C → Y )

  ❙❡❥❛ (C, c ✉♠❛ tr✐♣❧❛ ❡♠ C✳ ❉❡✜♥✐♠♦s ♦ g (X) ⊕ F g (Y ) ♠♦r✜s♠♦ α : C → F ♣♦r − −

  1

  1

  α = i F ◦ r ◦ c Y + i F ◦ s ◦ c X . g g

(Y ) g (X) g

◆♦t❡♠♦s q✉❡

  ◦ (γ ) X⊕Y ) =

  (X)⊕F g (Y ) s❡❣✉❡✱ s♦✲

  1 g

  ◦ s

  (X)

  ◦ c Y + i F g

  1 g

  ◦ r

  (Y )

  ♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s ❞✉❛s ✐❣✉❛❧❞❛❞❡s✱ q✉❡ β = i F g

  = I F g

  (X)

  (Y )

  ◦ π F g

  (Y )

  (X)

  ◦ π F g

  (X)

  ❈♦♠♦ i F g

  1 g ◦ c Y .

  ◦ r

  ◦ c X = α. P♦rt❛♥t♦✱ (F g (X) ⊕ F g (Y ), s g ◦ π F g

  , r g ◦ π F g

  ◦ β = i F g

  (Y )

  X,Y 1 G

  ) ◦ φ

  1 G

  ⊕ r

  1 G

  π X ◦ ((s

  Pr♦✈❡♠♦s q✉❡ (X ⊕ Y, s ⊕ r) é ✉♠ ♦❜❥❡t♦ ❡q✉✐✈❛r✐❛♥t❡✳ ❉❡ ❢❛t♦✱ ♥♦t❡♠♦s q✉❡

  ❡ ♥ã♦ ❢❛③❡♠♦s ♥❡♥❤✉♠❛ ♠❡♥çã♦ ❛ ❡❧❛s✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❢❛♠í❧✐❛ ❞❡ ✐s♦♠♦r✜s♠♦s s ⊕ r = {(s g ⊕ r g ) ◦ φ X,Y g : F g (X ⊕ Y ) → X ⊕ Y : g ∈ G}.

  ◦ φ g X,Y = F g (π Y ). ❊ss❛s ❞✉❛s ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ sã♦ t❛♠❜é♠ ♠✉✐t♦ ✉s❛❞❛s ♥♦ q✉❡ s❡❣✉❡

  ◦ φ g X,Y = F g (π X ) ❡ π F g

  (Y )

  (X)

  : F g (X ⊕ Y ) → F g (X) ⊕ F g (Y ) t❛❧ q✉❡ π F g

  ) é ✉♠ ♣r♦❞✉t♦ ❞❡ F g (X) ❡ F g (Y ) s❡❣✉✐♥❞♦ ❛ss✐♠ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ φ g X,Y

  (Y )

  , π F g

  (X)

  ❆❣♦r❛✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ ♦ ❢✉♥t♦r F g é ❛❞✐t✐✈♦ ❡ ❛ss✐♠✱ F g ♣r❡s❡r✈❛ ♣r♦❞✉t♦ ✭✈❡❥❛ Pr♦♣♦s✐çã♦ ✸✳✶✾✮ ❡ ♣♦rt❛♥t♦✱ (F g (X⊕Y ), F g (π X ), F g (π Y )) é ✉♠ ♣r♦❞✉t♦ ❞❡ F g (X) ❡ F g (Y ) ✳ ❚❛♠❜é♠✱ (F g (X)⊕F g (Y ), π F g

  X ❡ Y ❡♠ C ❡ ❞❛í✱ s g ⊕ r g é ✉♠ ✐s♦♠♦r✜s♠♦✳

  ) é ✉♠ ♣r♦❞✉t♦ ❞❡

  (Y )

  (Y )

  (s g ◦ π F g

  1 g

  ◦ i F g

  (X)

  ◦ c Y + s g ◦ π F g

  1 g

  ◦ r

  (Y )

  ◦ i F g

  (X)

  ◦ c X ) = s g ◦ π F g

  ◦ s

  ◦ s

  (X)

  ◦ c Y + i F g

  1 g

  ◦ r

  (Y )

  ) ◦ (i F g

  (X)

  ) ◦ α = = (s g ◦ π F g

  (X)

  (X)

  1 g

  ◦ π F g

  ) ◦ α = c Y ✳ P❛r❛ ♣r♦✈❛r♠♦s ❛ ✉♥✐❝✐❞❛❞❡✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ✉♠ ♠♦r✜s♠♦ β : C → F g (X) ⊕ F g (Y ) t❛❧ q✉❡

  (Y )

  ◦ c X ❡ i F g

  1 g

  ◦ π F g (X) ◦ β = i F g (X) ◦ s

  ) ◦ β = c Y ✳ ❉❡ss❛s ✐❣✉❛❧❞❛❞❡s✱ r❡s✉❧t❛ q✉❡ i F g (X)

  (Y )

  ) ◦ β = c X ❡ q✉❡ (r g ◦ π F g

  (X)

  (s g ◦ π F g

  (Y )

  ◦ c X = s g ◦ 0 F g (Y ) F g

  ❆♥❛❧♦❣❛♠❡♥t❡✱ (r g ◦ π F g

  ◦ c X = c X .

  1 g

  ◦ c X = s g ◦ s

  1 g

  ◦ c Y + s g ◦ I F g (X) ◦ s

  1 g

  ◦ r

  (X)

  • i F g

  1 G

  = (π X ◦ (s ⊕ r )) ◦ φ ◦ (γ ) X⊕Y

  1 G G X,Y

  1

  1 G

  = (s ◦ π ) ◦ φ ◦ (γ ) X⊕Y

  1 F (X) G X,Y 1G

  1 G

  = s ◦ (π F ◦ φ ) ◦ (γ ) X⊕Y

  1 G (X) X,Y 1G

  ◦ F = s

  1 G G 1 (π X ) ◦ (γ ) X⊕Y (∗)

  = s ◦ (γ ) X ◦ π X

  1 G (∗∗)

  =

  I X ◦ π X = π X = π X ◦ I X⊕Y ,

  ❡♠ ✭✯✮ ❡ ❡♠ ✭✯✯✮ ✉t✐❧✐③❛♠♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ ♦s ❞✐❛❣r❛♠❛s

  

(γ ) X⊕Y (γ )

X

  // F // F X ⊕ Y

  1 (X ⊕ Y ) G 1 (X)

  X G G G G G π F X (π 1G X ) G G s 1G I G X G G ##G

  // F (X)

  X X

  1 G (γ ) X

  ❝♦♠✉t❛♠✳ ❆ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ♣r✐♠❡✐r♦ ✈❡♠ ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ γ ❡ ❛ ❞♦ s❡❣✉♥❞♦✱ s❡❣✉❡ ❞♦ ❢❛t♦ ❞❡ q✉❡ X é ✉♠ ♦❜❥❡t♦ ❡q✉✐✈❛r✐❛♥t❡✳ Y ◦ ((s ⊕ r ) ◦ φ ◦ (γ ) X⊕Y ) = π Y ◦ I X⊕Y

  1 G

  ❆♥❛❧♦❣❛♠❡♥t❡✱ π

  1 G

  1 G X,Y ✳

  P♦rt❛♥t♦✱ ❞♦ ▲❡♠❛ 3.3✱ r❡s✉❧t❛ q✉❡

  1 G ((s ⊕ r ) ◦ φ ) ◦ (γ ) X⊕Y = I X⊕Y .

  1 G G X,Y

  1

  ❙❡❥❛♠ g, h ∈ G✳ ❊♥tã♦ ♣r♦✈❡♠♦s ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠ ❛❜❛✐①♦ F ((s ⊕r )◦φ ) g h h X,Y h

  (F g ◦ F h )(X ⊕ Y ) // F g (X ⊕ Y ) ⊕r g

  (γ ) (s g g )◦φ g,h X⊕Y X,Y F gh (X ⊕ Y ) // X ⊕ Y. gh (s ⊕r )◦φ gh gh X,Y

  ❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ g h π X ◦ (((s g ⊕ r g ) ◦ φ ) ◦ F g ((s h ⊕ r h ) ◦ φ )) = X,Y g X,Y h

  = (π X ◦ (s g ⊕ r g )) ◦ φ ◦ F g ((s h ⊕ r h ) ◦ φ ) g X,Y X,Y h ◦ π ◦ φ ◦ F ⊕ r

  = s g F g ((s h h ) ◦ φ ) g (X) X,Y X,Y g h = s g ◦ (π ◦ φ ) ◦ F g ((s h ⊕ r h ) ◦ φ ) F (X) g X,Y X,Y h = s g ◦ F g (π X ) ◦ F g ((s h ⊕ r h ) ◦ φ ) h X,Y = s g ◦ F g (π X ◦ (s h ⊕ r h ) ◦ φ ) h X,Y = s g ◦ F g (s h ◦ π ◦ φ ) F (X) h X,Y = s g ◦ F g (s h ◦ F h (π X ))

  ◦ F ◦ F = s g g (s h ) ◦ (F g h )(π X ). P♦r ♦✉tr♦ ❧❛❞♦✱ gh π X ◦ (((s gh ⊕ r gh ) ◦ φ ) ◦ (γ g,h ) X⊕Y ) = X,Y gh

  = (π X ◦ (s gh ⊕ r gh )) ◦ φ ◦ (γ g,h ) X⊕Y gh X,Y = s gh ◦ π F ◦ φ ◦ (γ g,h ) X⊕Y gh X,Y (X) gh

  = s gh ◦ (π F ◦ φ ) ◦ (γ g,h ) X⊕Y gh (X) X,Y ◦ F

  = s gh gh (π X ) ◦ (γ g,h ) X⊕Y

  (∗)

  = s gh ◦ (γ g,h ) X ◦ (F g ◦ F h )(π X )

  (∗∗)

  = s g ◦ F g (s h ) ◦ (F g ◦ F h )(π X ) ❡♠ ✭✯✮ ❡ ❡♠ ✭✯✯✮ ❢♦✐ ✉t✐❧✐③❛❞♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡

  ❞♦s ❞✐❛❣r❛♠❛s

  (γ ) F (s ) g,h X⊕Y g h

  (F g ◦ F h )(X ⊕ Y ) // F gh (X ⊕ Y ) (F g ◦ F h )(X) // F g (X) ◦F F s

  (F )(π ) (π ) (γ ) g g h X gh X g,h X

  ◦ F // F // X, (F g h )(X) gh (X) F gh (X) s gh

  (γ ) g,h X g,h

  t❛✐s ❝♦♠✉t❛t✐✈✐❞❛❞❡s s❡❣✉❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ γ ❡ ❞♦ ❢❛t♦ ❞❡ X s❡r ✉♠ ♦❜❥❡t♦ ❡q✉✐✈❛r✐❛♥t❡✳

  ❆♥❛❧♦❣❛♠❡♥t❡✱ g gh h ◦(((s ⊕r ⊕r ◦(((s ⊕r π Y g g )◦φ )◦F g ((s h h )◦φ )) = π Y gh gh )◦φ )◦(γ g,h ) X⊕Y ). X,Y X,Y X,Y

  ▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ 3.3✱ r❡s✉❧t❛ q✉❡ g h gh ((s g ⊕r g )◦φ )◦F g ((s h ⊕r h )◦φ ) = ((s gh ⊕r gh )◦φ )◦(γ g,h ) X⊕Y . X,Y X,Y X,Y ❖❜s❡r✈❛♠♦s t❛♠❜é♠ q✉❡✱ ♣❛r❛ t♦❞♦ g ∈ G, g g

  π X ◦ (s g ⊕ r g ) ◦ φ = s g ◦ π ◦ φ X,Y g X,Y F (X) X Y = s g ◦ F g (π X ).

  ❆ss✐♠✱ π é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✳ ❙✐♠✐❧❛r♠❡♥t❡✱ π é t❛♠❜é♠ ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✳

  P❛r❛ ✜♥❛❧✐③❛r♠♦s ❛ ♣r♦✈❛ ❞❡ss❛ ❛✜r♠❛çã♦✱ ♣r♦✈❡♠♦s q✉❡ ❛ tr✐♣❧❛ X Y G ((X ⊕ Y, s ⊕ r), π , π ) X , d Y ) é ✉♠ ♣r♦❞✉t♦ ❞❡ (X, s) ❡ (Y, r) ❡♠ C ✳ G X : (D, ι) → (X, s) Y : (D, ι) → (Y, r) ❙❡❥❛ ((D, ι), d ✉♠❛ tr✐♣❧❛ ❡♠ C ✱ ❡♠ q✉❡ d X ◦ ι g = ❡ d sã♦ ♠♦r✜s♠♦s ❡q✉✐✈❛r✐❛♥t❡s✱ ♦✉ s❡❥❛✱ d s g ◦ F g (d X ) Y ◦ ι g = s g ◦ F g (d Y )

  ❡ d ✱ ♣❛r❛ t♦❞♦ g ∈ G✳ X , π Y ) ❈♦♠♦ (X⊕Y, π é ✉♠ ♣r♦❞✉t♦ ❡♠ C✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦

  φ : D → X ⊕ Y X ◦ φ = d X Y ◦ φ = d Y ❡♠ C t❛❧ q✉❡ π ❡ π ✳

  Pr♦✈❡♠♦s q✉❡ φ é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✳ ❉❡ ❢❛t♦✱ π X ◦ (φ ◦ ι g ) = (π X ◦ φ) ◦ ι g

  = r g ◦ 0 F g (Ker(f )) F g

  99 k ::t t t t t t t t t t

  Ker(f ) Ker (f ) Y

  X f // Y

  %%

  $$J J J J J J J J J J w g Fg (ker(f )) Y

  (k)

  F g (ker(f )) s g ◦F g

  = F g (Ker(f )) Y , ❡♠ ✭✯✮ ✉s❛♠♦s q✉❡ f é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡ ❡ ❡♠ ✭✯✯✮ q✉❡ F g é ❛❞✐t✐✈♦ ✭♣r❡s❡r✈❛ ♣r♦❞✉t♦✮ ❡✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✼ ✭✐✐✮✱ s❡❣✉❡ ❛ ✐❣✉❛❧❞❛❞❡✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦

  (Y )

  (∗∗)

  = d X ◦ ι g = s g ◦ F g (d X ) = s g ◦ F g (π X ◦ φ) = s g ◦ F g (π X ) ◦ F g (φ) = s g ◦ π F g (X) ◦ φ g X,Y ◦ F g (φ) = π X ◦ (s g ⊕ r g ) ◦ φ g X,Y ◦ F g (φ) = π X ◦ ((s g ⊕ r g ) ◦ φ g X,Y ◦ F g (φ)).

  = (r g ◦ F g (f )) ◦ F g (k) = r g ◦ (F g (f ) ◦ F g (k)) = r g ◦ F g (f ◦ k) = r g ◦ F g (0 Ker (f ) Y )

  (∗)

  ❡♠ q✉❡ k : Ker(f) → X é ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ P❛r❛ t♦❞♦ g ∈ G✱ t❡♠♦s q✉❡ f ◦ (s g ◦ F g (k)) = (f ◦ s g ) ◦ F g (k)

  C G ✳ ◆❛ ❝❛t❡❣♦r✐❛ C✱ ❡①✐st❡ ♦ ♣❛r (Ker(f), k)✱ ♥ú❝❧❡♦ ❞❡ f : X → Y ✱

  ) ❡ f : (X, s) → (Y, r) ✉♠ ♠♦r✜s♠♦ ❡♠

  ❆✜r♠❛çã♦ ✭✐✈✮✿ ❚♦❞♦ ♠♦r✜s♠♦ ❡♠ C G ♣♦ss✉✐ ♥ú❝❧❡♦✳ ❙❡❥❛♠ (X, s), (Y, r) ∈ Ob(C G

  ⊕ r g ) ◦ φ g X,Y ◦ F g (φ) ✱ ♣❛r❛ t♦❞♦ g ∈ G✳

  ✳ P♦rt❛♥t♦✱ φ ◦ ι g = (s g

  ❆♥❛❧♦❣❛♠❡♥t❡✱ π Y ◦(φ◦ι g ) = π Y ◦((s g ⊕r g )◦φ g X,Y ◦F g (φ))

  ❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ♥ú❝❧❡♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ w g : F g (Ker(f )) → Ker(f ) t❛❧ q✉❡ k ◦ w g = s g ◦ F g (k).

  ◆♦✈❛♠❡♥t❡ ❛❧❡rt❛♠♦s ♣❛r❛ ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é ❜❡♠ ✉s❛❞❛ ♥♦ q✉❡ s❡❣✉❡ s❡♠ ❢❛③❡r♠♦s ♠❡♥çã♦ à ♠❡s♠❛✳ g : F g (Ker(f )) → Ker(f )

  Pr♦✈❡♠♦s q✉❡ w é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❈♦♥✲ s✐❞❡r❡♠♦s ♦ ♠♦r✜s♠♦ − − 1 1

  1 F g (w g ) ◦ (γ g,g ) ◦ (γ ) Ker : Ker(f ) → F g (Ker(f )). Ker

(f )

(f ) 1 − g (w g ) ◦ (γ g,g ) ◦ (γ ) Ker = w − − 1 1

  1

  ◆❡ss❡ ❝❛s♦✱ F (f ) g ✳ ❉❡ ❢❛t♦✱ Ker

  

(f )

  t❡♠♦s q✉❡ − − 1 1

  1

  (k ◦ w g ) ◦ F g (w ) ◦ (γ ) ◦ (γ ) = g g,g Ker (f ) Ker

  

(f )

− − 1 1

  1

  = (s g ◦ F g (k)) ◦ F g (w ) ◦ (γ ) ◦ (γ ) Ker g g,g (f ) Ker (f ) − − 1 1

  1

  = s g ◦ (F g (k) ◦ F g (w g )) ◦ (γ g,g ) ◦ (γ ) Ker Ker (f )

(f ) − − 1 1

  

1

  = s g ◦ F g (k ◦ w g ) ◦ (γ g,g ) ◦ (γ ) Ker

Ker

(f ) (f ) 1 − 1 − 1

  1

  = s g ◦ F g (s g ◦ F g (k)) ◦ (γ g,g ) ◦ (γ ) Ker Ker (f ) (f ) 1 − 1 − 1

  1

  = s g ◦ (F g (s g ) ◦ F g (F g (k))) ◦ (γ g,g ) ◦ (γ ) Ker Ker (f ) (f ) 1 − 1 − 1

  1

  ◦ F = s g g (s g ) ◦ (F g (F g (k)) ◦ (γ g,g ) ) ◦ (γ ) Ker Ker (f ) (f ) (∗) 1 − 1

  1

  ◦ F ◦ F = s g g (s g ) ◦ ((γ g,g ) X G (f ) 1 (k)) ◦ (γ ) Ker 1 − 1

  1

  = (s g ◦ F g (s g )) ◦ (γ g,g ) ◦ F (k) ◦ (γ ) Ker X

  1 G (f ) (∗∗)

− − 1 1

  

1

  = (s ◦ (γ ) X ) ◦ (γ ) ◦ F (k) ◦ (γ ) Ker

  1 g,g g,g G X G 1 (f )

  = s ◦ F (k) ◦ (γ ) Ker

  1 G G 1 (f ) (∗∗∗)

  = s ◦ (γ ) X ◦ k

  1 G (∗∗∗∗)

  =

  I X ◦ k = k = k ◦ I Ker ,

  (f )

  ❡♠ ✭✯✮✱ ✭✯✯✮✱ ✭✯✯✯✮ ❡ ✭✯✯✯✯✮ ✉t✐❧✐③❛♠♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s ❝♦♠✉t❛t✐✲ ✈✐❞❛❞❡s ❞♦s ❞✐❛❣r❛♠❛s F − − 1 1 g (s ) g−1

  // (F g ◦ F )(Ker(f )) (F g ◦ F )(X) // F F (Ker(f )) g g g (X)

  1 G 1 F ◦F s (γ ) g,g−1 Ker (f ) 1G g−1 g,g−1 (k) (F g )(k) (γ ) − − 1 X g 1 F (X) // (F g ◦ F g )(X) F gg (X)

  1 G − s 1 gg−1 // X (γ ) g,g−1 X (γ )

Ker (f ) (γ )

X Ker(f ) // F (Ker(f )) // F (X)

1 G

  X

  1 G

  G G G k (k) F G G s 1G 1G G I X G G ##G X. // F (X)

  X

  1 G (γ ) X 1

  ❆s ❝♦♠✉t❛t✐✈✐❞❛❞❡s s❡❣✉❡♠ ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ γ g,g ❡ ❞❡ γ ✱ ♣❛r❛ ♦s ❞♦✐s ♣r✐♠❡✐r♦s ♥❛ ✈❡rt✐❝❛❧ ❡ ❞♦ ❢❛t♦ ❞❡ X s❡r ✉♠ ♦❜❥❡t♦ ❡q✉✐✈❛r✐❛♥t❡

  ♣❛r❛ ♦s ❞♦✐s ú❧t✐♠♦s ♥❛ ✈❡rt✐❝❛❧✳ ❆ss✐♠✱ k ◦ (w g ◦ F g (w g 1 ) ◦ (γ g,g 1 )

  1 g

  1 g

  , ❡♠ ✭✯✮ ✉t✐❧✐③❛♠♦s q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ s g ◦ F g (s

  (Ker(f ))

  ◦ s g ◦ F g (k) = F g (k) = F g (k) ◦ I F g

  1 g

  ◦ I X ◦ k ◦ w g = s

  ◦ (γ ) X ◦ k ◦ w g = s

  1 G

  1 G

  ◦ s

  1 g

  = s

  (∗)

  ◦ ((γ ) X ◦ k) ◦ w g

  1 X

  ) = s

  ◦ (γ g,g 1 ) X é

  (f )

  1 Ker (f )

  (Ker(f )) .

  ◦ (γ ) Ker (f ) ) ◦ w g = I F g

  1 Ker (f )

  ❡ ❝♦♠♦ F g (k) é ♠♦♥♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ (F g (w g 1 ) ◦ (γ g,g 1 )

  (Ker(f ))

  ◦(γ ) Ker (f ) ◦w g )) = F g (k)◦I F g

  ✳ P♦rt❛♥t♦✱ F g (k)◦((F g (w g 1 )◦(γ g,g 1 )

  ❡q✉✐✈❛❧❡♥t❡ à F g (s

  1 G

  ◦ s

  1 g

  = s

  1 X

  ) ◦ (γ g,g 1 )

  1 g

  ) ◦ w g = F g (s g 1 ) ◦ (γ g,g 1 )

  (k) ◦ (γ ) Ker

  

1

Ker (f )

  ◦ w g ) = = F g (k) ◦ F g (w g 1 ) ◦ (γ g,g 1 )

  ◦ (γ ) Ker

  1 Ker

(f )

  ◦ w g = F g (k ◦ w g 1 ) ◦ (γ g,g 1 )

  (f )

  ◦ (γ ) Ker

  1 Ker (f )

  (f )

  ◦ w g = F g (s g 1 ◦ F g 1 (k)) ◦ (γ g,g 1 )

  ◦ (γ ) Ker

  

1

Ker (f )

  F g (k) ◦ (F g (w g 1 ) ◦ (γ g,g 1 )

  ♠♦♥♦♠♦r✜s♠♦ s❡❣✉❡✱ ❞❛ Pr♦♣♦s✐çã♦ 2.21✱ q✉❡ F g (k) é ✉♠ ♠♦♥♦♠♦r✲ ✜s♠♦✳ ❆❜❛✐①♦✱ ❡♠ ❛❧❣✉♠❛s ✐❣✉❛❧❞❛❞❡s✱ ✉s❛♠♦s ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❞✐❛❣r❛♠❛s ❞❛❞♦s ✐♠❡❞✐❛t❛♠❡♥t❡ ❛❝✐♠❛✳ ❚❡♠♦s q✉❡

  ◦ (γ ) Ker (f ) ) = I Ker (f ) . P❡❧♦ ▲❡♠❛ 4.2 s❛❜❡♠♦s q✉❡ F g é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❈♦♠♦ k é ✉♠

  

1

Ker (f )

  ◦ (γ ) Ker (f ) ) = k ◦ I Ker (f ) ❡ ❝♦♠♦ k é ♠♦♥♦♠♦r✜s♠♦ ✭✈❡❥❛ Pr♦♣♦s✐çã♦ 1.40✮✱ s❡❣✉❡ q✉❡ w g ◦ (F g (w g 1 ) ◦ (γ g,g 1 )

  (f )

  1 Ker (f )

  

1 G

  (f )

  ◦ (F

  1 X

  (k)) ◦ (γ ) Ker (f ) ◦ w g = F g (s g 1 ) ◦ (γ g,g 1 )

  

1

G

  ◦ F

  1 X

  ◦ w g = F g (s g 1 ) ◦ ((γ g,g 1 )

  ) ◦ (γ ) Ker

  ◦ (γ ) Ker

  1 Ker (f )

  ◦ w g = F g (s g 1 ) ◦ (F g (F g 1 (k)) ◦ (γ g,g 1 )

  (f )

  ◦ (γ ) Ker

  1 Ker (f )

  ◦ w g = (F g (s g 1 ) ◦ F g (F g 1 (k))) ◦ (γ g,g 1 )

  (f )

  ❆ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ❢❛♠í❧✐❛ ❞❡ ✐s♦♠♦r✜s♠♦s w = {w g : F g (Ker(f )) → Ker(f ) : g ∈ G}.

  ▼♦str❡♠♦s q✉❡ (Ker(f), w) ∈ Ob(C G )

  =

  (Ker(f )) F 1G

  (k)

  X

  (γ ) X

  // F

  1 G (X).

  ▲♦❣♦✱ k ◦(w

  ◦ (γ ) Ker

  (f )

  ) = k ◦ I Ker

  (F ) ❡ ❛ss✐♠✱ w

  1 G

  ◦ (γ ) Ker

  (f )

  I Ker

  // F

  (f ) ✭k é ✉♠ ♠♦♥♦♠♦r✜s♠♦✮✳

  ❙❡❥❛♠ g, h ∈ G✳ ❊♥tã♦ k ◦ (w g ◦ F g (w h )) = (k ◦ w g ) ◦ F g (w h ) = (s g ◦ F g (k)) ◦ F g (w h ) = s g ◦ (F g (k) ◦ F g (w h )) = s g ◦ F g (k ◦ w h ) = s g ◦ F g (s h ◦ F h (k)) = s g ◦ F g (s h ) ◦ F g (F h (k)). P♦r ♦✉tr♦ ❧❛❞♦✱ k ◦ (w gh

  ◦ (γ g,h ) Ker

  (f )

  ) = (k ◦ w gh ) ◦ (γ g,h ) Ker

  (f )

  = (s gh ◦ F gh (k)) ◦ (γ g,h ) Ker

  (f )

  = s gh ◦ (F gh (k) ◦ (γ g,h ) Ker

  (f )

  )

  (∗)

  = s gh ◦ ((γ gh ) X ◦ (F g ◦ F h )(k)) = (s gh ◦ (γ gh ) X ) ◦ (F g ◦ F h )(k) = (s g

  ◦ F g (s h )) ◦ (F g ◦ F h )(k), ❡♠ ✭✯✮ ✉t✐❧✐③❛♠♦s ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦✱ ❛ q✉❛❧ s❡❣✉❡

  1 G

  (γ ) Ker (f )

  ✳ ❉❡ ❢❛t♦✱ k ◦ (w

  1 G

  1 G

  ◦ (γ ) Ker

  (f )

  ) = (k ◦ w

  1 G

  ) ◦ (γ ) Ker

  (f )

  = (s

  1 G

  ◦ F

  1 G

  (k)) ◦ (γ ) Ker

  (f )

  = s

  ◦ (F

  Ker(f ) k

  1 G

  (k) ◦ (γ ) Ker

  (f )

  )

  (∗)

  = s

  1 G

  ◦ ((γ ) X ◦ k) = (s

  1 G

  ◦ (γ ) X ) ◦ k =

  I X ◦ k = k

  = k ◦ I Ker

  (f )

  , ❡♠ ✭✯✮ ✉t✐❧✐③❛♠♦s q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

1 G

  g,h

  ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ γ ✱

  

(γ )

g,h Ker (f )

  (F g ◦ F h )(Ker(f )) // F gh (Ker(f )) ◦F F

  

(F g h )(k) gh (k)

(F g ◦ F h )(X) // F gh (X).

  

(γ )

g,h X g g (w h )) = k ◦ (w gh g,h ) Ker ) ◦ F ◦ (γ

  P♦rt❛♥t♦✱ k ◦ (w (f ) ❡ ❝♦♠♦ k é ✉♠ g g (w h ) = w gh g,h ) Ker ◦ F ◦ (γ ♠♦♥♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ w (f ) ✳

  ❖❜s❡r✈❛♠♦s q✉❡ ❞❛ ❝♦♥str✉çã♦ ❞❛ ❢❛♠í❧✐❛ ❞❡ ✐s♦♠♦r✜s♠♦s w✱ ❡♠ g = s g ◦ F g (k) q✉❡ ♣❛r❛ ❝❛❞❛ g ∈ G✱ k ◦ w ✱ r❡s✉❧t❛ q✉❡ ♦ ♠♦r✜s♠♦ k : Ker(f ) → X

  é ❡q✉✐✈❛r✐❛♥t❡✳ ❋✐♥❛❧♠❡♥t❡✱ ♣r♦✈❡♠♦s q✉❡ ♦ ♣❛r ((Ker(f), w), k) é ✉♠ ♥ú❝❧❡♦ ❞♦ G ′ G

  ) ) ♠♦r✜s♠♦ f ❡♠ C ✳ ❙❡❥❛ ((Z, t), k ✉♠ ♣❛r✱ ❡♠ q✉❡ (Z, t) ∈ Ob(C ′ Z

  : (Z, t) → (X, s) = 0 ❡ k é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡ t❛❧ q✉❡ f ◦ k Y ✳ ❉♦ ❢❛t♦ ❞❡ (ker(f), k) s❡r ✉♠ ♥ú❝❧❡♦ ❞❡ f ❡♠ C✱ s❛❜❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u : Z → Ker(f) ❡♠ C t❛❧ q✉❡ k ◦ u = k ✳ ❇❛st❛ ♣r♦✈❛r♠♦s q✉❡ u é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✳ ❉❡ ❢❛t♦✱ k ◦ (u ◦ t g ) = (k ◦ u) ◦ t g

  = k ◦ t g = s g ◦ F g (k ) = s g ◦ F g (k ◦ u) = s g ◦ (F g (k) ◦ F g (u))

  ◦ F = (s g g (k)) ◦ F g (u) = (k ◦ w g ) ◦ F g (u)

  ◦ F g ) = k ◦ (w g ◦ F g (u)) g = w g ◦ F g (u) = k ◦ (w g g (u)). P♦rt❛♥t♦✱ k ◦ (u ◦ t ❡ ❛ss✐♠✱ u ◦ t ✱

  ♣♦✐s k é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ G ❆✜r♠❛çã♦ ✭✈✮✿ ❚♦❞♦ ♠♦r✜s♠♦ ❡♠ C ♣♦ss✉✐ ❝♦♥ú❝❧❡♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ ❢❛t♦ é s✐♠✐❧❛r ❛♦ ❝❛s♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ♥ú❝❧❡♦

  ♣❛r❛ ♠♦r✜s♠♦s✳ ❆♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ✐❞❡✐❛s ♣r✐♥❝✐♣❛✐s✳ G ) G ❙❡❥❛♠ (X, s), (Y, r) ∈ Ob(C ❡ f : (X, s) → (Y, r) ✉♠ ♠♦r✜s♠♦ ❡♠

  C ✳ ◆❛ ❝❛t❡❣♦r✐❛ C✱ ❡①✐st❡ ♦ ♣❛r (CoKer(f), q)✱ ❝♦♥ú❝❧❡♦ ❞♦ ♠♦r✜s♠♦ f : X → Y

  ✳ P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦✱ ♣❛r❛

  ❝❛❞❛ g ∈ G✱ Q = F g (CoKer(f )) − F 1

g

(q)◦r

  KK kk g ggOO OOO X OOO Q OOO f v g O X // Y. X o CoKer (f ) oo oo oo oo q wwooo rr

  CoKer(f ) ❈♦♥s✐❞❡r❛♠♦s ♦ ♠♦r✜s♠♦ − −

  1 1

  1

  ◦(γ ◦F u g = (γ ) g,g ) CoKer g (v ) : F g (CoKer(f )) → CoKer(f ). CoKer (f )

(f ) g

g = v

  1

  ❙✐♠✐❧❛r♠❡♥t❡ ❛♦ ❝❛s♦ ❞♦ ♥ú❝❧❡♦✱ ♣r♦✈❛✲s❡ q✉❡ u g ❡ ♣♦rt❛♥t♦✱ t❡♠♦s ✉♠ ✐s♦♠♦r✜s♠♦✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ❢❛♠í❧✐❛ ❞❡ ✐s♦✲ ♠♦r✜s♠♦s u = {u g : F g (CoKer(f )) → CoKer(f ) : g ∈ G}. G

  ) ◆❡ss❛s ❝♦♥❞✐çõ❡s (CoKer(f), u) ∈ Ob(C ❡ q é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✲

  ✈❛r✐❛♥t❡✳ ❋✐♥❛❧♠❡♥t❡✱ ♥ã♦ é ❞✐❢í❝✐❧ ♠♦str❛r q✉❡ ((CoKer(f), u), q) é ✉♠ G ❝♦♥ú❝❧❡♦ ❞❡ f ❡♠ C ✳ G

  ❆✜r♠❛çã♦ ✭✈✐✮✿ ❚♦❞♦ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C é ✉♠ ♥ú❝❧❡♦ ❞❡ ❛❧❣✉♠ G ♠♦r✜s♠♦ ❡♠ C ✳ G

  ❙❡❥❛ f : (X, s) → (Y, r) ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C ✳ ❈♦♠♦ C é ✉♠❛ ❝❛t❡❣♦r✐❛ k✲❧✐♥❡❛r s❡❣✉❡✱ ❞♦ ▲❡♠❛ 3.23✱ q✉❡ f é ✉♠ ♥ú❝❧❡♦ ❞♦ s❡✉ ❝♦♥ú❝❧❡♦ q : Y → CoKer(f)✳

  ❉❛ ❛✜r♠❛çã♦ ❛♥t❡r✐♦r✱ s❡❣✉❡ q✉❡ (CoKer(f), u) é ✉♠ ♦❜❥❡t♦ ❡q✉✐✲ ✈❛r✐❛♥t❡ ❡ q é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ à ❢❛♠í❧✐❛ u✳ ▼♦s✲ G tr❡♠♦s q✉❡ f é ✉♠ ♥ú❝❧❡♦ ❞❡ q ❡♠ C ✳ G

  ) ❙❡❥❛ ((D, l), d) ✉♠ ♣❛r✱ ❡♠ q✉❡ (D, l) ∈ Ob(C ❡ d : D → Y é ✉♠ D

  ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡ t❛❧ q✉❡ q ◦ d = 0 ✳ ❈♦♠♦ (X, f) é ✉♠ CoKer (f ) ♥ú❝❧❡♦ ❞❡ q ❡♠ C✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ι : D → X t❛❧ q✉❡ f ◦ι = d✳ ❘❡st❛✲♥♦s ✈❡r q✉❡ ι é ✉♠ ♠♦r✜s♠♦ ❡q✉✐✈❛r✐❛♥t❡✳ ❉❡ ❢❛t♦✱ f ◦ (ι ◦ l g ) = (f ◦ ι) ◦ l g

  = d ◦ l g = r g ◦ F g (d) = r g ◦ F g (f ◦ ι) = r g ◦ (F g (f ) ◦ F g (ι)) = (r g ◦ F g (f )) ◦ F g (ι) = (f ◦ s g ) ◦ F g (ι)

  ◦ F = f ◦ (s g g (ι)). g ) = f ◦ (s g ◦ F g (ι))

  P♦rt❛♥t♦✱ f ◦ (ι ◦ l ❡ ❝♦♠♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦ G g = s g ◦ F g (ι) ❡♠ C ✱ ♦ s❡rá ❡♠ C✳ ▲♦❣♦✱ ι ◦ l ✳

  ❆✜r♠❛çã♦ ✭✈✐✐✮✿ ❚♦❞♦ ❡♣✐♠♦r✜s♠♦ é ✉♠ ❝♦♥ú❝❧❡♦ ❞❡ ❛❧❣✉♠ ♠♦r✲ G ✜s♠♦ ❡♠ C ✳

  ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ ❢❛t♦ é s✐♠✐❧❛r à ❞❛ ❛✜r♠❛çã♦ ❛♥t❡r✐♦r✳ ❋✐♥❛❧✐③❛♠♦s ♦ tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ✉♠ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ ❛ r❡s♣❡✐t♦

  ❞❡st❡ ❛ss✉♥t♦✳ ❈♦♠♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ ❡r❛ s❡❣✉✐r ❛ r❡❢❡rê♥❝✐❛ [✶✻]✱ ♣♦✲ ❞❡♠♦s ❞✐③❡r q✉❡ ❡ss❡ ✏✜♠✑ ♥ã♦ é ❜❡♠ ✉♠ ✜♠✱ ♠❛s s✐♠ ✉♠❛ ♣r❡♣❛r❛çã♦ ♣❛r❛ ✉♠ ❡st✉❞♦ ♣♦st❡r✐♦r✱ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✳ ❊①✐st❡✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ♣❛r❛ ❝❛t❡❣♦r✐❛s t❡♥s♦r✐❛✐s ❛ ❝♦♥str✉çã♦ ❞❛ ❡q✉✐✲ ✈❛r✐❛♥t✐③❛çã♦✱ ✉t✐❧✐③❛♥❞♦ ♥♦çõ❡s ❞❡ ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s ❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ♠♦♥♦✐❞❛✐s ♣❛r❛ ❞❡✜♥✐r ❛ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦ ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ❞❡ss❡ t✐♣♦✳

  ◆♦ tr❛❜❛❧❤♦ ❝✐t❛❞♦✱ ♦ ♦❜❥❡t✐✈♦ é ❡st✉❞❛r r❡♣r❡s❡♥t❛çõ❡s ❞❡ ❝❛t❡❣♦✲ r✐❛s t❡♥s♦r✐❛s✳ P❛r❛ ✐ss♦ é ❢❡✐t❛ t♦❞❛ ❡ss❛ ♣r❡♣❛r❛çã♦✱ ❡♠ q✉❡ ❛ ♣❛rt❡ ✐♥✐❝✐❛❧ ❞❡❧❛ ❡stá ❡①♣♦st❛ ♥❡st❛ ❞✐ss❡rt❛çã♦✳ P❛r❛ ♦ ❧❡✐t♦r q✉❡ ❞❡s❡❥❛r ✉♠ ♠❛✐♦r ❛♣r♦❢✉♥❞❛♠❡♥t♦ ♥♦ ❛ss✉♥t♦ ❡ t❡♥❤❛ ❝✉r✐♦s✐❞❛❞❡ ❡♠ ❝♦♥t✐♥✉❛r ❡s✲ t✉❞❛♥❞♦ ♦ t❡♠❛✱ s✉❣❡r✐♠♦s ❛s ♥♦t❛s ❞❡ ❛✉❧❛ [✶✻]✳

  ❆♣ê♥❞✐❝❡ ❆ ➪❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡

  ◆❡ss❡ ❛♣ê♥❞✐❝❡ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s ♥♦ ❝♦♥✲ t❡①t♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❝♦♥str✉çã♦ ❞❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❈✐t❛♠♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛ [

  ✶✼]✳

  ❆✳✶ ➪❧❣❡❜r❛s ❞❡ ▲✐❡

  ❉❡✜♥✐çã♦ ❆✳✶ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❯♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ s♦❜r❡ k ♦✉ k✲ á❧❣❡❜r❛ ❞❡ ▲✐❡ ✭♦✉ s✐♠♣❧❡s♠❡♥t❡ á❧❣❡❜r❛ ❞❡ ▲✐❡✮ é ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ L

  ✱ ♠✉♥✐❞♦ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r [ , ] : L × L → L ❝❤❛♠❛❞❛ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✱ q✉❡ s❛t✐s❢❛③ ❛s 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✳

  ❆ ✐❣✉❛❧❞❛❞❡ ✭✐✐✮ é ❝❤❛♠❛❞❛ ❞❡ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✳ ❉❡♥♦t❛♠♦s (L, [ , ])

  ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ▲✳ →

  ❉❡✜♥✐çã♦ ❆✳✷ ❙❡❥❛♠ L

  1 ❡ L 2 á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ❯♠❛ ❢✉♥çã♦ φ : L

  1 L

2 é ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ s❡ φ é k✲❧✐♥❡❛r ❡ φ([x, y]) =

  [φ(x), φ(y)] ✱ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ L

  1 ✳

  ❊①❡♠♣❧♦ ❆✳✸ ❙❡❥❛ A ✉♠❛ k✲á❧❣❡❜r❛✳ ❉❡✜♥✐♠♦s ❡♠ A ♦ ❝♦❧❝❤❡t❡ [x, y] = xy − yx

  ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ A✳ ❊♥tã♦ (A, [ , ]) é ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳

  ❉❡ ❢❛t♦✱ ♥♦t❡♠♦s q✉❡ [x, x] = xx − xx = 0✱ ♣❛r❛ t♦❞♦ x ∈ A✳ ❙❡❥❛♠ x, y, z ∈ A ✳ ❊♥tã♦

  [x, [y, z]] = x(yz − zy) − (yz − zy)x = x(yz) − x(zy) − (yz)x + (zy)x. ❆♥❛❧♦❣❛♠❡♥t❡✱

  [z, [x, y]] = z(xy) − z(yx) − (xy)z + (yx)z ❡ [y, [z, x]] = y(zx) − y(xz) − (zx)y + (xz)y.

  ❙♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s três ✐❣✉❛❧❞❛❞❡s✱ ♦❜t❡♠♦s ❛ ✐❞❡♥t✐✲ ❞❛❞❡ ❞❡ ❏❛❝♦❜✐✳

  ❖ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❞❡✜♥✐❞♦ ♥❡ss❡ ❡①❡♠♣❧♦ é ❝❤❛♠❛❞♦ ❝♦♠✉t❛❞♦r✳ ❉❡♥♦t❛♠♦s ♣♦r L(A) ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ (A, [ , ]) ❝✉❥♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ é ♦ ❝♦♠✉t❛❞♦r✳

  ❈♦♥s✐❞❡r❡♠♦s B ✉♠❛ k✲á❧❣❡❜r❛ ❡ φ : A → B ✉♠ ♠♦r✜s♠♦ ❞❡ k✲ á❧❣❡❜r❛s✳ ❊♥tã♦ φ : L(A) → L(B) ✐♥❞✉③✐❞❛ é ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ❉❡ ❢❛t♦✱

  φ([x, y]) = φ(xy − yx) = φ(x)φ(y) − φ(y)φ(x) = [φ(x), φ(y)]

  ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ A✳ ❈♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ s❡❣✉❡ ♦ ♣ró①✐♠♦ ❡①❡♠✲

  ♣❧♦✳ ❊①❡♠♣❧♦ ❆✳✹ ❙❡❥❛ V ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡♥♦t❛♠♦s ♣♦r gl(V ) ♦ ❡s♣❛ç♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s k✲❧✐♥❡❛r❡s ❞❡ V ❡♠ V ✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s ❝♦♠♦ ♣r♦❞✉t♦✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ gl(V ) é ✉♠❛ k

  ✲á❧❣❡❜r❛✳ ❙❡ ❞❡✜♥✐r♠♦s ❡♠ gl(V ) ♦ ❝♦❧❝❤❡t❡ [ , ] : gl(V ) × gl(V ) → gl(V )

  (f, g) 7→ f ◦ g − g ◦ f ❡♥tã♦ gl(V ) t❡♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛ ❞❡ ▲✐❡✳

  

❆✳✷ ➪❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡

▲✐❡

  ❉❡✜♥✐çã♦ ❆✳✺ ❙❡❥❛♠ k ✉♠ ❝♦r♣♦ ❡ V ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❯♠❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧ ❞❡ V é ✉♠ ♣❛r (X, i)✱ ❡♠ q✉❡ X é ✉♠❛ k✲á❧❣❡❜r❛ ❡ i : V → X

  é ✉♠ ♠♦r✜s♠♦ ❞❡ k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s t❛❧ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r k ✲á❧❣❡❜r❛ A ❡ q✉❛❧q✉❡r f : V → A ♠♦r✜s♠♦ ❞❡ k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱

  ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s φ : X → A t❛❧ q✉❡ φ ◦ i = f✱ ♦✉ s❡❥❛✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦ i

  V // X A A φ

  A A f A A

  A ~~ A. ❖❜s❡r✈❛çã♦ ❆✳✻ ❆ á❧❣❡❜r❛ t❡♥s♦r✐❛❧ é ú♥✐❝❛✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ ❉❡ ❢❛t♦✱ s❡ (X, i) ❡ (Y, j) sã♦ á❧❣❡❜r❛s t❡♥s♦r✐❛✐s ❞❡ V ❡♥tã♦ s❡❣✉❡✱ ❞❛ ❞❡✜♥✐çã♦✱ q✉❡ ❡①✐st❡♠ ♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s φ : X → Y ❡ ψ : Y → X t❛✐s q✉❡ φ ◦ i = j ❡ ψ ◦ j = i✳

  ❆ss✐♠✱ j = φ ◦ i = (φ ◦ ψ) ◦ j✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ (ψ ◦ φ) ◦ i = i✳ ❉❡✈✐❞♦ Y X à ✉♥✐❝✐❞❛❞❡✱ r❡s✉❧t❛ q✉❡ φ ◦ ψ = I ❡ ψ ◦ φ = I ✱ ♦✉ s❡❥❛✱ X ❡ Y sã♦ ✐s♦♠♦r❢♦s✳

  ❆♣r❡s❡♥t❛♠♦s✱ r❡s✉♠✐❞❛♠❡♥t❡✱ ❛ ❝♦♥str✉çã♦ ❞❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧ ❞❡ ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ V ✳ ❉❡✜♥✐♠♦s r❡❝✉rs✐✈❛♠❡♥t❡ ♦ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ n n

  

1 +1

  T (V ) (V ) = k (V ) = V (V ) = T (V ) ⊗ V ❝♦♠♦ T ✱ T ❡ T ✱ ♣❛r❛ t♦❞♦ n ≥ 1✳

  ❈♦♥s✐❞❡r❛♥❞♦ ❛ s♦♠❛ ❞✐r❡t❛ ✐♥❞❡①❛❞❛ ❡♠ N✱ ❞❡✜♥✐♠♦s M M n ⊗ n n T (V ) = T (V ) = n> n> V , = V ⊗ · · · ⊗ V

  ❡♠ q✉❡ V ✭n ✈❡③❡s✮✳ ❋❛❧t❛ ❞❡♥✐✜r♠♦s ✉♠ ♣r♦❞✉t♦ ♣❛r❛ t♦r♥❛r T (V ) ✉♠❛ á❧❣❡❜r❛✳ ❙❡❥❛♠ v, w ∈ T (V ) s❛❜❡♠♦s q✉❡ v = ni mj v n +· · ·+v n m +· · ·+w m n ∈ V m ∈ V 1 ❡ w = w k l i j 1 ✱ ❡♠ q✉❡ v ❡ w ✱

  ♣❛r❛ i ∈ {1, · · · , k} ❡ j ∈ {1, · · · , l}✳ P❛r❛ ❞❡✜♥✐r♠♦s ♦ ♣r♦❞✉t♦ vw✱ ❜❛st❛ q✉❡ s❡❥❛ ❞❡✜♥✐❞♦ ♦ ♣r♦❞✉t♦

  ⊗ · · · ⊗ a ⊗ b ⊗ · · · ⊗ b v n w m = a

i j i j

1 n 1 m .

  ❊st❡♥❞❡♠♦s ❧✐♥❡❛r♠❡♥t❡ ♣❛r❛ ♦ ♣r♦❞✉t♦ vw✳ ❈♦♠ ❡ss❛ ❡str✉t✉r❛✱ T (V )

  é ✉♠❛ k✲á❧❣❡❜r❛✳

  Pr♦♣♦s✐çã♦ ❆✳✼ ❖ ♣❛r (T (V ), i)✱ ❡♠ q✉❡ T (V ) é ❝♦♠♦ ❛❝✐♠❛ ❡ i : V → T (V )

  é ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✱ é ❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧ ❞❡ V ✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s A ✉♠❛ k✲á❧❣❡❜r❛ ❡ f : V → A ✉♠

  ⊗ ♠♦r✜s♠♦ ❞❡ k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❉❡✜♥✐♠♦s φ : T (V ) → A ♣♦r φ(v

  1

  v ⊗· · ·⊗v n− ⊗v n ) = f (v )f (v ) · · · f (v n− )f (v n )

  2

  1

  1

  2 1 ❡ ❡st❡♥❞❡♠♦s ❧✐♥❡❛r✲

  ♠❡♥t❡✳ ❈❧❛r❛♠❡♥t❡✱ s❡❣✉❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ ❞❛ ❞❡✜♥✐çã♦✳ ◆♦t❡♠♦s q✉❡ ❛ ♠❛♥❡✐r❛ ❝♦♠♦ φ é ❞❡✜♥✐❞❛ é ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ t❡♥❤❛♠♦s ✉♠ ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ❡ ✐ss♦ ♥♦s ❣❛r❛♥t❡ ❛ ✉♥✐❝✐❞❛❞❡✳

  ❉❡✜♥✐çã♦ ❆✳✽ ❙❡❥❛ L ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❯♠❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ L é ✉♠ ♣❛r (U, ι)✱ ❡♠ q✉❡ U é ✉♠❛ k✲á❧❣❡❜r❛ ❡ ι : L → L

  (U ) é ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ t❛❧ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r k✲á❧❣❡❜r❛

  A ❡ q✉❛❧q✉❡r ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ φ : L → L(A)✱ ❡①✐st❡ ✉♠

  ú♥✐❝♦ ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ψ : U → A q✉❡ ✐♥❞✉③ ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ ψ : L(U) → L(A) q✉❡ ❝♦♠✉t❛ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ι

  // L(U ) L C C C C C C φ ψ C

  C !!C {{

  L (A). ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥♦ ❝❛s♦ ❞❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧✱ ❛

  á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ é ú♥✐❝❛✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳

  ❆❜❛✐①♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❝♦♥str✉çã♦ ❞❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❉❡✜♥✐çã♦ ❆✳✾ ❙❡❥❛♠ L ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❡ T (L) ❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧ ❞❡ L✳ ❈♦♥s✐❞❡r❡♠♦s ❡♠ T (L) ♦ ✐❞❡❛❧ I ❣❡r❛❞♦ ♣♦r ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛ [x, y] − x ⊗ y + y ⊗ x

  ✱ ♣❛r❛ x, y ∈ L✳ ❉❡✜♥✐♠♦s ✉♠❛ ♥♦✈❛ á❧❣❡❜r❛ ❝♦♠♦ s❡♥❞♦ ♦ q✉♦❝✐❡♥t❡ U(L) = T (L)/I✳ ❆ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♣❡❞✐r♠♦s q✉❡ ♦ ✐❞❡❛❧ I s❡❥❛ ❣❡r❛❞♦ ♣♦r ❡❧❡♠❡♥t♦s

  ❞❛q✉❡❧❛ ❢♦r♠❛ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❛♣❧✐❝❛çã♦ ι : L → L

  (U(L)) é ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ L

  ✱ t❡♠♦s [ι(x), ι(y)] = [x, y]

  = x y − y x = x ⊗ y − y ⊗ x = x ⊗ y − y ⊗ x

  (∗)

  = [x, y] = ι([x, y]),

  ❡♠ ✭✯✮ ✉s❛♠♦s ❡①♣❧✐❝✐t❛♠❡♥t❡ ❛ ❞❡✜♥✐çã♦ ❞♦ ✐❞❡❛❧ I✳ Pr♦♣♦s✐çã♦ ❆✳✶✵ ❖ ♣❛r (U(L), ι) é ❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ❞❡ L

  ✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ A ✉♠❛ k✲á❧❣❡❜r❛ ❡ φ : L → L(A) ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ◆♦t❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ φ : L → A é ✉♠ ♠♦r✜s♠♦ ❞❡ k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡ á❧❣❡❜r❛ t❡♥s♦r✐❛❧✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ψ : T (L) → A t❛❧ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ i

  // T (L) L

  > > ψ

  > >

  > φ > >

  > }}

  A, ✐st♦ é✱ ψ ◦ i = φ✳

  ❖❜s❡r✈❡♠♦s t❛♠❜é♠ q✉❡ ♦ ✐❞❡❛❧ I ❡stá ❝♦♥t✐♥❞♦ ♥♦ ♥ú❝❧❡♦ ❞♦ ♠♦r✲ ✜s♠♦ ψ✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ x, y ∈ L✳ ❊♥tã♦

  ψ([x, y]) = φ([x, y]) = [φ(x), φ(y)] = φ(x)φ(y) − φ(y)φ(x) = ψ(xy) − ψ(yx) = ψ(x ⊗ y) − ψ(y ⊗ x) = ψ(x ⊗ y − y ⊗ x).

  P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ✐♥❞✉③✐r ♦ ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ψ : U(L) → A ♣♦r ψ(z) = ψ(z)✱ ♣❛r❛ t♦❞♦ z ∈ T (L)✳ ◆❛t✉r❛❧♠❡♥t❡✱ ψ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ♣❡❧♦ ❛❝✐♠❛✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ é ❝♦♠✉t❛t✐✈♦ ι

  // L(U(L)) L C C C C C C φ C

  C ψ !!C zz

  L (A),

  ♣♦✐s (ψ ◦ ι)(x) = ψ(ι(x))

  = ψ(x) = ψ(x) = φ(x),

  ♣❛r❛ t♦❞♦ x ∈ L✳

  ❆♣ê♥❞✐❝❡ ❇ ❈♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s ❡ ❝♦❝❛❞❡✐❛s

  ◆❡ss❡ ❛♣ê♥❞✐❝❡ ❡st✉❞❛♠♦s ♦s ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛ ❡ ❝♦❝❛❞❡✐❛✳ ❈✐✲ t❛♠♦s ❝♦♠♦ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ [✽]✳ ❉❡✜♥✐çã♦ ❇✳✶ ❙❡❥❛ R ✉♠ ❛♥❡❧✳ ❯♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛ (C, d) é ✉♠ n } , {d n } ) n } ♣❛r ({C n∈Z n∈Z ❡♠ q✉❡ {C n∈Z é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ R✲♠ó❞✉❧♦s n } n : C n → C n− ❡ {d n∈Z é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ R✲❤♦♠♦♠♦r✜s♠♦s d n− ◦ d n = 0

  1 t❛✐s

  q✉❡ d

  1 ♣❛r❛ t♦❞♦ n ∈ Z✳

  ❆ ❝❛❞❡✐❛ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ ❞✐❛❣r❛♠❛ s❡❣✉✐♥t❡ d n +1 d n ...

  // C n // C n− // ... .

  • 1 // C n

  1

  ❊①❡♠♣❧♦ ❇✳✷ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❡ M ✉♠ R✲♠ó❞✉❧♦✳ P♦❞❡♠♦s ❝♦♥s✐✲ ❞❡r❛r ♦ ❝♦♠♣❧❡①♦ ...

  // ... . // 0 // 0 // M // 0 // 0

  ❊①❡♠♣❧♦ ❇✳✸ ❈♦♥s✐❞❡r❡♠♦s ♦ ♠♦r✜s♠♦ f : Z → Z

  8

  8 x 7→ 4x.

  ❊♥tã♦ t❡♠♦s ❛ ❝❛❞❡✐❛ f f ... // ...

  // Z // Z // Z

  8

  

8

  8

  ❉❡✜♥✐çã♦ ❇✳✹ ❙❡❥❛♠ (C, d) ❡ (D, δ) ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛✳ ❯♠ ♠♦r✲ ✜s♠♦ φ : C → D ❡♥tr❡ ❛s ❝❛❞❡✐❛s é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ R✲❤♦♠♦♠♦r✜s♠♦s {φ n : C n → D n } n ◦ d n = φ n ◦ δ n n∈Z t❛✐s q✉❡ φ ◦ d = δ ◦ φ✱ ♦✉ s❡❥❛✱ ♣❛r❛ ❝❛❞❛ n ∈ Z t❡♠♦s q✉❡ φ +1 +1 +1 ✳

  P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦ ♠♦r✜s♠♦ φ ♣❡❧❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛✲ ❣r❛♠❛ ❛❜❛✐①♦ d n n +1 d ... ...

  // // // // C n +1 C n C n−

φ φ φ

n +1 n− n

  1 1 ...

  // ... // D n +1 n // D n−

  1 δ δ n +1 n // D

  ❖❜s❡r✈❛çã♦ ❇✳✺ ❙❡❥❛♠ (A, α)✱ (B, β) ❡ (C, λ) ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛✳ ❉❛❞♦s f : A → B ❡ g : B → C ♠♦r✜s♠♦s ❡♥tr❡ ❝❛❞❡✐❛s✱ ❝♦♥s✐❞❡r❛♠♦s g ◦ f : A → C n ◦ f n : A n → C n }

  n∈Z

  ❝♦♠♦ ❛ ❢❛♠í❧✐❛ {g ✳ ❊♥tã♦✱ g ◦ f é ✉♠ ♠♦r✜s♠♦ ❡♥tr❡ ❝❛❞❡✐❛s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ ❝❛❞❛ n ∈ Z t❡♠♦s q✉❡

  λ n ◦ (g n ◦ f n ) = (λ n ◦ g n ) ◦ f n = (g n− ◦ β n ) ◦ f n

  1

  = g n− ◦ (β n ◦ f n )

  1

  = g n− ◦ (f n− ◦ α n )

  1

  1 = (g n− ◦ f n− ) ◦ α n .

  1

  1

  ❖❜s❡r✈❛♠♦s q✉❡ ❡ss❛ ❝♦♠♣♦s✐çã♦ é ❛ss♦❝✐❛t✐✈❛✱ ✐ss♦ s❡❣✉❡ ❞❛ ❛ss♦✲ ❝✐❛t✐✈✐❞❛❞❡ ❞❛ ❝♦♠♣♦s✐çã♦ ❡♥tr❡ ♠♦r✜s♠♦s ❞❡ R✲♠ó❞✉❧♦s✳ ❉❛❞♦ ✉♠ C : C → C ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛ (C, d) ♦ ♠♦r✜s♠♦ 1 ❞❡✜♥✐❞♦ ♣❡❧❛ ❢❛♠í❧✐❛ n : C n → C n } ❞❡ R✲❤♦♠♦♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡ {1 n∈Z é ❛ ✐❞❡♥t✐❞❛❞❡ ❝♦♠ r❡s♣❡✐t♦ ❛ ❡ss❛ ❝♦♠♣♦s✐çã♦ ❡♥tr❡ ♠♦r✜s♠♦s ❞❡ ❝❛❞❡✐❛✳ n

  ❉❛❞♦ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛ (C, d)✱ ♥♦t❡♠♦s q✉❡ ♦ ❢❛t♦ ❞❡ d d n +1 = 0 n +1 ) ⊆ ker(d n ) ✐♠♣❧✐❝❛ q✉❡ Im(d ❡ ❛ss✐♠ ♣♦❞❡♠♦s ❢♦r♠❛❧✐③❛r

  ❛ ❞❡✜♥✐çã♦ s❡❣✉✐♥t❡✳ n (C) = ker(d n )/Im(d n ) ❉❡✜♥✐çã♦ ❇✳✻ ❖ R✲♠ó❞✉❧♦ q✉♦❝✐❡♥t❡ H +1 é ❝❤❛♠❛❞♦ ❞❡ ♥✲é③✐♠♦ ❣r✉♣♦ ❞❡ ❤♦♠♦❧♦❣✐❛✳ ❉❡✜♥✐çã♦ ❇✳✼ ❙❡❥❛ R ✉♠ ❛♥❡❧✳ ❯♠ ❝♦♠♣❧❡①♦ ❞❡ ❝♦❝❛❞❡✐❛ (C, d) é ✉♠ n n n

  } , {d } ) } ♣❛r ({C n∈Z n∈Z ❡♠ q✉❡ {C n∈Z é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ R✲♠ó❞✉❧♦s n n n− n

  1

  } : C → C ❡ {d n∈Z é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ R✲❤♦♠♦♠♦r✜s♠♦s d t❛✐s n n−

  1

  ◦ d = 0 q✉❡ d ♣❛r❛ t♦❞♦ n ∈ Z✳

  ❯♠❛ ❝♦❝❛❞❡✐❛ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ ❞✐❛❣r❛♠❛ ❛ s❡❣✉✐r n n n d d n +1 n +2

  • 1 +2 ...

  // ... . // C

  // C // C ❉❡✜♥✐çã♦ ❇✳✽ ❙❡❥❛♠ (C, d) ❡ (D, δ) ❝♦♠♣❧❡①♦s ❞❡ ❝♦❝❛❞❡✐❛✳ ❯♠ ♠♦r✲ ✜s♠♦ φ : C → D ❡♥tr❡ ❛s ❝♦❝❛❞❡✐❛s é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ R✲❤♦♠♦♠♦r✜s♠♦s n n n {φ : C → D } n +1 n n +1 n +1 n∈Z t❛❧ q✉❡ φ ◦ d = δ ◦ φ✱ ♦✉ s❡❥❛✱ ♣❛r❛ ❝❛❞❛ n ∈ Z

  ◦ φ = φ ◦ d t❡♠♦s δ ✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦ ♠♦r✜s♠♦ φ ♣❡❧❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛✲

  ❣r❛♠❛ ❛❜❛✐①♦ n d d n +1 n +2 n n ... +1 +2 ...

  // // // // C n φ n n +1 n +2 C C n n φ φ

  ... +1 +2 // ...

  // D δ δ n +1 n +2 // D // D ❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ ❝❛s♦ ❢❡✐t♦ ♣❛r❛ ❝❛❞❡✐❛s✱ ❛ ❝♦♠♣♦s✐çã♦ ❡♥tr❡ ♠♦r✲

  ✜s♠♦s ❞❡ ❝♦❝❛❞❡✐❛s é ✉♠ ♠♦r✜s♠♦ ❞❡ ❝♦❝❛❞❡✐❛s✳ ❊ss❛ ❝♦♠♣♦s✐çã♦ é ❛ss♦❝✐❛t✐✈❛ ❡ t❡♠ ❝♦♠♦ ❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡ ❛ ❢❛♠í❧✐❛ ❞❛s ❛♣❧✐❝❛çõ❡s ✐❞❡♥t✐❞❛❞❡ ❡♥tr❡ R✲♠ó❞✉❧♦s✳ n

  ◦ n− ❉❛❞♦ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝♦❝❛❞❡✐❛ (C, d)✱ ♥♦t❡♠♦s q✉❡ ❛ ❝♦♥❞✐çã♦ d

  1 n− 1 n

  d = 0 ) ⊆ ker(d ) ❞❛ ❞❡✜♥✐çã♦✱ ♥♦s ❞✐③ q✉❡ Im(d ❡ ❛ss✐♠ t❡♠♦s

  ❛ ❞❡✜♥✐çã♦ s❡❣✉✐♥t❡✳ n n−

  1 n

  (C) = Im(d )/ker(d ) ❉❡✜♥✐çã♦ ❇✳✾ ❖ R✲♠ó❞✉❧♦ q✉♦❝✐❡♥t❡ H é ❝❤❛♠❛❞♦ ❞❡ ♥✲é③✐♠♦ ❣r✉♣♦ ❞❡ ❝♦❤♦♠♦❧♦❣✐❛✳ ❉❡✜♥✐çã♦ ❇✳✶✵ ❙❡❥❛ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡✳ ❯♠ R✲ n } ♠ó❞✉❧♦ Z✲❣r❛❞✉❛❞♦ é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ R✲♠ó❞✉❧♦s {M n∈Z ✳ ❉❛❞♦s ❞♦✐s R

  ✲♠ó❞✉❧♦s Z✲❣r❛❞✉❛❞♦s M ❡ N✱ ✉♠ ❤♦♠♦♠♦r✜s♠♦ φ : M → N ❞❡ n : M n → N n } ❣r❛✉ ❦ é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ R✲❤♦♠♦♠♦r✜s♠♦s {φ +k ✳

  ❆♣ê♥❞✐❝❡ ❈

❈♦♥str✉çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦

❞♦ ●r✉♣♦ ❞❡ ❚r❛♥ç❛s

  ❋❛r❡♠♦s ❛q✉✐ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ♣❛r❛ ♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s

  2

  × [0, 1] ♥♦ ❡s♣❛ç♦ R ✳ ❚❛❧ ♠♦❞❡❧♦ ✐rá s❛t✐s❢❛③❡r ❛s r❡❧❛çõ❡s ❞♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ♣❛r❛ ❞❡✜♥✐r t❛❧ ❣r✉♣♦ ✈✐❛ ❣❡r❛❞♦r❡s ❡ r❡❧❛çõ❡s✳ P♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çõ❡s✳ ❙❡❣✉✐♠♦s ❛ ❝♦♥tr✉çã♦ ❢❡✐t❛ ❡♠ [✶✷]✳

  ❈✳✶ ▲✐♥❦s n

  ❙❡❥❛ X ⊂ R ✉♠ s✉❜❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❝♦♥✈❡①♦✳ ❉❛❞♦ ✉♠❛ q✉❛♥✲ ∈ X

  1 , · · · , M k

  t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♣♦♥t♦s M ❝♦♠ k ∈ N✱ ❞❡♥♦t❛♠♦s [M

  1 , · · · , M k ]

  ❛ ❡♥✈♦❧tór✐❛ ❝♦♥✈❡①❛ ❢❡❝❤❛❞❛✱ ✐st♦ é✱ [M , · · · , M k ] = {λ M + · · · + λ k M k : λ i ∈ R i ≥ 0

  1

  1 1 ❡ λ + · · · + λ k = 1}.

  ♣❛r❛ i = 1, · · · , k ❝♦♠ λ

  1

  , · · · , M k [ ❉❡♥♦t❛♠♦s ♣♦r ]M

  1 ❛ ❡♥✈♦❧tór✐❛ ❝♦♥✈❡①❛ ❛❜❡rt❛ ❞♦s ♣♦♥✲

  1 ✳

  , · · · , M k ∈ X t♦s M

  ❉❡✜♥✐çã♦ ❈✳✶ ❯♠ ❛r❝♦ ♣♦❧✐❣♦♥❛❧ L ❡♠ X é ✉♠❛ ✉♥✐ã♦ n− [

  1 L = [M i , M i ] i +1 =1 i , M i [∩]M j , M j [= ∅

  ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ s❡❣♠❡♥t♦s t❛✐s q✉❡ ]M +1 +1 s❡♠♣r❡ q✉❡ i 6= j✳

  , · · · , M k ❖s ♣♦♥t♦s M i , M i ] 1 sã♦ ❝❤❛♠❛❞♦s ❞❡ ✈ért✐❝❡s ❞❛ ♣♦❧✐❣♦♥❛❧ ❡ ♦s s❡❣✲

  ♠❡♥t♦s [M +1 sã♦ ❝❤❛♠❛❞♦s ❞❡ ❛r❡st❛s ❞❛ ♣♦❧✐❣♦♥❛❧ L ♣❛r❛ ❝❛❞❛ i ∈ {1, · · · , n−1} ✳ ❉✐③❡♠♦s q✉❡ ♦ ❛r❝♦ ♣♦❧✐❣♦♥❛❧ é s✐♠♣❧❡s s❡ ♦s ✈ért✐❝❡s

  1 ❡ ♥❡ss❡

  = M k sã♦ ♣♦♥t♦s ❞✐st✐♥t♦s✳ ❖ ❛r❝♦ ♣♦❧✐❣♦♥❛❧ L é ❢❡❝❤❛❞♦ s❡ M

  6= M k ❝❛s♦ ❞❡✜♥✐♠♦s ❛ ❢r♦♥t❡✐r❛ ❝♦♠♦ ∂L = ∅✳ ❙❡ M

  1 ❡♥tã♦ ❞❡✜♥✐♠♦s

  ∂L = {M , M k }

  1 ✳ ❖ ♣♦♥t♦ M 1 é ❛ ♦r✐❣❡♠ ✭♦✉ ♣♦♥t♦ ✐♥✐❝✐❛❧✮ ❡ ♦ ♣♦♥t♦

  M k é ♦ ✜♥❛❧✳ i , M i +1 ]

  ❖❜s❡r✈❛çã♦ ❈✳✷ [M ❞❡✜♥❡ ✉♠❛ ♦r✐❡♥t❛çã♦✱ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ s❡t❛✿ // M i . M i +1

  ❉❡✜♥✐çã♦ ❈✳✸ ❯♠ ▲✐♥❦ L ❡♠ X é ✉♠❛ ✉♥✐ã♦ ✜♥✐t❛ ❞❡ m ∈ N ❛r❝♦s ♣♦❧✐❣♦♥❛✐s s✐♠♣❧❡s✱ ❢❡❝❤❛❞♦s ❡ ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s✳ ❖s ❛r❝♦s ❢❡❝❤❛❞♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞❡ L✳ ❖ ✐♥t❡✐r♦ m é ❝❤❛♠❛❞♦ ❞❡ ♦r❞❡♠ ❞♦ ▲✐♥❦✳ ❯♠ ❑♥♦t é ✉♠ ▲✐♥❦ ❞❡ ♦r❞❡♠ ✶✳

  ❆ ♦r✐❡♥t❛çã♦ ❞♦ ▲✐♥❦ é ❞❛❞❛ ♣❡❧❛ ✉♥✐ã♦ ❞❛s ♦r✐❡♥t❛çõ❡s ❞❡ ❝❛❞❛ ✉♠❛ ❞❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s✳ ❊①❡♠♣❧♦ ❈✳✹ ❯♠ ❡①❡♠♣❧♦✿

  ❋✐❣✉r❛ ❈✳✶✿ ▲✐♥❦ ❝♦♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❉❡✜♥✐çã♦ ❈✳✺ ✭✐✮ ❯♠❛ ✐s♦t♦♣✐❛ ❡♠ X é ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ♣♦r ♣❛rt❡s t h : [0, 1]×X → X (−) = h(t, −) : X → X t❛❧ q✉❡ ∀t ∈ [0, 1] t❡♠♦s q✉❡ h

  (−) = Id X (−) é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❝♦♠ h ✳

  ✭✐✐✮ ❉♦✐s ▲✐♥❦s L ❡ L sã♦ ❞✐t♦s ✐s♦tó♣✐❝♦s s❡ ❡①✐st✐r ✉♠❛ (L) = L

  ✐s♦t♦♣✐❛ h ❞❡ X q✉❡ ♣r❡s❡r✈❛ ♦r✐❡♥t❛çã♦ t❛❧ q✉❡ h ′ ′ 1 ✳ ❉❡♥♦t❛✲ ♠♦s L ∼ L ♣❛r❛ ❞❡s✐❣♥❛r q✉❡ L é ✐s♦tó♣✐❝♦ ❛ L ✳

  ■♥t✉✐t✐✈❛♠❡♥t❡ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ❛ ❢✉♥çã♦ s❡❥❛ ❧✐♥❡❛r ♣♦r ♣❛rt❡s ♥♦s ❞✐③ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ q✉❡ ❞❡❢♦r♠❛r❡♠♦s ♦ ❡s♣❛ç♦ ❞❡ ♠❛♥❡✐r❛ ❝♦♥tí♥✉❛ t❛❧ q✉❡ ♦ ♥♦ss♦ ▲✐♥❦ s❡❥❛ ❞❡❢♦r♠❛❞♦ s❡♠ ♣❡r❞❡r s✉❛ ❡str✉t✉r❛ ❞❡ ♣♦❧✐✲ ❣♦♥❛❧✳ ❆ ✐s♦t♦♣✐❛ ❞❡ ▲✐♥❦s é ✉♠❛ ❞❡❢♦r♠❛çã♦ q✉❡ s❡ ✐♥✐❝✐❛ ❝♦♠♦ ✉♠❛ ♣❛✐s❛❣❡♠ ❡stát✐❝❛ q✉❡ s❡ ❞❡❢♦r♠❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ❛té ❝❛rr❡❣❛r ✉♠ ▲✐♥❦ ❡♠ ♦✉tr♦ ▲✐♥❦✳ ▲❡♠❛ ❈✳✻ ❆ r❡❧❛çã♦ ❞❡ ✐s♦t♦♣✐❛ ∼ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ▲✐♥❦s✳ ❉❡♠♦♥str❛çã♦✿ ❘❡✢❡①✐✈❛✿ s❡❥❛ L ✉♠ ▲✐♥❦✳ ❊♥tã♦ ❞❡✜♥✐♠♦s

  → X h : [0, 1] × X 7→ t = Id X (t, x) x,

  ♦✉ s❡❥❛✱ h ♣❛r❛ t♦❞♦ t ∈ [0, 1]✳ ❊ss❛ ✐s♦t♦♣✐❛ tr✐✈✐❛❧✱ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ❞❡✐①❛r t♦❞♦ ♦ ❡s♣❛ç♦ ♣❛r❛❞♦✳ ◆❡ss❡ ❝❛s♦ L ∼ L✳ ′ ′

  ❙✐♠étr✐❝❛✿ s❡❥❛♠ L ❡ L ▲✐♥❦s t❛✐s q✉❡ L ∼ L ✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ (L) = L

  ✐s♦t♦♣✐❛ h : [0, 1] × X → X t❛❧ q✉❡ h

  1 ✳ ❈♦♠♦ ♣❛r❛ ❝❛❞❛

  t ∈ [0, 1] t ❛ ❛♣❧✐❝❛çã♦ h é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛

  ❛♣❧✐❝❛çã♦ g : [0, 1] × X →

  X

  1

  7→ h (t, x) (x) ′ − t

  1 ′ −

  1 1 (L ) = h (L ) = L t

  q✉❡ é ✉♠❛ ✐s♦t♦♣✐❛ t❛❧ q✉❡ g

  1 ✱ ♣♦✐s h é ✉♠ 1 ′

  = h = Id X ∼ L ❤♦♠❡♦♠♦r✜s♠♦ ♣❛r❛ ❝❛❞❛ t ∈ [0, 1] ❡ g ✳ ▲♦❣♦✱ L ✳ ′ ′′ ′ ′ ′′

  ∼ L ❚r❛♥s✐t✐✈❛✿ s❡❥❛♠ L✱ L ❡ L ▲✐♥❦s t❛✐s q✉❡ L ∼ L ❡ L ✳

  : [0, 1] × X → X ❊♥tã♦ ❡①✐st❡♠ ✐s♦t♦♣✐❛s h : [0, 1] × X → X ❡ h t❛✐s ′ ′ ′ ′′

  1 ❡ h ✳ ❉❡✜♥✐♠♦s

  1

  (L) = L (L ) = L q✉❡ h

  → g : [0, 1] × X X h(2t, x), s❡ 0 ≤ t ≤ 1/2

  (t, x) 7→ ′ h (2t − 1, h(1, x)), s❡ 1/2 ≤ t ≤ 1.

  ◆♦t❡ q✉❡ g ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣♦✐s ′ ′

  1

  h (2( ) − 1, h (x)) = h (h (x))

  1

  1

  2 = h (x). t =

  1 P❛r❛ ❝❛❞❛ t ∈ [0, 1] ❛ ❛♣❧✐❝❛çã♦ g é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❝♦♠ g

  h = Id X g (L) = h (h (L))

  1

  1 ′ ′

  1

  = h (L )

  1 ′′ = L .

  P♦rt❛♥t♦ L ∼ L ✳

  ❈✳✷ ❚❛♥❣❧❡s

  ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛ ✐r❡♠♦s r❡str✐♥❣✐r ❛ ♥♦ss❛ ❛♥á❧✐s❡ ♣❛r❛ ♦ s✉❜❡s♣❛ç♦

  2

  3

  × [0, 1] ❝♦♥✈❡①♦ X = R ❞❡ R ✳ ❉❡✜♥✐♠♦s ✉♠ ♦❜❥❡t♦ ❝❤❛♠❛❞♦ ❚❛♥❣❧❡ q✉❡ ♥♦s s❡r✈✐rá ♣❛r❛ ❝♦♥str✉✐r ♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s ♠❛✐s ❛❞✐❛♥t❡✳

  P❛r❛ ❝❛❞❛ n ∈ N ❞❡✜♥✐♠♦s [n] = {1, · · · , n} ❡ ♣❛r❛ n = 0 ❢❛③❡♠♦s [n] = ∅

  ✳ ❉❡✜♥✐çã♦ ❈✳✼ ❙❡❥❛♠ k, l ∈ N✳ ❯♠ ❚❛♥❣❧❡ ❞♦ t✐♣♦ (k, l) é ✉♠❛ ✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❛r❝♦s ♣♦❧✐❣♦♥❛✐s s✐♠♣❧❡s✱ ♦r✐❡♥t❛❞♦s ❡ ❞♦✐s ❛

  2

  × [0, 1] ❞♦✐s ❞✐s❥✉♥t♦s ❡♠ R t❛❧ q✉❡ ❛ ❢r♦♥t❡✐r❛ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❝♦♥✲ ❞✐çã♦✿

  2

  ∂L = L ∩ (R × {0, 1}) = ([k] × 0 × 0) ∪ ([l] × 0 × 1). P❡❧❛ ❝♦♥✈❡♥çã♦ ❢❡✐t❛ ❛❝✐♠❛✱ ♦s ▲✐♥❦s ♣♦❞❡♠ s❡r ✈✐st♦s ❝♦♠♦ ❚❛♥❣❧❡s

  ❞♦ t✐♣♦ (0, 0)✱ ♣♦✐s ♥❡ss❛s ❝♦♥❞✐çõ❡s ❛ ❢r♦♥t❡✐r❛ é ✈❛③✐❛✱ ❡ ♣♦rt❛♥t♦ t♦❞♦s ♦s ❛r❝♦s ♣♦❧✐❣♦♥❛✐s sã♦ ❢❡❝❤❛❞♦s✳ ❆ss✐♠ ❝♦♠♦ ✜③❡♠♦s ♣❛r❛ ▲✐♥❦s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ r❡❧❛çã♦ ❞❡ ✐s♦t♦♣✐❛ ❡♥tr❡ ❚❛♥❣❧❡s ❛♣❡♥❛s ❢❛③❡♥❞♦ ✉♠ ❛❥✉st❡✳ ❊①❡♠♣❧♦ ❈✳✽ ❯♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ ❚❛♥❣❧❡ ❞♦ t✐♣♦ (2, 4) ✈✐st♦ ♣♦r ❞♦✐s â♥❣✉❧♦s ❞✐st✐♥t♦s

  2

  × [0, 1] ❉❡✜♥✐çã♦ ❈✳✾ ✭✐✮ ❯♠❛ ✐s♦t♦♣✐❛ ❡♠ X = R é ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ♣♦r ♣❛rt❡s h : [0, 1] × X → X t❛❧ q✉❡ ∀t ∈ [0, 1] t❡♠♦s q✉❡ h t (−) : X → X

  é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ t❛❧ q✉❡ r❡str✐t♦ ❛ ❢r♦♥t❡✐r❛

  2

  ∂X = R × {0, 1} (−) = Id X (−) é ❛ ✐❞❡♥t✐❞❛❞❡ ❡ h ✳

  

  ✭✐✐✮ ❉♦✐s ❚❛♥❣❧❡s L ❡ L sã♦ ❞✐t♦s ✐s♦tó♣✐❝♦s s❡ ❡①✐st✐r ✉♠❛ (L) = L

  ✐s♦t♦♣✐❛ h ❞❡ X q✉❡ ♣r❡s❡r✈❛ ♦r✐❡♥t❛çã♦ t❛❧ q✉❡ h ′ ′ 1 ✳ ❉❡♥♦t❛✲ r❡♠♦s L ∼ L ♣❛r❛ ❞❡s✐❣♥❛r q✉❡ L é ✐s♦tó♣✐❝♦ ❛ L ✳ ❖❜s❡r✈❛çã♦ ❈✳✶✵ ◆♦t❡♠♦s q✉❡ ❛ r❡❧❛çã♦ ❞❡ ✐s♦t♦♣✐❛ s♦❜r❡ ♦s ❚❛♥❣❧❡s é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❆ ✈❡r✐✜❝❛çã♦ ❞❡ss❡ ❢❛t♦ é ❛♥á❧♦❣♦ ❛♦ ❝❛s♦ ❞♦s ▲✐♥❦s✱ ❝♦♠ ❛ r❡ss❛❧✈❛ ❞❡ q✉❡ ♥❛ ❢r♦♥t❡✐r❛ ❛ ✐s♦t♦♣✐❛ ♥ã♦ s❡ ♠♦✈❡✳ ❉❡✜♥✐çã♦ ❈✳✶✶ ❯♠❛ ♣r♦❥❡çã♦ θ ❞❡ ✉♠ ❚❛♥❣❧❡ ❞♦ t✐♣♦ (k, l) é ✉♠❛

  2

  ✉♥✐ã♦ ✜♥✐t❛ ❞❡ ❛r❝♦s ♣♦❧✐❣♦♥❛✐s ❡♠ R t❛❧ q✉❡ ♥❡♥❤✉♠ ✈ért✐❝❡ s❡ ❡♥✲ ❝♦♥tr❛ ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠❛ ❛r❡st❛ ❡ t❛❧ q✉❡ ❛ ❢r♦♥t❡✐r❛ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ∂θ = ([k] × 0) ∪ ([l] × 1).

  ❯♠ ♣♦♥t♦ ❞❡ ❝r✉③❛♠❡♥t♦ ❞❡ θ é ✉♠ ♣♦♥t♦ ❞❛ ♣r♦❥❡çã♦ q✉❡ s❡ ❡♥✲ ❝♦♥tr❛ ♥♦ ✐♥t❡r✐♦r ❞❡ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ❛r❡st❛s ❞✐st✐♥t❛s✳ ❆ ♦r❞❡♠ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❝r✉③❛♠❡♥t♦ é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞✐st✐♥t❛s às q✉❛✐s ♦ ♣♦♥t♦ ♣❡rt❡♥❝❡✳ ❯♠❛ ♣r♦❥❡çã♦ é ❞✐t❛ r❡❣✉❧❛r s❡ ❛ ♦r❞❡♠ ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❡ ❝r✉③❛♠❡♥t♦ é ❡①❛t❛♠❡♥t❡ ✷✳ ❉❛❞♦ ✉♠ ♣♦♥t♦ ❞❡ ❝r✉③❛♠❡♥t♦ P ❡♠ θ P

  ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ E ❢♦r♠❛❞♦ ♣❡❧❛s ❛r❡st❛s q✉❡ ❝♦♥té♠ P ✳ ❉❡✜♥✐çã♦ ❈✳✶✷ ❯♠ ❞✐❛❣r❛♠❛ ❞❡ ✉♠ ❚❛♥❣❧❡ ❞♦ t✐♣♦ (k, l) é ✉♠❛ ♣r♦✲ P ❥❡çã♦ r❡❣✉❧❛r ❡♠ R × [0, 1] t❛❧ q✉❡ ♦ ❝♦♥❥✉♥t♦ E é ♦r❞❡♥❛❞♦✱ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ ❞❡ ❝r✉③❛♠❡♥t♦ P ✱ ❛ ♣r✐♠❡✐r❛ ❛r❡st❛ ❝♦♠ r❡s♣❡✐t♦ ❛ ♦r❞❡♥❛çã♦ ❞✐③❡♠♦s q✉❡ ❡stá ♣♦r ❝✐♠❛✱ ❡ ❛ s❡❣✉♥❞❛ ❛r❡st❛ ❞✐③❡♠♦s q✉❡ ❡stá ♣♦r ❜❛✐①♦✳

  ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❞✐❛❣r❛♠❛✳ ❊①❡♠♣❧♦ ❈✳✶✸

  ◆♦ss♦ ♦❜❥❡t✐✈♦ é ❞❡✜♥✐r ✉♠❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♠♣♦s✐çã♦ ❡♥tr❡ ❛❧❣✉♥s ❚❛♥❣❧❡s✱ ❡ss❛ ♦♣❡r❛çã♦ é ❛ q✉❡ ✈❛✐ ❞❛r ♦r✐❣❡♠ ❛♦ ♣r♦❞✉t♦ ♥♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s✳ P❛r❛ ✐ss♦✱ s❡❥❛ L ✉♠ ❚❛♥❣❧❡ ❞♦ t✐♣♦ (k, l)✱ ❞❡✜♥✐♠♦s ❞✉❛s s❡q✉ê♥❝✐❛s s(L) ❡ b(L) ❝♦♥s✐st✐♥❞♦ ❞❡ s✐♥❛✐s + ❡ −✳ ❈❛s♦ k = 0 ❞❡✜✲ ♥✐♠♦s s(L) = ∅ ❡ s❡ l = 0 ❢❛③❡♠♦s b(L) = ∅✳ ❙✉♣♦♥❞♦ q✉❡ k ❡ l ♥ã♦

  , ..., ε k ) , ..., δ l ) i = + sã♦ ♥✉❧♦s ❞❡✜♥✐♠♦s s(L) = (ε i = + 1 ❡ b(L) = (δ 1 ♦♥❞❡ ε ✭r❡s♣✳ δ ✮ s❡ ♦ ♣♦♥t♦ (i, 0, 0) ❬r❡s♣✳ (i, 0, 1)❪ é ✉♠ ♣♦♥t♦ t❡r♠✐♥❛❧ i = − i = − ✭r❡s♣✳ ❞❡ ♦r✐❣❡♠✮ ❞❡ L ✳ ❈❛s♦ ❝♦♥trár✐♦✱ t❡♠♦s ε ❡ δ ✳

  ❈♦♥s✐❞❡r❡♠♦s ❞✉❛s ❛♣❧✐❝❛çõ❡s✿

  2

  2

  2

  2

  α : R × [0, 1] → R × [0, 1] α : R × [0, 1] → R × [0, 1]

  1 z z

  2

  ❡ +1 (p, z) 7→ (p, ) (p, z) 7→ (p, ).

′ ′

  2

  2

  ) ❙❡❥❛♠ L ❡ L ❞♦✐s ❚❛♥❣❧❡s ♦r✐❡♥t❛❞♦s t❛✐s q✉❡ b(L) = s(L ✳ ❊♥tã♦

  ❞❡✜♥✐♠♦s ′ ′ L ◦ L = α (L) ∪ α (L ).

  1 ′ ′ ′

  2

  ◦L) = s(L) ◦L) = b(L ) ❊ss❡ ♥♦✈♦ ❚❛♥❣❧❡ é ♦r✐❡♥t❛❞♦ ❝♦♠ s(L ❡ b(L ✱

  ❝❤❛♠❛♠♦s ❞❡ ❝♦♠♣♦s✐çã♦ ❞❡ L ❝♦♠ L✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ♣rát✐❝♦ ❞❡ss❛ ❝♦♠♣♦s✐çã♦✳ ❊①❡♠♣❧♦ ❈✳✶✹ ❈♦♥s✐❞❡r❡♠♦s ♦s ❚❛♥❣❧❡s✿

  ❈❤❛♠❡♠♦s ♦ ❚❛♥❣❧❡ à ❡sq✉❡r❞❛ ❞❡ L ❡ ♦ ❚❛♥❣❧❡ à ❞✐r❡✐t❛ ❞❡ G✳ ❊♥tã♦ ❛ ❝♦♠♣♦s✐çã♦ L ◦ G é ✈✐st❛ ♣♦r ❞♦✐s â♥❣✉❧♦s

  ♦✉ s❡❥❛✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❡ss❛ ❝♦♠♣♦s✐çã♦ é ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ❝♦❧❛❣❡♠ ❡♥tr❡ ♦s ❚❛♥❣❧❡s ❡♠ ✉♠❛ ❞❡rt❡r♠✐♥❛❞❛ ♦r❞❡♠✳

  ▲❡♠❛ ❈✳✶✺ ❙❡❥❛♠ L

  3

  ❡ g : [0, 1] × X → X t❛✐s q✉❡ h

  ✱ ❡①✐st❡♠ ✐s♦t♦♣✐❛s h : [0, 1]× X → X

  4

  ∼ L

  

2

  ❡ L

  ∼ L

  3

  1

  ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ L

  ∼ α 1 ::

  && f //

  α 1 1

  88

  1 (L 1 ) = L

  ❡ g

  ✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❆ ✐❞❡✐❛ ❞❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ♣❡❧♦ ✏❞✐❛❣r❛♠❛✑ ❛❜❛✐①♦

  2 α

  2

  ❡♠ q✉❡ X = R

  2 , 1

  1

  (g t (p, 2λ − 1)), s❡ λ ∈

  

2

  1

  1 (L 2 ) = L

  (h t (p, 2λ)), s❡ λ ∈ 0,

  

1

  α

  X (t, (p, λ)) 7→

  ✳ ❉❡✜♥✐♠♦s f : [0, 1] × X →

  4

  ∼ α 2 $$ α 2 1

  1 )

  1 ✱ L 2 ✱ L 3 ❡ L 4 ❚❛♥❣❧❡s ♦r✐❡♥t❛❞♦s ❝♦♠ b(L

  ) ✳ ❊♥tã♦

  ∼ L

  2

  3 ❡ L

  ∼ L

  1

  ✭✐✮ ❙❡ L

  4

  

2

  ) = s(L

  3

  ) ❡ b(L

  2

  ) = s(L

  1

  4 ❡♥tã♦ L

  ◦ L

  ◦ L

  ◦ L

  2

  ◦ (L

  3

  ∼ L

  1

  2 ) ◦ L

  3

  1

  ❡♥tã♦ (L

  2 ) = s(L 3 )

  ✭✐✐✮ ❙❡ b(L

  3 ❀

  ◦ L

  4

  ∼ L

  × [0, 1] ✳ ◆♦t❡♠♦s q✉❡ f ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s ♣❛r❛

  (t, (p,

  )) ∪ f

  1

  ◦ α

  1

  (h

  1

  = α

  (∗)

  )))

  2

  (L

  2

  (α

  1

  1

  (α

  (L

  1

  (α

  1

  = f

  1 ) = f 1 (α 1 (L 1 ) ∪ α 2 (L 2 ))

  ◦ L

  2

  1 (L

  = Id X ✳ ❋✐♥❛❧♠❡♥t❡ f

  , 1 ✳ P♦rt❛♥t♦✱ f

  2

  1

  = (p, λ) s❡ λ ∈

  1

  

1

  (g (p, 2λ − 1)) = α

  1

  ◦ L

  4

  = L

  1 (L 3 ) ∪ α

2 (L

4 )

  )) = α

  2

  (L

  1

  (g

  2

  )) ∪ α

  1

  (L

  (h

  (L

  1

  ))) = α

  2

  (L

  2

  (α

  2

  1

  ◦ α

  1

  (g

  2

  ))) ∪ α

  1

  2 (p, 2λ − 1)

  

2

  1

  2

  ×

  2

  ❡ r❡str✐t❛ ❛ R

  1

  1

  ◦ h ◦ α

  1

  é ❛ ❝♦♠♣♦s✐çã♦ α

  2

  1

  × 0,

  2

  (g t (p, 0)) ❡♠ ✭✯✮ ❡ ❡♠ ✭✯✯✮ ✉t✐❧✐③❛♠♦s q✉❡ ❛s ✐s♦t♦♣✐❛s h ❡ g sã♦ ❛ ✐❞❡♥t✐❞❛❞❡ ♥❛ ❢r♦♥t❡✐r❛ ❞♦ ❡s♣❛ç♦ X✳ ❖❜s❡r✈❛♠♦s q✉❡ f r❡str✐t❛ ❛ R

  = α

  2

  

(∗∗)

  (p, 0)

  2

  ) = α

  2

  1

  = (p,

  1 (p, 1)

  = α

  

(∗)

  (p, 1))

  1 (h t

  )) ∈ X t❡♠♦s α

  2

  1

  , 1 é ❛ ❝♦♠♣♦s✐çã♦

  ❡ f (p, λ) = α

  (g

  2

  1

  (p, 2λ) = (p, λ) s❡ λ ∈ 0,

  1

  (h (p, 2λ)) = α

  1

  f (p, λ) = α

  1 t ♦ sã♦✳ ❖❜s❡r✈❛♠♦s t❛♠❜é♠ q✉❡

  1 t ❡ g

  2 , 1 q✉❡ é ❝♦♥tí♥✉❛ ❥á q✉❡ h

  1

  (p, 2λ − 1)), s❡ λ ∈

  1

t

  2

  α

  2 α

  1

  (p, 2λ)), s❡ λ ∈ 0,

  1

t

  (h

  1

  α

  X (p, λ) 7→

  ψ t : X →

  ✳ ◆♦t❡♠♦s t❛♠❜é♠ q✉❡✱ ♣❛r❛ ❝❛❞❛ t ∈ [0, 1] ❛ ❛♣❧✐❝❛çã♦ f t é ✉♠ ❤♦♠❡♦♠♦✜s♠♦✱ ♣♦✐s é ❝♦♠♣♦st❛ ❞❡ ❝♦♥tí♥✉❛s ❡ ♣♦rt❛♥t♦ ❝♦♥tí♥✉❛✱ ❝♦♠ ✐♥✈❡rs❛

  2

  1

  ◦ g ◦ α

  2

  3

  2

  1

  (L ) ⊆ R × 0, (L ) ⊆ ❡♠ ✭✯✮ ✉t✐❧✐③❛♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ α

  1 1 ❡ α

  2

  2

  2

  1

2 R × , 1 ◦ L ∼ L ◦ L

  ✳ P♦rt❛♥t♦✱ L

  2

  1

  

4

3 ✳

  2

  ◦ L ) ◦ L ∼ L ◦ (L ◦ L ) ✭✐✐✮ P❛r❛ ♣r♦✈❛r q✉❡ (L

  3

  2

  1

  3

  2 1 ❞❡✜♥✐♠♦s ❛ s❡❣✉✐♥t❡

  ✐s♦t♦♣✐❛ h : [0, 1] × X →

  X (t, (p, λ)) 7→ (p, φ t (λ)),

  ❡♠ q✉❡ φ t : [0, 1] → [0, 1]  λ(1 − ) t

  1 ✱ s❡ 0 ≤ λ ≤

  2

  2 t

  1

  3 λ 7→

  λ − ≤ λ ≤ ✱ s❡

  4

  2

  4

  3 (1 + t)λ − t ≤ λ ≤ 1. ✱ s❡ t

  4 ❋✐①❛❞♦ ✉♠ t ∈ [0, 1] ❛ ❢✉♥çã♦ φ é ✐♥✈❡rtí✈❡❧ ❡ s✉❛ ✐♥✈❡rs❛ é ❛ ❢✉♥çã♦

  ψ t : [0, 1] → [0, 1] 2λ 1 t  2 − t  ✱ s❡ 0 ≤ λ ≤

  2

  4 t 1 t 3 − t

  λ 7→ λ + − ≤ λ ≤ ✱ s❡  λ + t 3 − t

  4

  2

  4

  4 ≤ λ ≤ 1

  ✱ s❡ t 1 + t

  4 ❛ss✐♠ h é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ♣❛r❛ ❝❛❞❛ t ∈ [0, 1]✱ ❥á q✉❡ é ❝♦♥t✐♥✉❛

  1

  (p, λ) = (p, ψ t (λ)) ❡ t❡♠ ✐♥✈❡rs❛✱ ❛ s❛❜❡r h t q✉❡ t❛♠❜é♠ é ❝♦♥tí♥✉❛✳

  (p, λ) = (p, φ (λ)) = (p, λ) ◆♦t❡♠♦s q✉❡ h ✳

  ◦ L ◦ (L ◦ L

  1 ((L

  3 2 ) ◦ L 1 ) = L

  3

  2 1 )

  ▼♦str❡♠♦s q✉❡ h ✳ Pr✐♠❡✐r♦ ♦❜s❡r✲

  2

  × [0, 1] ✈❛♠♦s q✉❡ ❞❛❞♦ (p, λ) ∈ R ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s

  λ h (α (p, λ)) = h (p, )

  1

  1

  1

  2

  (∗) λ

  = (p, )

  4 = α (α (p, λ)). λ

  1 1 ✭❈✳✶✮

  

1

  ∈ 0,

  ❊♠ ✭✯✮ ✉t✐❧✐③❛♠♦s q✉❡ ♣❛r❛ λ ∈ [0, 1]✳

  2

  

2

  λ h (α (α (p, λ))) = h (α (p, ))

  1

  2

  1

  1

  2

  2

  • 2
  • 3

  (L

  1

  (α

  1

  (L

  1

  )) C.

  1,C.2 ❡ C.3

  = α

  2

  

3

  2

  ) ∪ α

  1

  (α

  2

  (L

  2

  )) ∪ α

  1

  (α

  ))) ∪ h

  (L

  (L

  (α

  1

  (L

  2

  ))) ∪ α

  1

  (L

  1

  )) = h

  1

  

2

  1

  (α

  2

  (L

  3

  ))) ∪ h

  1

  (α

  2

  (α

  1

  1

  2

  (L) ✱

  1

  , ..., ε n ) ✱ ❡♥tã♦

  ❞❡✜♥✐♠♦s Id s

  

(L) ❝♦♠♦ s❡♥❞♦ ♦ t❛♥❣❧❡ ❞❛❞♦ ♣♦r {1, ..., n} × {0} × [0, 1]

  ❝♦♠ s(Id s

  (L)

  ) = b(Id s

  (L)

  ) = s(L) ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥✐♠♦s Id b

  ❝♦♠ ❛ r❡ss❛❧✈❛ q✉❡ s(Id b

  ) ✳

  (L)

  ) = b(Id b

  (L)

  ) = b(L) ✳

  ▲❡♠❛ ❈✳✶✻ ❙❡ L é ✉♠ ❚❛♥❣❧❡ t❛❧ q✉❡ s(L) = (ε

  1

  , ..., ε n ) ❡♥tã♦ (Id b (L)

  ◦ L) ∼ L

  ❡ L ∼ (L ◦ Id s (L) )

  ❉❡✜♥✐♠♦s ✉♠ t❛♥❣❧❡ q✉❡ s❡rá ❛ ✐❞❡♥t✐❞❛❞❡ ❝♦♠ r❡s♣❡✐t♦ ❛ ❡ss❛ ♦♣❡✲ r❛çã♦ ❞❡ ❝♦♠♣♦s✐çã♦✳ ❙❡❥❛ L ✉♠ t❛♥❣❧❡ t❛❧ q✉❡ s(L) = (ε

  1

  )) = α

  2

  2 (L 3 ) ∪ α 1 (α 2 (L 2 ) ∪ α 1 (L 1 ))

  = α

  2 (L 3 ) ∪ α 1 (L

  2

  ◦ L

  1 )

  = L

  3

  ◦ (L

  ◦ L 1 ). ▲♦❣♦✱ (L

  ◦ L

  3

  ◦ L

  2

  ) ◦ L

  1

  ∼ L

  3

  ◦ (L

  2

  (α

  ))) ∪ (α

  = h

  2 ))

  (α

  2

  (α

  2

  (p, λ))) = h

  

1

  (α

  2

  (p, λ + 1

  = h

  ♣❛r❛ λ ∈ [0, 1]✳ h

  

1 (p,

  λ + 3

  4 )

  (∗)

  = (p, λ + 3

  2 − 1)

  = (p, λ + 1

  2 )

  = α

  1

  4

  ✭❈✳✸✮ ❊♠ ✭✯✮ ✉t✐❧✐③❛♠♦s q✉❡ λ

  4 )

  

1

  (p, λ + 2

  4 )

  (∗)

  = (p, λ + 2

  4 −

  1

  4 )

  = (p, λ + 1

  = α

  3

  

1

  (α

  2 (p, λ)).

  ✭❈✳✷✮ ❊♠ ✭✯✮ ✉t✐❧✐③❛♠♦s q✉❡ λ

  4

  ∈

  1

  2

  ,

  

2

(p, λ).

  4

  3

  )) ∪ α

  

2

  (α

  2

  (L

  3

  ) ∪ α

  1

  (L

  2

  1

  1

  (L

  1

  )) = h

  1

  ((α

  

2

  (α

  2

  (L

  (α

  )) = h

  ∈

  1

  3

  4

  , 1 ♣❛r❛ λ ∈ [0, 1]✳ ❆ss✐♠✱ t❡♠♦s q✉❡ h

  1

  ((L

  3

  ◦ L

  2

  ) ◦ L

  ) = h

  1

  1

  (α

  

2

  (L

  3

  ◦ L

  2

  ) ∪ α

  1

  (L

  ✳

  ❚❛❧ ❧❡♠❛ é ❡♥❝♦♥tr❛❞♦ ❡♠ ✭❬✶✷❪✱ ▲❡♠♠❛ ❳✳✺✳✶✶✱ ♣✳ ✷✻✷✮✳ ◆ã♦ ✐r❡♠♦s ❞❡♠♦♥str❛r ❞❡✈✐❞♦ ❛♦s ❛r❣✉♠❡♥t♦s ❛❧t❛♠❡♥t❡ ❣❡♦♠étr✐❝♦s q✉❡ ♥ã♦ ❡stã♦ ♥♦ ❡s❝♦♣♦ ❞❡st❡ tr❛❜❛❧❤♦✳

  ❈✳✸ ❚r❛♥ç❛s

  ■r❡♠♦s ♥♦s ❛t❡r ❛ ✉♠ t✐♣♦ ❡s♣❡❝✐❛❧ ❞❡ ❚❛♥❣❧❡ ❝❤❛♠❛❞♦ ❞❡ ❜r❛✐❞ q✉❡ ❛q✉✐ ✐r❡♠♦s ♥♦s r❡❢❡r✐r ❝♦♠♦ tr❛♥ç❛✳ ❋✐①❡♠♦s 1 ≤ n✳ ❉❡✜♥✐çã♦ ❈✳✶✼ ❯♠❛ tr❛♥ç❛ L ❝♦♠ n ✜♦s é ✉♠ ❚❛♥❣❧❡ ❞♦ t✐♣♦ (n, n) t❛❧ q✉❡ ✭✐✮ s(L) = b(L) = (+, +, ..., +)❀ ✭✐✐✮ L ♥ã♦ ❝♦♥té♠ ❛r❝♦s ♣♦❧✐❣♦♥❛✐s ❢❡❝❤❛❞♦s❀

  2

  × {z}) ✭✐✐✐✮ P❛r❛ ❝❛❞❛ z ∈ [0, 1] ✈❛♠♦s t❡r q✉❡ L ∩ (R t❡♠ ❡①❛t❛♠❡♥t❡ n

  ♣♦♥t♦s ❞✐st✐♥t♦s✳ ❈❤❛♠❛♠♦s ✉♠❛ tr❛♥ç❛s L ❝♦♠ n ✜♦s s✐♠♣❧❡s♠❡♥t❡ ❞❡ tr❛♥ç❛✱ ✜✲

  ❝❛♥❞♦ s✉❜❡♥t❡♥❞✐❞♦ q✉❡ t❡♠♦s ✉♠ ♥❛t✉r❛❧ n ✜①♦✳ ❆ r❡❧❛çã♦ ❞❡ ✐s♦t♦♣✐❛ ❡♥tr❡ ❚❛♥❣❧❡s ✐♥❞✉③ s♦❜r❡ ❛s tr❛♥ç❛s ✉♠❛ r❡❧❛çã♦ ❞❡ ✐s♦t♦♣✐❛✳

  ❆ ♦♣❡r❛çã♦ ❞❡ ❝♦♠♣♦s✐çã♦ ❞❡✜♥✐❞❛ ❡♥tr❡ ❚❛♥❣❧❡s s❡♠♣r❡ s❡rá ♣♦s✲ n = sí✈❡❧ ♥♦ ❝❛s♦ ❞❛s tr❛♥ç❛s ❡ ♣❛r❛ q✉❛❧q✉❡r tr❛♥ç❛ ♦ ❡❧❡♠❡♥t♦ 1 {1, ..., n} × {0} × [0, 1] n ) = b(1 n ) =

  ❝♦♠ ❛ ♦r✐❡♥t❛çã♦ ❞❛❞❛ ♣♦r s(1 (+, ..., +) s b

  é ♦ ❡❧❡♠❡♥t♦ Id (L) ❡ Id (L) ♣❛r❛ q✉❛❧q✉❡r tr❛♥ç❛ L✳

  1

  ❉❛❞❛ ✉♠❛ tr❛♥ç❛ L ❞❡✜♥✐♠♦s ✉♠❛ ♥♦✈❛ tr❛♥ç❛✱ ❞❡♥♦t❛❞❛ ♣♦r L

  1

  2

  × { } q✉❡ é ❛ ✐♠❛❣❡♠ ❞❡ L ♣❡❧❛ r❡✢❡①ã♦ ❛tr❛✈és ❞♦ ♣❧❛♥♦ R ✱ ♦✉ s❡❥❛✱

  2

1 L = h(L)

  ❡♠ q✉❡✿

  2

  2 R

  h : × [0, 1] → R × [0, 1] (p, z) 7→ (p, 1 − z). ❊①❡♠♣❧♦ ❈✳✶✽ ◆♦ ❡①❡♠♣❧♦ C.14 ♦ ❚❛♥❣❧❡ L é ✉♠❛ tr❛♥ç❛ ❡ t❡♠ ✉♠ ✐♥✈❡rs♦ q✉❡ é r❡♣r❡s❡♥t❛❞♦ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♣♦r

  −

  1

  ◆♦t❡♠♦s q✉❡ s❡ ✜③❡r♠♦s ❛ ❝♦♠♣♦s✐çã♦ L ◦ L ♥❡ss❡ ❡①❡♠♣❧♦ ♦❜t❡✲ ♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦

  1

  − 1 −

  ) ∼ 1 n ∼ (L ◦ L) ❖❜s❡r✈❛çã♦ ❈✳✶✾ ❖❜s❡r✈❛♠♦s q✉❡ (L ◦ L ✳ ❉❡ ❢❛t♦✱ s❡ L é ✉♠❛ tr❛♥ç❛ t❛❧ q✉❡ ♦ ♣♦♥t♦ (i, 0, 1) é ❧❡✈❛❞♦ ♣♦r ✉♠ ❛r❝♦

  1

  ♣♦❧✐❣♦♥❛❧ ♥♦ ♣♦♥t♦ (j, 0, 0) ❡♥tã♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ h ❛ r❡✢❡①ã♦ L ❧❡✈❛ ♦ ♣♦♥t♦ (j, 0, 1) ♣♦r ✉♠ ❛r❝♦ ♣♦❧✐❣♦♥❛❧ ♥♦ ♣♦♥t♦ (i, 0, 0) ❡ ❛ ❝♦♠♣♦s✐çã♦ ✈❛✐ t❡r ✉♠❛ ♥♦✈❛ ♣♦❧✐❣♦♥❛❧ ❞❡ (i, 0, 1) ❡♠ (i, 0, 0)✳ ■ss♦ ♦❝♦rr❡ ❝♦♠ t♦❞❛s ❛s ♣♦❧✐❣♦♥❛✐s✱ ❡ ♣♦rt❛♥t♦ ❛ ❝♦♠♣♦s✐çã♦ é ✐s♦tó♣✐❝❛ à ✐❞❡♥t✐❞❛❞❡✳

  ❆❣♦r❛ q✉❡ t❡♠♦s t♦❞♦s ♦s ✐♥❣r❡❞✐❡♥t❡s✱ ♣♦❞❡♠♦s ❢❛③❡r ❛ ❝♦♥str✉çã♦ n ❞♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s✳ P❛r❛ ❝❛❞❛ n ∈ N ❞❡♥♦t❛♠♦s ♣♦r B ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝❧❛ss❡s ❞❡ tr❛♥ç❛s ✈✐❛ ✐s♦t♦♣✐❛✳ ❉❡♥♦t❛♠♦s ♣♦r [L] ❛ ❝❧❛ss❡ ❞❛ tr❛♥ç❛ L✳ n , ◦) Pr♦♣♦s✐çã♦ ❈✳✷✵ ❖ ♣❛r (B é ✉♠ ❣r✉♣♦✱ ❡♠ q✉❡ ◦ é ♦ ♣r♦❞✉t♦ ′ ′ ′

  ] = [L ◦ L ] ❞❡✜♥✐❞♦ ♣♦r [L] ◦ [L ✱ ❡♠ q✉❡ L ❡ L sã♦ tr❛♥ç❛s q✉❛✐sq✉❡r✳ ❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❛♠♦s q✉❡ ❡ss❛ ♦♣❡r❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ′ ′

  ]) ]) ∈ B n × B n ♣♦✐s ❞❛❞♦s ([L], [L ✱ ([G], [G t❡♠♦s ′ ′ ′ ′

  ⇔ ([L], [L ]) = ([G], [G ]) [L] = [G] ] = [G ]

  ❡ [L ′ ′ ⇔ ∼ G

  L ∼ G ❡ L

  ▲❡♠❛ C.15 ′ ′

  ⇒ (L ◦ L ) ∼ (G ◦ G ) ′ ′

  ⇔ [L ◦ L ] = [G ◦ G ] ′ ′

  ⇔ ([L] ◦ [L ]) = ([G] ◦ [G ]). ❱❡r✐✜q✉❡♠♦s ♦s ❛①✐♦♠❛s ❞❡ ❣r✉♣♦✳

  1 ]

2 ]

3 ] ∈ B n

  ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ s❡❥❛♠ [L ✱[L ❡ [L ✳ ❊♥tã♦✱ ([L

  1 ] ◦ [L 2 ]) ◦ [L

3 ] = [L

1 ◦ L 2 ] ◦ [L 3 ]

  ◦ L = [(L

  1 2 ) ◦ L 3 ]

▲❡♠❛ C.15

  = [L

  1 ◦ (L 2 ◦ L 3 )]

  = [L

  1 ] ◦ ([L 2 ] ◦ [L 3 ])

  

n ]

  ❊❧❡♠❡♥t♦ ♥❡✉tr♦✿ ♦ ❡❧❡♠❡♥t♦ [1 é ❛ ✉♥✐❞❛❞❡ ♥❡ss❛ ♦♣❡r❛çã♦✳ ❉❡ ❢❛t♦✱

  [L] ◦ [1 n ] = [L ◦ 1 n ] [1 n ] ◦ [L] = [1 n ◦ L]

  

▲❡♠❛ C.16 ❡ ▲❡♠❛ C.16

  = [L] = [L] n ♣❛r❛ q✉❛❧q✉❡r [L] ∈ B ✳

  1

  ] ■♥✈❡rs♦✿ s❡❥❛ [L] ✉♠❛ tr❛♥ç❛✳ ❖ ❡❧❡♠❡♥t♦ [L é s❡✉ ✐♥✈❡rs♦ ❝♦♠ r❡s♣❡✐t♦ ❛ ❡ss❡ ♣r♦❞✉t♦✳ ❉❡ ❢❛t♦✱ − − − −

  1

  1

  1

  1

  [L] ◦ [L ] = [L ◦ L ] [L ] ◦ [L] = [L ◦ L]

  ❖❜s✳ C.19 ❡ ❖❜s✳ C.19 − − = [1 n ] = [1 n ].

  1

  1

  = [L ] P♦rt❛♥t♦✱ [L] ✳ n ❖ ❣r✉♣♦ B é ❝❤❛♠❛❞♦ ❞❡ ❣r✉♣♦ ❞❡ tr❛♥ç❛s✳ ❉❡♥♦t❛♠♦s ♦ ♣r♦❞✉t♦

  [L] ◦ [G] s✐♠♣❧❡s♠❡♥t❡ ♣♦r L ◦ G ♦✉ LG✱ ✜❝❛♥❞♦ s✉❜❡♥t❡♥❞✐❞♦ q✉❡ sã♦ ❝❧❛ss❡s✳

  1 , · · · , σ n−

  1

  ❉❡✜♥✐♠♦s ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ❡s♣❡❝✐❛✐s✱ ❞❡♥♦t❛❞♦s ♣♦r σ ✳ i = ([(i, 0, 1); ( , − , )]∪[( , − , ); (i+1, 0, 0)]) 2i+1

  1 1 2i+1

  1

  1

  ❙❡ ❝❤❛♠❛r♠♦s M i = [(i + 1, 0, 1); (i, 0, 0)]

  2

  2

  2

  2

  2

  2

  ❡ N ✱ ❢❛③❡♠♦s σ i = [(1, 0, 1), (1, 0, 0)] ∪ · · · ∪ M i ∪ N i ∪ · · · ∪ [(n, 0, 1), (n, 0, 0)] i ∈ B n

  ❈❧❛r♦ q✉❡ σ ♣❛r❛ t♦❞♦ i ∈ {1, · · · , n − 1}✳ ❚❛✐s ❡❧❡♠❡♥t♦s sã♦ ✐♠♣♦rt❛♥t❡s ♣♦r ❝❛r❛❝t❡r✐③❛r❡♠ ♦ ❣r✉♣♦ ❞❡ tr❛♥ç❛s ✈✐❛ r❡❧❛çõ❡s s♦❜r❡ ❣❡r❛❞♦r❡s✳ ❊ss❡ ❡❧❡♠❡♥t♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ❞✐❛❣r❛♠❛ Pr♦♣♦s✐çã♦ ❈✳✷✶ ❙ã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s n , · · · , σ n− ✭✐✮ ❖ ❣r✉♣♦ B é ❣❡r❛❞♦ ♣♦r σ

  1 1 ✳

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

  ❚❛♠❜é♠ é ✈á❧✐❞♦ q✉❡ σ i σ i σ i = σ i σ i σ i

  • 1 +1 +1

  ♣❛r❛ t♦❞♦ i ∈ {1, · · · , n − 1}✳ ❆s ❞✉❛s r❡❧❛çõ❡s ❞❡s❝r✐t❛s ♥♦ ✐t❡♠ ✭✐✐✮ sã♦ ❝❤❛♠❛❞❛s r❡❧❛çõ❡s ❞♦

  ❣r✉♣♦ ❞❡ tr❛♥ç❛s✳ ❚❛❧ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ✭❬✶✷❪✱▲❡♠♠❛ ❳✳✻✳✹✱ ♣✳

  ✷✻✹✮✳ ❆ ❞❡♠♦♥str❛çã♦ ❡♥✈♦❧✈❡ ❛r❣✉♠❡♥t♦s ❣❡♦♠étr✐❝♦s q✉❡ ♥ã♦ ❡stã♦ ♥♦ ❡s❝♦♣♦ ❞❡st❡ tr❛❜❛❧❤♦✳

  ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

  ❬✶❪ ❆❲❖❉❊❨✱ ❙✳ ❈❛t❡❣♦r② ❚❤❡♦r②✱ ❖①❢♦r❞✱ ✷✺✻♣✳ ✭✷✵✵✻✮✳ ❬✷❪ ❉˘A❙❈˘A▲❊❙❈❯✱ ❙✳❀ ◆˘A❙❚˘A❙❊❙❈❯✱ ❈✳❀ ❘❆■❆◆❯✱ ❙✳✳ ❍♦♣❢ ❆❧✲

  ❣❡❜r❛s✿ ❆♥ ■♥tr♦❞✉❝t✐♦♥✱ ◆❡✇ ❨♦r❦✿ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ✹✵✶♣✳ ✭✷✵✵✶✮✳

  ❬✸❪ ❊■▲❊◆❇❊❘●✱ ❙✳ ❡ ▼❆❈▲❆◆❊✱ ❙✳ ◆❛t✉r❛❧ ■s♦♠♦r♣❤✐s♠s ✐♥ ●r♦✉♣ ❚❤❡♦r②✱ Pr♦❝✳ ◆❛t✳ ❆❝❛✳ ❙❝✐✳✱ ♣♣✳ ✺✸✼✲✺✹✸ ✭✶✾✹✷✮✳

  ❬✹❪ ❊■▲❊◆❇❊❘●✱ ❙✳ ❡ ▼❆❈▲❆◆❊✱ ❙✳ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ◆❛t✉r❛❧ ❊q✉✐✈❛❧❡♥❝❡s✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱♦❧✳ ✺✽✱ ◆♦✳ ✷ ✱ ♣♣✳ ✷✸✶✲✷✾✹ ✭✶✾✹✺✮✳

  ❬✺❪ ❊❚■◆●❖❋✱ P✳❀ ●❊▲❆❑■✱ ❙✳❀ ◆■❑❙❍❨❈❍✱ ❉✳❀ ❖❙❚❘■❑✱ ❱✳✳ ❚❡♥✲ s♦r ❈❛t❡❣♦r✐❡s✱ ▲❡❝t✉r❡ ♥♦t❡s ❢♦r t❤❡ ❝♦✉rs❡ ✶✽✳✼✻✾ ✏❚❡♥s♦r ❝❛✲ t❡❣♦r✐❡✑✱ ▼■❚✱ ✭✷✵✵✾✮✳

  ❬✻❪ ❊❚■◆●❖❋✱ P✳❀ ●❖▲❇❊❘●✱ ❖✳❀ ❍❊◆❙❊▲✱ ❙✳❀ ▲■❯✱ ❚✳❀ ❙❈❍❲❊◆❉◆❊❘✱ ❆✳❀ ❱❆■◆❚❘❖❇✱ ❉✳❀ ❨❯❉❖❱■◆❆✱ ❊✳✳ ■♥tr♦✲ ❞✉❝t✐♦♥ t♦ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦✲ ❝✐❡t②✱ ✭✷✵✶✶✮✳

  ❬✼❪ ❋❘❊❨❉✱ P✳ ❏✳✳ ❆❜❡❧✐❛♥ ❈❛t❡❣♦r✐❡s ❘❡♣r✐♥ts ✐♥ ❚❤❡♦r② ❛♥❞ ❆♣✲ ♣❧✐❝❛t✐♦♥s ♦❢ ❈❛t❡❣♦r✐❡s✱ ◆♦✳ ✸✱ ✭✷✵✵✸✮✳

  ❬✽❪ ❍■▲❚❖◆✱ P✳❏✳❀ ❙❚❆▼▼❇❆❈❍✱❯✳✳ ❆ ❈♦✉rs❡ ✐♥ ❍♦♠♦❧♦❣✐❝❛❧ ❆❧❣❡❜r❛✱ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✭✶✾✾✼✮✳

  ❬✾❪ ❍❯◆●❊❘❋❖❘❉✱ ❚✳ ❲✳✳ ❆❧❣❡❜r❛✱ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✱ ✺✵✷♣✳ ✭✷✵✵✵✮✳

  ❬✶✵❪ ❏❆❈❖❇❙❖◆✱ ◆✳✳ ❇❛s✐❝ ➪❧❣❡❜r❛ ■■✱ ◆❡✇ ❨♦r❦✿ ❲✳❍✳ ❋❘❊❊✲ ▼❆◆ ❆◆❉ ❈❖▼P❆◆❨✱ ✻✽✻♣✳ ✭✶✾✽✾✮✳

  ❬✶✶❪ ❑❆◆✱ ❉✳ ❆❞❥♦✐♥t ❋✉♥❝t♦rs✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱♦❧✳ ✽✼✱ ◆♦✳ ✷ ✱ ♣♣✳ ✷✾✹✲✸✷✾ ✭✶✾✺✽✮✳

  ❬✶✷❪ ❑❆❙❙❊▲✱ ❈✳✳ ◗✉❛♥t✉♠ ●r♦✉♣s✱ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✺✸✶♣✳ ✭✶✾✾✺✮✳

  ❬✶✸❪ ▼❆❈▲❆◆❊✱ ❙✳✳ ❉✉❛❧✐t② ❢♦r ●r♦✉♣s✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱♦❧✳ ✺✻✱ ◆♦✳ ✻ ✱ ♣♣✳ ✹✽✺✲✺✶✻ ✭✶✾✺✵✮✳

  ❬✶✹❪ ▼❆❈▲❆◆❊✱ ❙✳✳ ❈❛t❡❣♦r✐❡s ❢♦r t❤❡ ❲♦r❦✐♥❣ ▼❛t❤❡♠❛t✐❝✐❛♥✱ ❙♣r✐♥❣❡r✱ ✭✶✾✼✶✮✳

  ❬✶✺❪ ▼❆❈▲❆◆❊✱ ❙✳✳ ❈♦♥❝❡♣ts ❛♥❞ ❝❛t❡❣♦r✐❡s ✐♥ ♣❡rs♣❡❝t✐✈❡ ❆▼❙ ❍✐st♦r② ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✶✿ ❆ ❈❡♥t✉r② ♦❢ ▼❛t❤❡♠❛t✐❝s ✐♥ ❆♠❡r✐❝❛✱ P❛rt ■✱ ♣♣✳ ✸✷✸✲✸✻✺ ✭✶✾✽✽✮✳

  ❬✶✻❪ ▼❖▼❇❊▲▲■✱ ❏✳ ▼✳ ❯♥❛ ✐♥tr♦❞✉❝✐ó♥ ❛ ❧❛s ❝❛t❡❣♦rí❛s t❡♥s♦✲ r✐❛❧❡s ② s✉s r❡♣r❡s❡♥t❛❝✐♦♥❡s✱ ◆♦t❛s ❞❡ ❛✉❧❛✱ ✽✵♣✳ ❬✶✼❪ ❙❊❘❘❊✱ ❏✳ P✳ ▲✐❡ ❆❧❣❡❜r❛s ❛♥❞ ▲✐❡ ●r♦✉♣s✱ ▲❡❝t✉r❡ ♥♦t❡s✱

  ❍❛✈❛r❞✱ ✭✶✾✻✹✮✳

Novo documento

Tags

Documento similar

Universidade Federal de Santa Catarina Programa de Pós-Graduação em Engenharia e Gestão do Conhecimento – EGC
0
10
368
Universidade Federal de Santa Catarina Programa de Pós-Graduação em Engenharia e Gestão do Conhecimento
0
0
243
Uiversidade Federal de Santa Catarina Programa de Pós-Graduação em Engenharia e Gestão do Conhecimento
0
0
340
Universidade Federal de Santa Catarina Centro Tecnológico Departamento de Engenharia Civil
0
4
207
Universidade Federal de Santa Catarina
0
6
183
Universidade Federal de Santa Catarina UFSC Curso de Ciências Contábeis Maiara Camargo da Cruz
0
0
99
Universidade Federal de Santa Catarina – UFSC Rodrigo Garcia Silveiro
0
0
43
Universidade Federal de Santa Catarina – UFSC Centro Sócio Econômico Curso de Graduação em Ciências Econômicas
0
1
81
Universidade Federal de Santa Catarina – UFSC Centro Sócio Econômico Departamento de Ciências Econômicas Curso de Graduação em Ciências Econômicas
0
0
43
Universidade Federal de Santa Catarina – UFSC Centro Sócio Econômico Departamento de Ciências Econômicas Curso de Graduação em Ciências Econômicas
0
0
165
Universidade Federal de Santa Catarina
0
0
21
Universidade Federal de Santa Catarina Centro de Comunicação e Expressão UFSC
0
0
154
Universidade Federal de Santa Catarina Curso de Graduação em Odontologia
0
0
75
Universidade Federal de Santa Catarina Curso de Graduação em Odontologia
0
0
80
Universidade Federal de Santa Catarina Centro Sócio-Econômico Departamento de Serviço Social
0
0
88
Show more