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

Livre

0
0
137
1 year ago
Preview
Full text

  

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

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  

❈❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❡ ♦

❚❡♦r❡♠❛ ❞❡ ▼❛❝ ▲❛♥❡ ♣❛r❛ ❛

❝♦♥❞✐çã♦ ❡str✐t❛

  

●❛❜r✐❡❧ ❙❛♠✉❡❧ ❞❡ ❆♥❞r❛❞❡

❖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❛ ❡ ❆♣❧✐❝❛❞❛

  

❈❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ▼❛❝

▲❛♥❡ ♣❛r❛ ❛ ❝♦♥❞✐çã♦ ❡str✐t❛

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

  ❈♦♥❝❡♥tr❛çã♦ ❡♠ ➪❧❣❡❜r❛✳ ●❛❜r✐❡❧ ❙❛♠✉❡❧ ❞❡ ❆♥❞r❛❞❡

  ❋❧♦r✐❛♥ó♣♦❧✐s ▼❛rç♦ ❞❡ ✷✵✶✻

  

❈❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ▼❛❝

▲❛♥❡ ♣❛r❛ ❛ ❝♦♥❞✐çã♦ ❡str✐t❛

  ♣♦r

  ●❛❜r✐❡❧ ❙❛♠✉❡❧ ❞❡ ❊st❛ ❉✐ss❡rt❛çã♦ ❢♦✐ ❥✉❧❣❛❞❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ✏▼❡str❡✑✱

  ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ➪❧❣❡❜r❛✱ ❡ ❛♣r♦✈❛❞❛ ❡♠ s✉❛ ❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✳

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

  ❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛ Pr♦❢✳➟ ❉r❛✳ ❱✐r❣í♥✐❛ ❙✐❧✈❛ ❘♦❞r✐❣✉❡s

  ✭❖r✐❡♥t❛❞♦r❛ ✲ ❯❋❙❈✮ ❆❜❞❡❧♠♦✉❜✐♥❡ ❆♠❛r ❍❡♥♥✐

  ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮ ▲✉③ ❆❞r✐❛♥❛ ▼❡❥í❛ ❈❛st❛ñ♦

  ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮ ❘❡❣✐♥❛ ▼❛r✐❛ ❞❡ ❆q✉✐♥♦

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

  ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✲ ❯❋❙❈✮ ❋❧♦r✐❛♥ó♣♦❧✐s✱ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻✳

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

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

  ❘❡s✉♠♦

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

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

  ❆❜str❛❝t

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

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

  ❙✉♠ár✐♦

   ✶

   ✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

   ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

   ✻✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

   ✾✽ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹

  ■♥tr♦❞✉çã♦

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

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

  C (X, Y ) ❣♦r✐❛ C✱ ♦ ❝♦♥❥✉♥t♦ Hom ❞❡ ♠♦r✜s♠♦s ❞❡ X ❡♠ Y ❢♦r♠❛ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳

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

  ❯♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é✱ ❜❛s✐❝❛♠❡♥t❡✱ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠✉♥✐❞❛ ❞❡ ✉♠ ❢✉♥t♦r ⊗ ❡ ✉♠ ♦❜❥❡t♦ 1 t❛✐s q✉❡ ♦s ♦❜❥❡t♦s (X ⊗Y )⊗Z✱ X ⊗(Y ⊗Z) ❡ 1⊗X✱ X✱ X ⊗1 ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ♣♦r ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛✐s✳ ◗✉❛♥❞♦ t❛✐s ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s sã♦ ❛s r❡s♣❡❝t✐✈❛s ✐❞❡♥t✐❞❛❞❡s✱ ❞✐③❡♠♦s q✉❡ ❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ❡str✐t❛✳ ❖ ❚❡♦r❡♠❛ ❞❡ ▼❛❝ ♣❛r❛ ❛ ❝♦♥❞✐çã♦ ❡str✐t❛ ❛✜r♠❛ q✉❡ ♣♦❞❡♠♦s✱ ❡♠ ✉♠ ❝❡rt♦ s❡♥t✐❞♦✱ ❝♦♥s✐❞❡r❛r q✉❛✐sq✉❡r ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❝♦♠♦ ❡str✐t❛s✳

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

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

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

  ❛ss❡❣✉r❛✲s❡ q✉❡ ❛❧❣✉♠❛s ❝❛t❡❣♦r✐❛s s❡❥❛♠ ✏♠❛✐s ❡str✐t❛s✑ ❞♦ q✉❡ ❛ ❝❛✲ ω t❡❣♦r✐❛ ❛♣r❡s❡♥t❛❞❛ ♥♦ ❡①❡♠♣❧♦ ❞❛❞♦✱ ❛ s❛❜❡r✱ V ec G ✳ ❊st✉❞❛♠♦s ❡ss❛ ❝❛t❡❣♦r✐❛ ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✱ ♣♦ré♠ ❛ ❞❡♥♦t❛♠♦s ♣♦r C(G, ω)✳

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

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

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

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

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

  ❈❛♣ít✉❧♦ ✶ Pré✲r❡q✉✐s✐t♦s

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

  ✶✳✶ ❈❛t❡❣♦r✐❛s

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

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

  C (X, X) ✭✐✐✐✮ ♣❛r❛ q✉❛❧q✉❡r ♦❜❥❡t♦ X ❡♠ Ob(C)✱ ✉♠ ♠♦r✜s♠♦ id X ❡♠ Hom ✱ ❝❤❛♠❛❞♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ X❀ ✭✐✈✮ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ♦❜❥❡t♦s ❡♠ Ob(C)✱ ✉♠❛ ❢✉♥çã♦

  Hom C (X, Y ) × Hom C (Y, Z) → Hom C (X, Z) (f, g) 7→ g ◦ f

  ❝❤❛♠❛❞❛ ❝♦♠♣♦s✐çã♦✱ q✉❡ s❛t✐s❢❛③ ♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s✿ ✭❛✮ ♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦s X ❡ Y ❡♠ Ob(C)✱ ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ id

  X C (X, X) ❡♠ Hom s❛t✐s❢❛③ f ◦ id = f ◦ g = g,

  X ❡ id

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

  C (X, Y ) ✭❜✮ ❞❛❞♦s ♦❜❥❡t♦s X, Y, Z, W ❡♠ Ob(C) ❡ ♠♦r✜s♠♦s f ❡♠ Hom ✱ g C (Y, Z) C (Z, W )

  ❡♠ Hom ✱ h ❡♠ Hom ✱ ❛ ❝♦♠♣♦s✐çã♦ é ❛ss♦❝✐❛t✐✈❛✱ ♦✉ s❡❥❛✱ h ◦ (g ◦ f ) = (h ◦ g) ◦ f. ❆ r❡❢❡rê♥❝✐❛ é ❜ás✐❝❛ ♥♦ ❡st✉❞♦ ❞❛ ❚❡♦r✐❛ ❞❡ ❈❛t❡❣♦r✐❛s✳ ◆❡ss❛ r❡❢❡rê♥❝✐❛ sã♦ ❛♣r❡s❡♥t❛❞❛s três ❞❡✜♥✐çõ❡s ❞❡ ❝❛t❡❣♦r✐❛✱ ❛ q✉❡ ❛♣r❡s❡♥✲ t❛♠♦s✱ ✉♠❛ q✉❡ ❡♥✈♦❧✈❡ ❛♣❡♥❛s ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♠♦r✜s♠♦s ❡ ✉♠❛ ♦✉tr❛ q✉❡ ❝♦♥s✐❞❡r❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠♦ ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ ❝♦♠ ❞✉❛s ❢✉♥çõ❡s✱ ✐❞❡♥t✐❞❛❞❡ ❡ ❝♦♠♣♦s✐çã♦✳ ❊s❝♦❧❤❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❛❞❛ ♣♦r s❡r ❛ ♠❛✐s ❝♦♠✉♠ ❡ ❝ô♠♦❞❛ ♣❛r❛ tr❛❜❛❧❤❛r✳

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

  ❆♣❡s❛r ❞❛s ❝♦❧❡çõ❡s ❡♥✈♦❧✈✐❞❛s ❡♠ ❝❛t❡❣♦r✐❛s ♥ã♦ s❡r❡♠ s❡♠♣r❡ ❝♦♥❥✉♥t♦s✱ ✈❛♠♦s ✉s❛r ♦s sí♠❜♦❧♦s ❡ t❡r♠♦s ❥á ❝♦♥❤❡❝✐❞♦s✱ ❝♦♠♦ ∈✱ ⊆✱ ✏❢✉♥çã♦✑✱ ✏❛♣❧✐❝❛çã♦✑✱ ♣❛r❛ r❡❧❛❝✐♦♥❛r ❝♦❧❡çõ❡s ❡ s❡✉s ❡❧❡♠❡♥t♦s✳ ❙❛✲ ❜❡♥❞♦ ❞✐ss♦✱ ✜①❛♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ♥♦t❛çõ❡s✳

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

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

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

  ✳ ❆♣❡s❛r ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ s❡r ❝♦♥st✐t✉í❞❛ ♣♦r ♦❜❥❡t♦s✱ ♠♦r✜s♠♦s✱

  ♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡ ❡ ✉♠❛ ❝♦♠♣♦s✐çã♦✱ ❣❡r❛❧♠❡♥t❡ s❡ ❛♣r❡s❡♥t❛♠ ❛♣❡♥❛s ♦s ♦❜❥❡t♦s ❡ ♠♦r✜s♠♦s✱ ✜❝❛♥❞♦ s✉❜❡♥t❡♥❞✐❞♦s ♦s ♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡ ❡ ❛ ❝♦♠♣♦s✐çã♦✳ ➱ ❞❡ss❛ ❢♦r♠❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛ ♠❛✐♦r✐❛ ❞♦s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r✳ ❊①❡♠♣❧♦ ✶✳✶✳✷ ❆ ❝❛t❡❣♦r✐❛ Set é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❝♦♥❥✉♥✲ t♦s ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s ❢✉♥çõ❡s✳

  ❊①❡♠♣❧♦ ✶✳✶✳✸ ❆ ❝❛t❡❣♦r✐❛ Rel é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❝♦♥❥✉♥✲ t♦s ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s r❡❧❛çõ❡s✳ ▲❡♠❜r❛♥❞♦✱ ♣❛r❛ X, Y ❝♦♥❥✉♥t♦s✱ ✉♠❛ r❡❧❛çã♦ R ❡♥tr❡ X ❡ Y é

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

  ❝♦♠♦ ❛ s❡♥❞♦ ❛ r❡❧❛çã♦ S◦R := {(x, z) ∈ X×Z : ❡①✐st❡ y ∈ Y t❛❧ q✉❡ (x, y) ∈ R, (y, z) ∈ S}.

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

  ❊①❡♠♣❧♦ ✶✳✶✳✹ ❆ ❝❛t❡❣♦r✐❛ Grp é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❣r✉♣♦s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s✳ ❊①❡♠♣❧♦ ✶✳✶✳✺ ❆ ❝❛t❡❣♦r✐❛ Ab é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s✳ ❊①❡♠♣❧♦ ✶✳✶✳✻ ❆ ❝❛t❡❣♦r✐❛ Div é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❣r✉♣♦s ❞✐✈✐sí✈❡✐s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦s✳

  ▲❡♠❜r❛♥❞♦✱ s❡ G é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ G é ❞✐✈✐sí✈❡❧ s❡✱ ♣❛r❛ t♦❞♦ x ∈ G ❡ t♦❞♦ ✐♥t❡✐r♦ ♥ã♦✲♥✉❧♦ n✱ ❡①✐st❡ y ∈ G t❛❧ q✉❡ x = ny✳ ❆❧✲ ❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❣r✉♣♦s ❞✐✈✐sí✈❡✐s sã♦ Q✱ Q/Z✱ ♦ ❣r✉♣♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦

  ∗ ∞ )

  ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s C ❡ ♦ ❣r✉♣♦ ❞❡ Prü❢❡r Z(p ✳ ❊①❡♠♣❧♦ ✶✳✶✳✼ ❆ ❝❛t❡❣♦r✐❛ Ring é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❛♥é✐s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❛♥é✐s✳

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

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

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

  R ♠❡♥t❡ M ✮ ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ R

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

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

  ❉❡♥♦t❛♠♦s ♣♦r vect ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ ♦s ♠♦r✜s♠♦s sã♦ ❛s tr❛♥s❢♦r♠❛çõ❡s k✲ ❧✐♥❡❛r❡s✳ k ❊①❡♠♣❧♦ ✶✳✶✳✶✵ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r Alg ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ❛s k✲á❧❣❡❜r❛s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s✳

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

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

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

  ❆ ❝❛t❡❣♦r✐❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ m ✮ é ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮✳

  P❛r❛ ♦s ❞♦✐s ♣ró①✐♠♦s ❡①❡♠♣❧♦s✱ ❧❡♠❜r❛♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❡ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❡ ❞❡ ❜✐á❧❣❡❜r❛✳ ❉❡✜♥✐çã♦ ✶✳✶✳✶✷ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❯♠❛ k✲á❧❣❡❜r❛ ❞❡ ▲✐❡ é ✉♠ ♣❛r (L, [−, −])

  ✱ ❡♠ q✉❡ L é ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ [−, −] : L⊗L → L é ✉♠❛ ❛♣❧✐❝❛çã♦ k✲❧✐♥❡❛r✱ ❝❤❛♠❛❞❛ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✱ q✉❡ s❛t✐s❢❛③ às s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✭✐✮ [x, x] = 0✱ ♣❛r❛ t♦❞♦ x ∈ L❀ ✭✐✐✮ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0✱ ♣❛r❛ q✉❛✐sq✉❡r x, y, z ∈ L✳

  ❊ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ■❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✳ k

  ❊①❡♠♣❧♦ ✶✳✶✳✶✸ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r Lie ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ❛s k✲á❧❣❡❜r❛s ❞❡ ▲✐❡ ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✲ ✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s ❞❡ ▲✐❡✱ ♦✉ s❡❥❛✱ ❛♣❧✐❝❛çõ❡s k✲❧✐♥❡❛r❡s q✉❡ ♣r❡s❡r✈❛♠ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✳ ❉❡✜♥✐çã♦ ✶✳✶✳✶✹ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❯♠❛ k✲❜✐á❧❣❡❜r❛ é ✉♠❛ q✉í♥t✉♣❧❛ (H, M, µ, ∆, ε)

  ✱ ❡♠ q✉❡ (H, M, µ) é ✉♠❛ k✲á❧❣❡❜r❛✱ (H, ∆, ε) é ✉♠❛ k✲ ❝♦á❧❣❡❜r❛ ❡ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮ M ❡ µ sã♦ ❤♦♠♦♠♦r✜s♠♦s ❞❡ k✲❝♦á❧❣❡❜r❛s❀ ✭✐✐✮ ∆ ❡ ε sã♦ ❤♦♠♦♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s✳ k

  ❊①❡♠♣❧♦ ✶✳✶✳✶✺ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r Bialg ❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ❛s k✲❜✐á❧❣❡❜r❛s ❡ ♦s ♠♦r✜s♠♦s sã♦ ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡ k✲❜✐á❧❣❡❜r❛s✱ ♦✉ s❡❥❛✱ ❛♣❧✐❝❛çõ❡s k✲❧✐♥❡❛r❡s q✉❡ sã♦ ❤♦♠♦♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s ❡ k✲❝♦á❧❣❡❜r❛s✳

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

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

  ∗ = 1 A ♣r♦❞✉t♦ ❞❡ A ❡ id ✳ ❉❡✜♥✐çã♦ ✶✳✶✳✶✾ ❙❡❥❛♠ C✱ D ❝❛t❡❣♦r✐❛s✱ D é ❞✐t❛ ✉♠❛ s✉❜❝❛t❡❣♦r✐❛

  D (X, Y ) ⊆ Hom C (X, Y ) ❞❡ C s❡ Ob(D) ⊆ Ob(C)✱ Hom ✱ ♣❛r❛ q✉❛✐sq✉❡r

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

  ❡♠ C✳ D (X, Y ) =

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

  ✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ D✳ ❊①❡♠♣❧♦ ✶✳✶✳✷✶ ◆♦t❡♠♦s q✉❡✱ ♣❛r❛ X✱Y ❝♦♥❥✉♥t♦s ❡ f : X → Y ❢✉♥çã♦✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r f ❝♦♠♦ ❛ r❡❧❛çã♦ {(x, f (x)) : x ∈ X} ⊆ X × Y.

  ❉❡ss❛ ❢♦r♠❛✱ Set é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❞❡ Rel✱ ♠❛s ♥ã♦ é ✉♠❛ s✉❜✲ ❝❛t❡❣♦r✐❛ ♣❧❡♥❛✱ ♣♦✐s ♥❡♠ t♦❞❛ r❡❧❛çã♦ é ✉♠❛ ❢✉♥çã♦✳ ❊①❡♠♣❧♦ ✶✳✶✳✷✷ ❆ ❝❛t❡❣♦r✐❛ Div é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❡ Ab✱ q✉❡ ♣♦r s✉❛ ✈❡③ é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❡ Grp✳ ❊①❡♠♣❧♦ ✶✳✶✳✷✸ ❆ ❝❛t❡❣♦r✐❛ ring é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❞❡ Ring q✉❡ ♥ã♦ é ♣❧❡♥❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ R ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s f : R → R × R r 7→ (r, 0).

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

  ❊①❡♠♣❧♦ ✶✳✶✳✷✹ P❛r❛ k ✉♠ ❝♦r♣♦✱ vect é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣❧❡♥❛ ❞❡ m V ect k A

  ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣❛r❛ A ✉♠❛ k✲á❧❣❡❜r❛✱ é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ M A

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

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

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

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

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

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

  ) = Ob(C) ✭✐✮ Ob(C ❀ op

  ✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C ✱

  op

  Hom C (X, Y ) = Hom C (Y, X);

  op op

  C (X, Y ) C (Y, Z) ✭✐✐✐✮ ♣❛r❛ ♠♦r✜s♠♦s f ∈ Hom ✱ g ∈ Hom ✱ ❛ ❝♦♠♣♦✲ s✐çã♦ é ❞❛❞❛ ♣♦r op g ◦ f = f ◦ g. op op

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

  ♣❛r❛ ❡st✉❞❛r ❞✉❛❧✐❞❛❞❡ ❡ ❞❡✜♥✐r ❢✉♥t♦r❡s ❝♦♥tr❛✈❛r✐❛♥t❡s✳ ◆♦ ❡st✉❞♦ ❞❡ ❝❛t❡❣♦r✐❛s✱ ❝❛❞❛ ❝♦♥❝❡✐t♦ é ❛❝♦♠♣❛♥❤❛❞♦ ❞♦ s❡✉ ❝♦♥❝❡✐t♦ ❞✉❛❧✱ ♦❜t✐❞♦ ✏✐♥✈❡rt❡♥❞♦ ❛s ✢❡❝❤❛s✑ ♥❛ ❞❡✜♥✐çã♦ ❞♦ ❝♦♥❝❡✐t♦ ♦r✐❣✐♥❛❧✳ ■ss♦ ✈❛✐ ✜❝❛r ♠❛✐s ❝❧❛r♦ ♥❛s s❡çõ❡s s❡❣✉✐♥t❡s✳ P❛r❛ ♠❛✐s ✐♥❢♦r♠❛çõ❡s✱ ♦ ❧❡✐t♦r ♣♦❞❡ ♣❡sq✉✐s❛r ❡♠ s❡❝t✐♦♥ ✸✳✶✮✳ ❉❡✜♥✐çã♦ ✶✳✶✳✷✾ ❙❡❥❛♠ C✱ D ❝❛t❡❣♦r✐❛s✳ ❉❡♥♦t❛♠♦s ♣♦r C × D ❛ ❝❛✲ t❡❣♦r✐❛ ♣r♦❞✉t♦ ❞❡ C ❡ D✱ ❞❡✜♥✐❞❛ ❝♦♠♦ s❡❣✉❡✿ ✭✐✮ Ob(C × D) = Ob(C) × Ob(D)❀

  ′ ′ , Y ) ∈ C × D

  ✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r (X, Y ), (X ✱ ′ ′ ′ ′

  Hom C ((X, Y ), (X , Y )) = Hom C (X, X ) × Hom D (Y, Y ); ×D

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

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

  ′ ′ ′ ′ C (X, X ) × Hom D (Y, Y ) , g ) ∈

  ✭✐✈✮ ♣❛r❛ ♠♦r✜s♠♦s (f, g) ∈ Hom ✱ (f ′ ′′ ′ ′′

  Hom C (X , X ) × Hom D (Y , Y ) ✱ ❛ ❝♦♠♣♦s✐çã♦ é ❞❛❞❛ ♣♦r

  ′ ′ ′ ′ (f , g ) ◦ (f, g) = (f ◦ f, g ◦ g).

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

  

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

✜s♠♦s

  ❋✉♥çõ❡s ✐♥❥❡t♦r❛s ❡ s♦❜r❡❥❡t♦r❛s ♣♦❞❡♠ s❡r ❞❡✜♥✐❞❛s ❡♠ t❡r♠♦s ❞❡ ❡❧❡♠❡♥t♦s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ é ❛♣r❡s❡♥t❛❞❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳ ❉❡✜♥✐çã♦ ✶✳✷✳✶ ❙❡❥❛♠ X, Y ❝♦♥❥✉♥t♦s✳ ❯♠❛ ❢✉♥çã♦ f : X → Y é ✭✐✮ ✐♥❥❡t♦r❛ s❡✱ ♣❛r❛ x, y ∈ X t❛✐s q✉❡ f(x) = f(y)✱ ❡♥tã♦ x = y❀ ✭✐✐✮ s♦❜r❡❥❡t♦r❛ s❡ ♣❛r❛ ❝❛❞❛ y ∈ Y ✱ ❡①✐st❡ x ∈ X t❛❧ q✉❡ f(x) = y✳

  ❊♠ ✉♠❛ ❝❛t❡❣♦r✐❛ q✉❛❧q✉❡r✱ ♥❡♠ s❡♠♣r❡ ♦s ♦❜❥❡t♦s sã♦ ❝♦♥❥✉♥t♦s✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r ♥ã♦ ❢❛r✐❛ s❡♥t✐❞♦ ♣❛r❛ ♠♦r✜s♠♦s✳ P♦r ❡ss❛ r❛③ã♦✱ ♥❛ t❡♥t❛t✐✈❛ ❞❡ ❣❡♥❡r❛❧✐③❛r ❡ss❡s ❝♦♥❝❡✐t♦s✱ é ♣r❡❝✐s♦ ❡♥t❡♥❞❡r ❝♦♠♦ ❛s ❢✉♥çõ❡s ✐♥❥❡t♦r❛s ❡ s♦❜r❡❥❡t♦r❛s s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ❛s ♦✉tr❛s ❢✉♥çõ❡s✳ ❯♠ r❡s✉❧t❛❞♦ ❝♦♥❤❡❝✐❞♦ é ❛♣r❡s❡♥t❛❞♦ ♣❡❧❛ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡✳ Pr♦♣♦s✐çã♦ ✶✳✷✳✷ ❙❡❥❛♠ X, Y ❝♦♥❥✉♥t♦s✳ ❯♠❛ ❢✉♥çã♦ f : X → Y é ✭✐✮ ✐♥❥❡t♦r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ g : Y → X t❛❧ q✉❡ g ◦ f = id

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

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

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

  ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ❝❛t❡❣♦r✐❛ ✳ ❙❡❥❛♠ M ∈ ✱ N ✉♠ R✲ s✉❜♠ó❞✉❧♦ ❞❡ M ❡ f : N → M ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✳ ❊♥tã♦✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❡①✐st❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s g : M → N t❛❧ q✉❡ g ◦ f = id N s❡✱ ❡ s♦♠❡♥t❡ s❡✱ N é ✉♠ s♦♠❛♥❞♦ ❞✐r❡t♦ ❞❡ M✳ P♦r ❡ss❛ r❛③ã♦✱ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❣❡r❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❛❞❛ é ❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s

  ❛❣♦r❛✳

  ❉❡✜♥✐çã♦ ✶✳✷✳✸ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❊♥tã♦✱ ♦ ♠♦r✜s♠♦ f é ❞✐t♦ ✉♠ ✭✐✮ ♠♦♥♦♠♦r✜s♠♦ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♠♦r✜s♠♦s g, h : Z → X t❛✐s q✉❡ f ◦ g = f ◦ h

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

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

  X ❡ f ◦ g = id Y ✳

  ❆❧é♠ ❞✐ss♦✱ X ❡ Y sã♦ ❞✐t♦s ✐s♦♠♦r❢♦s✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r X ≃ Y ✱ s❡ ❡①✐st✐r ✉♠ ✐s♦♠♦r✜s♠♦ f : X → Y ✳

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

  ♠♦s ❞❡ C ✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ✐s♦♠♦r✜s♠♦ é ❛✉t♦✲❞✉❛❧✳ ❚❛♠❜é♠✱ t♦❞♦ ✐s♦♠♦r✜s♠♦ é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡ ✉♠ ❡♣✐♠♦r✜s♠♦✳ ❆ r❡❝í♣r♦❝❛ ♥❡♠ s❡♠♣r❡ é ✈❡r❞❛❞❡✐r❛✱ ❝♦♠♦ s❡rá ♠♦str❛❞♦ ♥♦s ❡①❡♠♣❧♦s✳ ❆q✉✐✱ ❢❛③❡✲ ♠♦s ♦ ❝♦♠❡♥tár✐♦ ❞❡ q✉❡ q✉❛♥❞♦ C é ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✱ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ❛ r❡❝í♣r♦❝❛ é ✈❡r❞❛❞❡✐r❛✳

  ◆❛ ❝❛t❡❣♦r✐❛ Set✱ ❢✉♥çõ❡s ✐♥❥❡t♦r❛s ✭s♦❜r❡❥❡t♦r❛s✮ sã♦ ❡①❛t❛♠❡♥t❡ ♦s ♠♦♥♦♠♦r✜s♠♦s ✭❡♣✐♠♦r✜s♠♦s✮✳ ❊ss❡s ❢❛t♦s s❡❣✉❡♠ ❞✐r❡t❛♠❡♥t❡ ❞❛ Pr♦♣♦s✐çã♦ ❆❜❛✐①♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ♠♦♥♦♠♦r✜s♠♦s ♥ã♦ ✐♥✲ ❥❡t♦r❡s ❡ ❡♣✐♠♦r✜s♠♦s ♥ã♦ s♦❜r❡❥❡t♦r❡s✳ ❊①❡♠♣❧♦ ✶✳✷✳✹ ❊♠ Div✱ ❛ ♣r♦❥❡çã♦ π : Q → Q/Z é ✉♠ ♠♦♥♦♠♦r✲ ✜s♠♦ ♥ã♦ ✐♥❥❡t♦r✳

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

  ♣❡rt❡♥❝❡♠ à ♠❡s♠❛ ❝❧❛ss❡ ❡♠ Q/Z✱ ♦✉ s❡❥❛✱ g(x)−h(x) ∈ Z✱ ♣❛r❛ t♦❞♦ x ∈ G✳ ❈❤❛♠❛♠♦s k : G → Q ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ❞❛❞♦ ♣♦r k = g − h✳

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

  (k(x) + 1)k(y) ✳ ❊♥tã♦ k(x)

  0 ≤ = k(y) < 1. k(x) + 1

  ❈♦♠♦ k(y) ∈ Z✱ s❡❣✉❡ q✉❡ k(y) = 0✳ ❈♦♠♦ x = ny✱ t❡♠♦s k(x) = 0✳ ❙❡ k(x) < 0✱ ✉s❛♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦✱ ❝❤❡❣❛♠♦s à ❞❡s✐❣✉❛❧❞❛❞❡ k(x)

  0 < = k(y) < 1, k(x) + 1

  ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s k(y) ∈ Z✳ ■ss♦ ✐♠♣❧✐❝❛ k = 0✱ ❧♦❣♦ g = h ❡ π é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ ❈❧❛r❛♠❡♥t❡✱ π ♥ã♦ é ✐♥❥❡t♦r✳ ❊①❡♠♣❧♦ ✶✳✷✳✺ ❊♠ Ring✱ ❛ ✐♥❝❧✉sã♦ i : Z → Q é ✉♠ ❡♣✐♠♦r✜s♠♦ ♥ã♦ s♦❜r❡❥❡t♦r✳

  = h(n)g(m −1 m)h(m

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

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

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

  ❝♦♥tí♥✉❛✱ ❧♦❣♦ f ♥ã♦ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ T op✳ ❆❜❛✐①♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s ❡♠ q✉❡ ♠♦♥♦♠♦r✜s♠♦s sã♦

  ✱ t ∈ [0, 2π) ✱ é ✉♠❛ ❜✐❥❡çã♦ ❝♦♥tí♥✉❛✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ✐♥✈❡rs❛ ❞❡ f ♥ã♦ é

  1 ❞❛❞❛ ♣♦r f(t) = e it

  P♦rt❛♥t♦✱ g = h ❡ i é ✉♠ ❡♣✐♠♦r✜s♠♦ q✉❡ ♥ã♦ é s♦❜r❡❥❡t♦r✳ ❊①❡♠♣❧♦ ✶✳✷✳✻ P♦❞❡ ❛✐♥❞❛ ❛❝♦♥t❡❝❡r ❞❡ ✉♠❛ ❜✐❥❡çã♦ ♥ã♦ s❡r ✉♠ ✐s♦✲ ♠♦r✜s♠♦✳ ❊♠ T op✱ ❛ ❢✉♥çã♦ f : [0, 2π) → S

  −1 ) = h(q).

  ) = h(nm

  = h(n1m −1

  −1 )

  ) = h(n)h(1)h(m

  = h(n)g(1)h(m −1

  −1 )

  −1 )

  ❉❡ ❢❛t♦✱ s❡❥❛♠ R ✉♠ ❛♥❡❧ ❡ g, h : Q → R ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❛♥é✐s t❛✐s q✉❡ g ◦ i = h ◦ i✳ ■ss♦ q✉❡r ❞✐③❡r q✉❡ g ❡ h ❝♦✐♥❝✐❞❡♠ ♥♦s ✐♥t❡✐r♦s✱ ♦✉ s❡❥❛✱ g(n) = h(n)✱ ♣❛r❛ t♦❞♦ n ∈ Z✳

  −1 )g(m)h(m

  ) = h(n)g(m

  )h(m)h(m −1

  = h(n)g(m −1

  −1 )

  −1 )h(mm

  )h(1) = h(n)g(m

  = h(n)g(m −1

  −1 )g(1)

  1) = g(n)g(m

  = g(nm −1

  −1 )

  ✱ ♣❛r❛ n, m ∈ Z✱ m 6= 0 ✳ ❊♥tã♦ g(q) = g(nm

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

  G , ♣❛r❛ t♦❞♦ x ∈ K,

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

  G } ✐♠♣❧✐❝❛ K = {e ✳ ▲♦❣♦✱ f é ✐♥❥❡t♦r✳

  Pr♦✈❛s ❛♥á❧♦❣❛s ❛♦ ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ♠♦str❛♠ q✉❡ ♦s ♠♦♥♦♠♦r✲ M R

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

  ❱❛♠♦s ♠♦str❛r ❛ ❝♦♥tr❛♣♦s✐t✐✈❛✱ ♦✉ s❡❥❛✱ ✉♠ ♠♦r✜s♠♦ ♥ã♦ ✐♥❥❡t♦r ❡♠ ring ♥ã♦ é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ ❙❡❥❛♠ f : R → S ✉♠ ♠♦r✜s♠♦ ❡♠ ring ❡ r, s ∈ R✱ r 6= s✱ t❛✐s q✉❡ f(r) = f(s)✳ ❙❡ R[x] é ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s s♦❜r❡ R✱ ❡①✐st❡♠ g, h : R[x] → R ♠♦r✜s♠♦s ❡♠ ring t❛✐s q✉❡ g(x) = r ❡ h(x) = s✳ P♦rt❛♥t♦✱ f ◦ g = f ◦ h✱ ♠❛s g 6= h✱ ♦ q✉❡ ♣r♦✈❛ q✉❡ f ♥ã♦ é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳

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

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

  N → N/f (M ) ❞❛❞❛s ♣♦r g(n) = n + f (M )

  ❡ h(n) = 0 + f(M), ♣❛r❛ t♦❞♦ n ∈ N, ♦✉ s❡❥❛✱ ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛ ❡ ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦ g ◦ f = h ◦ f✳ ❈♦♠♦ f é ✉♠ ❡♣✐♠♦r✜s♠♦✱ t❡♠♦s g = h ❡ ✐ss♦ ✐♠♣❧✐❝❛ f(M) = N✳ ▲♦❣♦✱ f é s♦❜r❡❥❡t♦r✳

  ❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♠♦str❛ q✉❡ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦♥♦♠♦r✜s♠♦s ❡ ❡♣✐♠♦r✜s♠♦s é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡ ✉♠ ❡♣✐♠♦r✜s♠♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ′ : Y → Z

  Pr♦♣♦s✐çã♦ ✶✳✷✳✶✵ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ f : X → Y ✱ f ♠♦r✜s♠♦s ❡♠ C✳

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

  ′ ◦ f

  ♠❡♥t❡ s❡✱ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ ′

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

  ◦ f s❡✱ f é ✉♠ ❡♣✐♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ✭⇒✮ ❙❡❥❛♠ g, h : W → X ♠♦r✜s♠♦s ❡♠ C t❛✐s q✉❡

  ′ ′ ′ ◦ f ◦ g = f ◦ f ◦ h f

  ✳ ❈♦♠♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦s f ◦g = f ◦h✳ ❈♦♠♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦s g = h✳

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

  ◦ f ◦ g = f ◦ f ◦ h ◦ f tã♦ f ❡ ❝♦♠♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦s g = h✳ op

  ✭✐✐✮ ❙❡❣✉❡ ❞♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳

  ✶✳✸ ❋✉♥t♦r❡s

  ❆♦ ❞❡✜♥✐r♠♦s ✉♠❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛✱ é ♥❛t✉r❛❧ q✉❡ s❡❥❛♠ ❞❡✜♥✐✲ ❞♦s t❛♠❜é♠ ♦s ♠♦r✜s♠♦s q✉❡ ♣r❡s❡r✈❛♠ t❛❧ ❡str✉t✉r❛✳ ◆♦ ♥♦ss♦ ❝❛s♦✱ ❞❡✜♥✐♠♦s ❛❣♦r❛ ♦s ✏♠♦r✜s♠♦s✑ ❡♥tr❡ ❝❛t❡❣♦r✐❛s✱ ❝❤❛♠❛❞♦s ❢✉♥t♦r❡s✳ ❉❡✜♥✐çã♦ ✶✳✸✳✶ ❙❡❥❛♠ C✱ D ❝❛t❡❣♦r✐❛s✳ ❯♠ ❢✉♥t♦r ❡♥tr❡ C ❡ D✱ ❞❡✲ ♥♦t❛❞♦ ♣♦r F : C → D✱ ❝♦♥s✐st❡ ❞❡ ❞✉❛s ❛♣❧✐❝❛çõ❡s✿ ✭✐✮ ✉♠❛ ❛♣❧✐❝❛çã♦ F : Ob(C) → Ob(D) q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ♦❜❥❡t♦ X ∈ C ❛ ✉♠ ♦❜❥❡t♦ F (X) ∈ D❀

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

  ❡♠ D t❛❧ q✉❡ F (id ) = id

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

  ♣❛r❛ X ∈ C ❡ f, g ♠♦r✜s♠♦s ❡♠ C t❛❧ q✉❡ ❛ ❝♦♠♣♦s✐çã♦ g ◦ f ❡①✐st❛✳ ❖❜s❡r✈❛çã♦ ✶✳✸✳✷ ❖ ❢✉♥t♦r ❞❡✜♥✐❞♦ ❛❝✐♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❢✉♥t♦r ❝♦✈❛r✐❛♥t❡✳ ❯♠ ❢✉♥t♦r ❝♦♥tr❛✈❛r✐❛♥t❡ é ❝♦♠♣❧❡t❛♠❡♥t❡ ❛♥á❧♦❣♦✱ ❡①❝❡t♦ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ✏✐♥✈❡rt❡ ✢❡❝❤❛s✑✱ ♦✉ s❡❥❛✱ s❡ f : X → Y é ✉♠ ♠♦r✜s♠♦ ❡♠ C✱ ❡♥tã♦ F (f) : F (Y ) → F (X) é ✉♠ ♠♦r✜s♠♦ ❡♠ D✳ P♦rt❛♥t♦✱ ♣❛r❛ f, g ♠♦r✜s♠♦s ❡♠ C t❛❧ q✉❡ ❛ ❝♦♠♣♦s✐çã♦ g ◦ f ❡①✐st❛✱ t❡♠✲s❡ F (g ◦ f ) = F (f ) ◦ F (g).

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

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

  ❡♠ C ✱ ❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r s❡ t♦r♥❛ op

  F (f ◦ g) = F (g ◦ f ) = F (f ) ◦ F (g). ➱ ♣♦r ❡ss❛ r❛③ã♦ q✉❡ é s✉✜❝✐❡♥t❡ ❡st✉❞❛r♠♦s ❢✉♥t♦r❡s ❝♦✈❛r✐❛♥t❡s✳

  ❉❡✜♥✐çã♦ ✶✳✸✳✸ ❙❡❥❛♠ C✱D✱E ❝❛t❡❣♦r✐❛s ❡ F : C → D✱ G : D → E ❢✉♥t♦r❡s✳ ❆ ❝♦♠♣♦s✐çã♦ G ◦ F : C → E é ❞❡✜♥✐❞❛ ♣♦r

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

  ♣❛r❛ X ∈ C ❡ f ✉♠ ♠♦r✜s♠♦ ❡♠ C✳

  P❛r❛ C, D ❝❛t❡❣♦r✐❛s✱ ❞❡♥♦t❛♠♦s ♣♦r F un(C, D) ❛ ❝♦❧❡çã♦ ❞❡ ❢✉♥t♦✲ r❡s ❡♥tr❡ C ❡ D✳ ◆❛ ♣ró①✐♠❛ s❡çã♦✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛ ❝♦❧❡çã♦ F un(C, D) ❝♦♠♦ ✉♠❛ ❝❛t❡❣♦r✐❛✳ P❛r❛ ✉♠❛ ❝❛t❡❣♦r✐❛ C✱ F un(C, C) = End(C)✳

  ❆♣r❡s❡♥t❛♠♦s ❛❣♦r❛ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❢✉♥t♦r❡s✳ ❈♦♠❡ç❛♠♦s ❛♣r❡✲ s❡♥t❛♥❞♦ ❢✉♥t♦r❡s q✉❡ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ❡♠ q✉❛❧q✉❡r ❝❛t❡❣♦r✐❛✳ C :

  ❊①❡♠♣❧♦ ✶✳✸✳✹ ❚♦❞❛ ❝❛t❡❣♦r✐❛ C ♣♦ss✉✐ ✉♠ ❢✉♥t♦r ✐❞❡♥t✐❞❛❞❡ Id C

  → C C (X) = X C (f ) = f ❞❡✜♥✐❞♦ ♣♦r Id ❡ Id ✱ ♣❛r❛ X ∈ C ❡ f ✉♠

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

  D : D → C D (X) = X D (f ) = f ❖ ❢✉♥t♦r ✐♥❝❧✉sã♦ I é ❞❡✜♥✐❞♦ ♣♦r I ❡ I ✱ ♣❛r❛ X ∈ D ❡ f ✉♠ ♠♦r✜s♠♦ ❡♠ D✳ ❊①❡♠♣❧♦ ✶✳✸✳✻ ❙❡❥❛♠ C, D ❝❛t❡❣♦r✐❛s ❡ Z ∈ D✳ ❖ ❢✉♥t♦r ❝♦♥st❛♥t❡ C Z : C → D Z (X) = Z Z (f ) = id Z

  é ❞❡✜♥✐❞♦ ♣♦r C ❡ C ✱ ♣❛r❛ X ∈ C ❡ f ✉♠ ♠♦r✜s♠♦ ❡♠ C✳

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

  C (X, Y ) = X C (f, g) = f ❝❤❛♠❛❞♦ ♣r♦❥❡çã♦ s♦❜r❡ C✱ é ❞❡✜♥✐❞♦ ♣♦r P ❡ P ✱ ♣❛r❛ (X, Y ) ∈ C × D ❡ (f, g) ✉♠ ♠♦r✜s♠♦ ❡♠ C × D✳

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

  ❝❤❛♠❛❞♦ ♣r♦❥❡çã♦ s♦❜r❡ D✳ X : C →

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

  é ❞❡✜♥✐❞♦ ♣♦r L X (Y ) = Hom C (X, Y ),

  ♣❛r❛ Y ∈ C, ❡ ♣❛r❛ f : Y → Z ✉♠ ♠♦r✜s♠♦ ❡♠ C✱

  → Hom L X (f ) : Hom C (X, Y ) C (X, Z)

  7→ α f ◦ α.

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

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

  R X (Y ) = Hom C (Y, X), ♣❛r❛ Y ∈ C,

  ❡ ♣❛r❛ f : Y → Z ✉♠ ♠♦r✜s♠♦ ❡♠ C✱ R X (f ) : Hom C (Z, X) → Hom C (Y, X)

  7→ α α ◦ f. X ◆❡ss❡ ❝❛s♦✱ ♦ ❢✉♥t♦r R é ❝♦♥tr❛✈❛r✐❛♥t❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ Y ∈ C ❡

  C (Y, X) ♣❛r❛ t♦❞♦ α ∈ Hom ✱ t❡♠♦s

  R X (id Y )(α) = α ◦ id Y = α = id R (α).

  X (Y )

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

  ❊♥tã♦ R X (g ◦ f )(α) = α ◦ (g ◦ f )

  = (α ◦ g) ◦ f = (R X (g)(α)) ◦ f = R X (f )(R X (g)(α)) = (R X (f ) ◦ R X (g))(α).

  X X C (X, −) ❯♠❛ ♥♦t❛çã♦ ♠❛✐s ❝♦♥❤❡❝✐❞❛ ♣❛r❛ L ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ R ✮ é Hom

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

  ∗ C (X, −)(f ) C (−, X)(f ) t✐✈❛♠❡♥t❡ f ✮ ♣❛r❛ Hom ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ Hom ✮✳ ❊ss❡s ❞♦✐s ❢✉♥t♦r❡s ♣♦❞❡♠ ❛✐♥❞❛ s❡r ❝♦♠❜✐♥❛❞♦s✱ ❝♦♠♦ ♠♦str❛ ♦ ♣ró✲ ①✐♠♦ ❡①❡♠♣❧♦✳

  C (−, −) : ❊①❡♠♣❧♦ ✶✳✸✳✶✵ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ X ∈ C✳ ❖ ❢✉♥t♦r Hom op

  C × C → Set

  é ❞❡✜♥✐❞♦ ♣♦r op

  Hom C (−, −)(X, Y ) = Hom C (X, Y ), × C, ♣❛r❛ (X, Y ) ∈ C op

  ′ ′ , Y ) × C

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

  Hom C (−, −)(f, g) : Hom C (X, Y ) → Hom C (X , Y ) α 7→ g ◦ α ◦ f.

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

  ∗ ∗ f ◦ g = g ◦ f

  ∗ ∗ ✳

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

  ❊①❡♠♣❧♦ → V ect k k k

  ❊①❡♠♣❧♦ ✶✳✸✳✶✶ ❉❡♥♦t❛♠♦s ♣♦r D : V ect ♦ ❢✉♥t♦r R ✳ P♦rt❛♥t♦✱ t❡♠♦s

  ∗ ∗ D(V ) = V ,

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

  ∗ = Hom k (V, k)

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

  T : W →

  V f 7→ f ◦ T é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ k✲❧✐♥❡❛r✱ ❝❤❛♠❛❞❛ tr❛♥s♣♦st❛ ❞❡ T : V → W ✳

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

  M R → Ab

  ❡sq✉❡❝❡ ❛ ❛çã♦ ❞❡ R s♦❜r❡ ♦s R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❊①❡♠♣❧♦ ✶✳✸✳✶✺ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❖ ❢✉♥t♦r ❞❡ ❡sq✉❡❝✐♠❡♥t♦ U : Alg k → V ect k

  ❡sq✉❡❝❡ ♦ ♣r♦❞✉t♦ ❞❛s k✲á❧❣❡❜r❛s✳ ❯s❛♥❞♦ ♦s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s q✉❡ ❛♣r❡s❡♥t❛♠♦s✱ ♣♦❞❡rí❛♠♦s

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

  X , ι X )

  X X : X → F

  X é ✉♠ ♣❛r (F ✱ ❡♠ q✉❡ F é ✉♠ ❣r✉♣♦ ❡ ι é ✉♠❛ ❢✉♥çã♦✱ s❛t✐s❢❛③❡♥❞♦ à s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✿ ♣❛r❛ q✉❛❧q✉❡r ♣❛r (G, f)✱ ❡♠ q✉❡ G é ✉♠ ❣r✉♣♦ ❡ f : X → G é ✉♠❛ ❢✉♥çã♦✱ ❡①✐st❡

  → G

  X X = f ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s g : F t❛❧ q✉❡ g ◦ ι ✳

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

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

  X ❞✐③❡♠♦s q✉❡ F é ♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r X✳ ❉❡✈✐❞♦ à ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦s ❣r✉♣♦s ❧✐✈r❡s✱ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f : X → Y ✱ ❡①✐st❡ ✉♠ f : F

  X → F Y f ◦ ι X = ι Y ◦ f ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ¯ t❛❧ q✉❡ ¯ ✳ P❛r❛ ♠❛✐s ✐♥❢♦r♠❛çõ❡s✱ ✈❡❥❛ ♣á❣s✳ ✻✹✲✻✻✮✳

  ❊①❡♠♣❧♦ ✶✳✸✳✶✼ ❙❡❥❛ F : Set → Grp ♦ ❢✉♥t♦r ❞❡✜♥✐❞♦ ♣♦r F (X) = F

  X X ✱ ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ X✱ ❡♠ q✉❡ F é ♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r X✱ f f : F

  X → F Y ❡ F (f) = ¯ ✱ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f : X → Y ✱ ❡♠ q✉❡ ¯ é ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ❝✐t❛❞♦ ❛❝✐♠❛✳

  ▲❡♠❜r❛♠♦s ❛❣♦r❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠✉t❛❞♦r ❡♥tr❡ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❣r✉♣♦✳

  ❙❡❥❛ G ✉♠ ❣r✉♣♦✳ P❛r❛ q✉❛✐sq✉❡r g, h ∈ G✱ ❝♦♥s✐❞❡r❡♠♦s [g, h] = −1 −1 ghg h

  ✱ t❛❧ ❡❧❡♠❡♥t♦ é ❝❤❛♠❛❞♦ ❝♦♠✉t❛❞♦r✳ ❉❡♥♦t❛♠♦s ♣♦r [G, G] = < [g, h] : g, h ∈ G > ♦ s✉❜❣r✉♣♦ ❣❡r❛❞♦

  ♣❡❧♦s ❝♦♠✉t❛❞♦r❡s✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ [G, G] é ♦ ♠❡♥♦r s✉❜❣r✉♣♦ ♥♦r♠❛❧ K t❛❧ q✉❡ G/K é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳

  G : G → G/[G, G] ❉❡♥♦t❛♠♦s ♣♦r π ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳ ➱ ♣♦ssí✈❡❧

  ♠♦str❛r q✉❡ G/[G, G] s❛t✐s❢❛③ à s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✿ ♣❛r❛ q✉❛❧q✉❡r ♣❛r (H, f)✱ ❡♠ q✉❡ H é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡ f : G → H é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s g : G/[G, G] → H G = f t❛❧ q✉❡ g ◦ π ✳

  ❉❡✈✐❞♦ ❛ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✱ ♣❛r❛ f : G → H ✉♠ ❤♦✲ f : ♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ¯ G/[G, G] → H/[H, H] f ◦ π G = π H ◦f f : G/[G, G] → t❛❧ q✉❡ ¯ ✳ ❆ s❛❜❡r✱ ¯ H/[H, H] f ([G, G]g) = [H, H]f (g)

  é ❞❛❞♦ ♣♦r ¯ ✱ ♣❛r❛ g ∈ G✳ P❛r❛ ♠❛✐s ✐♥❢♦r♠❛çõ❡s✱ ♦ ❧❡✐t♦r ♣♦❞❡ ❝♦♥s✉❧t❛r ♣á❣s✳ ✶✵✷✲✶✵✸✮✳ ❊①❡♠♣❧♦ ✶✳✸✳✶✽ ❖ ❢✉♥t♦r ❛❜❡❧✐❛♥✐③❛çã♦ A : Grp → Ab é ❞❡✜♥✐❞♦ ♣♦r A(G) = G/[G, G] f

  ✱ ♣❛r❛ G ✉♠ ❣r✉♣♦ ❡ A(f) = ¯ ✱ ♣❛r❛ f : G → H f : G/[G, G] → H/[H, H] ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ❡♠ q✉❡ ¯ é ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ❥á ❝✐t❛❞♦✳

  ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡ ❡ G ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ ❚❡♠♦s q✉❡ R

  ❡ G sã♦ Z✲♠ó❞✉❧♦s ❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ Z Z

  R ⊗ G G : G → R ⊗ G ✳ ❉❡♥♦t❛♠♦s ♣♦r ι ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s

  G (g) = 1 R ⊗ g ❞❛❞♦ ♣♦r ι ✱ ♣❛r❛ g ∈ G✳

  ➱ s❛❜✐❞♦ q✉❡✱ ♣❛r❛ G, H ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ❡ f : G → H ✉♠ ❤♦✲ ♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s

  Z Z ◦ f f : R ⊗ G → R ⊗ H f ◦ ι G = ι H

  à ❡sq✉❡r❞❛ ¯ t❛❧ q✉❡ ¯ ✳ ❆ s❛❜❡r✱ ¯

  Z Z f : R ⊗ G → R ⊗ H f (r ⊗ g) = r ⊗ f (g)

  é ❞❛❞♦ ♣♦r ¯ ✱ ♣❛r❛ q✉❛❧q✉❡r Z r ⊗ g ∈ R ⊗ G

  ✳ M

  Z − : Ab → R ❊①❡♠♣❧♦ ✶✳✸✳✶✾ ❙❡❥❛ R ⊗ ♦ ❢✉♥t♦r q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ G Z G Z −)(f ) = ¯ f

  ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❛♦ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ R ⊗ ❡ (R ⊗ ✱ f : ♣❛r❛ f : G → H ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s✱ ❡♠ q✉❡ ¯ R ⊗ Z G → R ⊗ Z H

  é ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ❞❛❞♦ ❛❝✐♠❛✳

  ❙❡❥❛♠ k ✉♠ ❝♦r♣♦ ❡ V ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡♥♦t❛♠♦s ♣♦r T (V ) ❛ k✲á❧❣❡❜r❛ t❡♥s♦r✐❛❧ s♦❜r❡ V ✳ ❈♦♠♦ k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ t❡♠♦s

  ∞ M

  ⊗n T (V ) = V , n

  =0 n ⊗0 ⊗1 ⊗n −1

  = k = V = V ⊗ V ❡♠ q✉❡ V ✱ V ❡ V ✱ ♣❛r❛ n ≥ 2✳

  ⊗n ⊗m ⊗ · · · ⊗ v ∈ V ⊗ · · · ⊗ w ∈ V

  1 n 1 m ❆❣♦r❛✱ ♣❛r❛ ❡❧❡♠❡♥t♦s v ❡ w ✱

  ❞❡✜♥✐♠♦s ❛ ♦♣❡r❛çã♦ (v 1 ⊗ · · · ⊗ v n ) · (w 1 ⊗ · · · ⊗ w m ) = v 1 ⊗ · · · ⊗ v n ⊗ w 1 ⊗ · · · ⊗ w m .

  ❊st❡♥❞❡♥❞♦ ❡ss❛ ♦♣❡r❛çã♦ k✲❧✐♥❡❛r♠❡♥t❡ ♣❛r❛ T (V )✱ ♦❜t❡♠♦s ✉♠❛ V : V → T (V )

  ❡str✉t✉r❛ ❞❡ k✲á❧❣❡❜r❛✳ ❉❡♥♦t❛♠♦s ♣♦r ι ❛ tr❛♥s❢♦r♠❛çã♦ ⊗1 k V (v) = v ∈ V

  ✲❧✐♥❡❛r ❞❛❞❛ ♣♦r ι ✱ ♣❛r❛ v ∈ V ✳ V )

  ❙❛❜❡♠♦s q✉❡ (T (V ), ι s❛t✐s❢❛③ à s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✿ ♣❛r❛ q✉❛❧q✉❡r ♣❛r (A, f)✱ ❡♠ q✉❡ A é ✉♠❛ k✲á❧❣❡❜r❛ ❡ f : V → A é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ k✲❧✐♥❡❛r✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s g : T (V ) → A V = f t❛❧ q✉❡ g ◦ ι ✳ ❱❡❥❛ ♣á❣✳ ✶✺✾✮✳

  ❉❡✈✐❞♦ ❛ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✱ ♣❛r❛ f : V → W ✉♠❛ tr❛♥s✲ f : ❢♦r♠❛çã♦ k✲❧✐♥❡❛r✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ¯ T (V ) → T (W ) f ◦ ι V = ι W ◦ f f : T (V ) → T (W ) t❛❧ q✉❡ ¯ ✳ ❆ s❛❜❡r✱ ¯ é ❞❛❞♦ ♣♦r

  ¯ f (v ⊗ · · · ⊗ v n ) = f (v ) ⊗ · · · ⊗ f (v n ),

  1

  1 , ..., v n ∈ V

  ❡♠ q✉❡ v 1 ✳ → Alg k k

  ❊①❡♠♣❧♦ ✶✳✸✳✷✵ ❙❡❥❛ T : V ect ♦ ❢✉♥t♦r q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ k f

  ✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ V à á❧❣❡❜r❛ t❡♥s♦r✐❛❧ T (V ) s♦❜r❡ V ❡ T (f) = ¯ ✱ ♣❛r❛ f : V → W f : T (V ) → T (W ) ✉♠❛ tr❛♥s❢♦r♠❛çã♦ k✲❧✐♥❡❛r✱ ❡♠ q✉❡ ¯ é ♦

  ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ❝✐t❛❞♦ ❛❝✐♠❛✳ Pr♦ss❡❣✉✐♠♦s ❛♣r❡s❡♥t❛♥❞♦ ♠❛✐s ❡①❡♠♣❧♦s ❞❡ ❢✉♥t♦r❡s✳

  ❊①❡♠♣❧♦ ✶✳✸✳✷✶ ❉❡♥♦t❛♠♦s ♣♦r ∆ : Set → Set ♦ ❢✉♥t♦r✱ ❝❤❛♠❛❞♦ ❞✐❛❣♦♥❛❧✱ ❞❡✜♥✐❞♦ ♣♦r ∆(X) = X × X ❡ ∆(f) = f × f✱ ♣❛r❛ X ✉♠ ❝♦♥❥✉♥t♦ ❡ f : X → Y ✉♠❛ ❢✉♥çã♦✱ ❡♠ q✉❡

  → f × f : X × X Y × Y. 7→ (f (x), f (y))

  (x, y) ▲❡♠❜r❛♠♦s ❛❣♦r❛ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❣r✉♣♦ ♦♣♦st♦ ❡ ❞❡ ❤♦♠♦♠♦r✜s♠♦

  ❞❡ ❣r✉♣♦s ♦♣♦st♦✳ op ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❝♦♠ ♦♣❡r❛çã♦ ∗✳ ❉❡♥♦t❛♠♦s ♣♦r G ♦ ❣r✉♣♦ ❝♦♠ op

  ♦♣❡r❛çã♦ ∗ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ♦s ♠❡s♠♦s ❞❡ G ❡ op op x ∗ y = y ∗ x, .

  ♣❛r❛ x, y ∈ G op

  ❖ ❣r✉♣♦ G é ❝❤❛♠❛❞♦ ♦ ❣r✉♣♦ ♦♣♦st♦ ❞❡ G✳ P❛r❛ f : G → H op op op

  → H : G

  ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ❞❡✜♥❡✲s❡ f ♦ ❤♦♠♦♠♦r✜s♠♦ op op op

  → (x) = f (x) : G

  ❞❡ ❣r✉♣♦s ❞❛❞♦ ♣♦r f ✱ ♣❛r❛ x ∈ G✳ ❊♥tã♦✱ f op

  H é ❝❤❛♠❛❞♦ ♦ ♦♣♦st♦ ❞❡ f : G → H✳ op

  : Grp → Grp ❊①❡♠♣❧♦ ✶✳✸✳✷✷ ❉❡♥♦t❛♠♦s ♣♦r (−) ♦ ❢✉♥t♦r q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ❣r✉♣♦ G ❡ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s f : G → H ❛♦s s❡✉s op op op op

  : G → H ♦♣♦st♦s G ❡ f ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ n : Cring →

  ❊①❡♠♣❧♦ ✶✳✸✳✷✸ ❙❡❥❛ n ∈ N✱ n ≥ 1✳ ❉❡♥♦t❛♠♦s ♣♦r GL Grp

  ♦ ❢✉♥t♦r ❞❡✜♥✐❞♦ ♣♦r GL n (R) = {A ∈ M n (R) : A é ✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧}. P❛r❛ f : R → S ✉♠ ♠♦r✜s♠♦ ❡♠ Cring✱ t❡♠♦s

  → GL n (f ) : GL n (R) GL n (S). 7→ (f (r

  (r ij ) i,j ij )) i,j ×

  : Cring → Grp ❊①❡♠♣❧♦ ✶✳✸✳✷✹ ❉❡♥♦t❛♠♦s ♣♦r (−) ♦ ❢✉♥t♦r q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡ R ❛♦ s❡✉ ❣r✉♣♦ ❞❡ ✐♥✈❡rtí✈❡✐s

  × R = {r ∈ R : r

  é ✐♥✈❡rtí✈❡❧} ❡ ❝❛❞❛ ♠♦r✜s♠♦ f : R → S ❡♠ Cring ❛♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s

  × × × × × f : R → S

  ✱ ❞❛❞♦ ♣❡❧❛ r❡str✐çã♦ ❡ ❝♦r❡str✐çã♦ ❞❡ f ❛ R ❡ S ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛♠ k ✉♠ ❝♦r♣♦ ❡ A ✉♠❛ k✲á❧❣❡❜r❛✳ ❖ ❝♦♠✉t❛❞♦r ❞❡ a, b ∈ A✱

  A A = ab − ba ❞❡♥♦t❛❞♦ ♣♦r [a, b] ✱ é ❞❛❞♦ ♣♦r [a, b] ✳ ◆♦t❡♠♦s q✉❡ s❡ f : A → B

  é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✱ ❡♥tã♦ f ♣r❡s❡r✈❛ ♦ A ) = [f (a), f (b)] B

  ❝♦♠✉t❛❞♦r✱ ♦✉ s❡❥❛✱ f([a, b] ✳ → Lie k k

  ❊①❡♠♣❧♦ ✶✳✸✳✷✺ ❉❡♥♦t❛♠♦s ♣♦r L : Alg ♦ ❢✉♥t♦r ❞❡✜♥✐❞♦ A )

  ♣♦r L(A) = (A, [−, −] ❡ L(f) = f✱ ♣❛r❛ A ✉♠❛ k✲á❧❣❡❜r❛ ❡ f : A → B ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳

  ❙❡❥❛♠ k ✉♠ ❝♦r♣♦ ❡ L ✉♠❛ k✲á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❆ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ L✱ ❞❡♥♦t❛❞❛ ♣♦r U(L)✱ é ❞❡✜♥✐❞❛ ❝♦♠♦ s❡♥❞♦ ❛ k✲á❧❣❡❜r❛ U(L) =

  T (L)/I ✱ ❡♠ q✉❡ I é ♦ ✐❞❡❛❧ ❞❡ T (L) ❣❡r❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛ x ⊗ y − y ⊗ x − [x, y] ∈ T (L)

  ✱ ♣❛r❛ x, y ∈ L✳ ❙❡ f : L → K é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ❞❡ ▲✐❡✱ ❡①✐st❡ ✉♠ f : U(L) → U(K) f ◦ ι L =

  ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ¯ t❛❧ q✉❡ ¯ ι K ◦ f

  ✳ → Alg k k

  ❊①❡♠♣❧♦ ✶✳✸✳✷✻ ❉❡♥♦t❛♠♦s ♣♦r U : Lie ♦ ❢✉♥t♦r q✉❡ ❛ss♦✲ f ❝✐❛ ❝❛❞❛ k✲á❧❣❡❜r❛ ❞❡ ▲✐❡ L à á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ U(L) ❞❡ L ❡ U(f) = ¯ ✱ ♣❛r❛ f : L → K ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ❞❡ ▲✐❡✱ ❡♠ q✉❡

  ¯ f : U(L) → U(K) é ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ❞❡ ▲✐❡ ❝✐t❛❞♦ ♥♦

  ❝♦♠❡♥tár✐♦ ❛♥t❡r✐♦r✳ k → Lie k

  ❊①❡♠♣❧♦ ✶✳✸✳✷✼ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r P : Bialg ♦ ❢✉♥t♦r q✉❡ t♦♠❛ ♣r✐♠✐t✐✈♦s✱ ♦✉ s❡❥❛✱ ♣❛r❛ H ✉♠❛ k✲❜✐á❧❣❡❜r❛ ❝♦♠ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ∆✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ♣r✐♠✐t✐✈♦s ❞❡ H é ❞❡✜♥✐❞♦ ♣♦r P (H) = {h ∈ H : ∆(h) = h ⊗ 1 H + 1 H ⊗ h}.

  ❊♥tã♦✱ P (H) é ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❡♠ P (H) ♣❡❧♦ ❝♦♠✉t❛❞♦r [h, k] = hk − kh ∈ P (H)✱ ♣❛r❛ h, k ∈ P (H)

  ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ h, k ∈ P (H)✱ t❡♠♦s ∆([h, k]) = ∆(hk − kh)

  = ∆(h)∆(k) − ∆(k)∆(h) = hk ⊗ 1 H + h ⊗ k + k ⊗ h + 1 H ⊗ hk

  −kh ⊗ 1 H − k ⊗ h − h ⊗ k − 1 H ⊗ kh = (hk − kh) ⊗ 1 H + 1 H ⊗ (hk − kh) = [h, k] ⊗ 1 H + 1 H ⊗ [h, k],

  ♣r♦✈❛♥❞♦ q✉❡ [h, k] ∈ P (H)✳ ❙❡ f : H → K é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲❜✐á❧❣❡❜r❛s✱ ❡♥tã♦ P (f) :

  P (H) → P (K) é ❛ r❡str✐çã♦ ❡ ❝♦r❡str✐çã♦ ❞❡ f ❛ P (H) ❡ P (K)✱ r❡s✲

  ♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡ss❛ ❢♦r♠❛✱ P (f) é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ◆♦t❡♠♦s q✉❡ P (f) ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ h ∈ P (H)✱ t❡♠♦s

  (∗) ∆(f (h)) = (f ⊗ f )(∆(h))

  = (f ⊗ f )(h ⊗ 1 H + 1 H ⊗ h) = f (h) ⊗ 1 K + 1 K ⊗ f (h),

  ❧♦❣♦ f(h) ∈ P (K)✳ ❆ ✐❣✉❛❧❞❛❞❡ ✭∗✮ s❡❣✉❡ ❞♦ ❢❛t♦ ❞❡ q✉❡ ∆ é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳

  ❊①❡♠♣❧♦ ✶✳✸✳✷✽ ❈♦♠ ❛ ♥♦t❛çã♦ ❞♦ ❊①❡♠♣❧♦ s❡ ❝♦♥s✐❞❡r❛r♠♦s D ◦ D : V ect k → V ect k

  ✱ ♦❜t❡♠♦s ✉♠ ❢✉♥t♦r ❝♦✈❛r✐❛♥t❡✱ ❝❤❛♠❛❞♦ ❞✉♣❧♦ ∗∗

  ❞✉❛❧✳ ❊ss❡ ❢✉♥t♦r ❛ss♦❝✐❛ ❝❛❞❛ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ V ❛♦ ❞✉♣❧♦ ❞✉❛❧ V ∗∗

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

  ∗∗ ∗∗ ∗∗ ∗ ∗∗ V → W (Φ) = Φ ◦ T

  ❞❡✜♥✐❞❛ ♣♦r T ✱ ♣❛r❛ t♦❞♦ Φ ∈ V ✳ ❚❡r♠✐♥❛♠♦s ❡ss❛ s❡çã♦ ❞❡✜♥✐♥❞♦ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s q✉❡ ✉♠ ❢✉♥✲ t♦r ♣♦❞❡ t❡r✳

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

  F : Hom C (X, Y ) → Hom D (F (X), F (Y )) é ✐♥❥❡t✐✈❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ s♦❜r❡❥❡t✐✈❛✳

  ❖ ❢✉♥t♦r F é ❞✐t♦ ❞❡♥s♦ s❡✱ ♣❛r❛ t♦❞♦ ♦❜❥❡t♦ Z ∈ D✱ ❡①✐st❡ ✉♠ ♦❜❥❡t♦ X ∈ C t❛❧ q✉❡ F (X) ≃ Z✳

  ❊ss❛s ♣r♦♣r✐❡❞❛❞❡s s❡rã♦ ✉s❛❞❛s ♥❛ ♣ró①✐♠❛ s❡çã♦ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ❢✉♥t♦r❡s q✉❡ ❞❡✜♥❡♠ ❡q✉✐✈❛❧ê♥❝✐❛s ❡♥tr❡ ❝❛t❡❣♦r✐❛s✳

  ✶✳✹ ❚r❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s

  ◆❡st❛ s❡çã♦ ♦❝♦rr❡✱ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✱ ✉♠❛ ❞❡✜♥✐çã♦ ✉s❛♥❞♦ ❞✐❛❣r❛✲ ♠❛s ❝♦♠✉t❛t✐✈♦s✳ ❯♠ ❞✐❛❣r❛♠❛ ❡♠ C é ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ ❝✉❥♦s ♥ós sã♦ ♦❜❥❡t♦s ❞❡ C ❡ ❝✉❥❛s ✢❡❝❤❛s sã♦ ♠♦r✜s♠♦s ❡♠ C✳ ◆❡ss❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ✉♠ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ ♦✉ é ❝♦♠✉t❛t✐✈♦ s❡ s❡♠♣r❡ q✉❡ s❡ ✈❛✐ ❞❡ ✉♠ ♦❜❥❡t♦ ❛ ♦✉tr♦ s❡❣✉✐♥❞♦ ❛s ✢❡❝❤❛s✱ s❡♠♣r❡ s❡ ♦❜té♠ ♦ ♠❡s♠♦ ♠♦r✜s♠♦✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ f

  X Y

  ′

  g f Z W

  

  g é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r q✉❡

  ′ ′ g ◦ f = g ◦ f ∈ Hom C (X, W ).

  ❙❛❜❡♠♦s q✉❡ ❢✉♥t♦r❡s ♣r❡s❡r✈❛♠ ❝♦♠♣♦s✐çõ❡s✳ P♦rt❛♥t♦✱ ❢✉♥t♦r❡s t❛♠❜é♠ ♣r❡s❡r✈❛♠ ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✈♦s✳ P♦r ❡①❡♠♣❧♦✱ s❡ F : C → D é ✉♠ ❢✉♥t♦r✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡

  F (f )

  F (X) F (Y )

  ′

  F (f ) F (g) F (Z) F (W )

  

  F (g )

  é ✉♠ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦ ❡♠ D✳ ❉✐③❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛ ❛❝✐♠❛ ❢♦✐ ♦❜t✐❞♦ ❛♣❧✐❝❛♥❞♦ F ❛♦ ❞✐❛❣r❛♠❛ ♦r✐❣✐♥❛❧ ❡♠ C✳ ❉❡✜♥✐çã♦ ✶✳✹✳✶ ❙❡❥❛♠ F, G : C → D ❢✉♥t♦r❡s✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❡♥tr❡ F ❡ G✱ ❞❡♥♦t❛❞❛ ♣♦r µ : F → G✱ é ✉♠❛ ❝♦❧❡çã♦ ❞❡

  : F (X) → G(X) : X ∈ C} ♠♦r✜s♠♦s {µ X ❡♠ D t❛❧ q✉❡✱ ♣❛r❛ ❝❛❞❛ ♠♦r✜s♠♦ f : X → Y ❡♠ C✱ ♦ ❞✐❛❣r❛♠❛

  µ

  

X

F (X) G(X)

  F G (f ) (f )

  F (Y ) G(Y ) µ

  

Y

  é ❝♦♠✉t❛t✐✈♦✱ ♦✉ s❡❥❛✱ G(f ) ◦ µ = µ ◦ F (f ).

  X Y ❆ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ µ : F → G é ❞✐t❛ ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧

  X ◆❡ss❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ F é ❡q✉✐✈❛❧❡♥t❡ ❛ G ❡ ❞❡♥♦t❛♠♦s ♣♦r F ≃ G✳

  : F (X) → G(X) s❡ ♦s ♠♦r✜s♠♦s µ sã♦ ✐s♦♠♦r✜s♠♦s✱ ♣❛r❛ X ∈ C✳

  X : F (X) → ❖❜s❡r✈❛çã♦ ✶✳✹✳✷ ➱ ❝♦♠✉♠ ❛♣r❡s❡♥t❛r ♦s ♠♦r✜s♠♦s µ

G(X)

  ✱ ✜❝❛♥❞♦ ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❡ ♦s ❢✉♥t♦r❡s ❡♥✈♦❧✈✐❞♦s s✉❜❡♥✲ t❡♥❞✐❞♦s✳ ❆❞♦t❛♠♦s ❡ss❛ ♣rát✐❝❛✳ ❚❛♠❜é♠ ✈❛♠♦s ❞❡s❝r❡✈❡r ❛ ❝♦♠✉✲ t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ ♥❛ ❞❡✜♥✐çã♦ ❝♦♠♦ s❡♥❞♦ ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ µ ♣❛r❛ ♦ ♠♦r✜s♠♦ f✳

  ❯♠❛ ❢♦r♠❛ ❞❡ ✐♥t❡r♣r❡t❛r♠♦s ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ s❡r✐❛ ❝♦♠♦ ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♠♦r✜s♠♦s q✉❡ tr❛♥s❢♦r♠❛ ♦s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✈♦s ♦❜✲ t✐❞♦s ❛♣❧✐❝❛♥❞♦ ✉♠ ❢✉♥t♦r ♥❛q✉❡❧❡s ♦❜t✐❞♦s ❛♣❧✐❝❛♥❞♦ ♦✉tr♦ ❢✉♥t♦r✱ ♦✉ s❡❥❛✱ ❝♦♠♦ s❡ ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ tr❛♥s❧❛❞❛ss❡ ❝❡rt♦s ❞✐❛❣r❛♠❛s✳ ❈♦♥s✐❞❡r❛♥❞♦ µ : F → G ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✱ ✐ss♦ ♣♦❞❡ s❡r ✈✐st♦ ♣❡❧♦s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛t✐✈♦s

  µ

  X F (X) G(X)

  X F G (f ) (f ) f

  µ

  Y

  F (Y ) G(Y ) Y F

  (h) G (h) h g

  F G (g) (g)

  µ

  Z

  F (Z) G(Z) Z

  ❡♠ q✉❡ X, Y, Z ∈ C ❡ f, g, h sã♦ ♠♦r✜s♠♦s ❡♠ C✳ ❊①❡♠♣❧♦ ✶✳✹✳✸ ❚♦❞♦ ❢✉♥t♦r F : C → D ♣♦ss✉✐ ✉♠❛ tr❛♥s❢♦r♠❛çã♦

  : F → F ) = id ♥❛t✉r❛❧ ✐❞❡♥t✐❞❛❞❡ id F ❞❡✜♥✐❞❛ ♣♦r (id F

  X F ✱ ♣❛r❛ (X) t♦❞♦ X ∈ C✳ ❊①❡♠♣❧♦ ✶✳✹✳✹ ❆ ❝♦❧❡çã♦ τ : ∆ → ∆ ❞❛❞❛ ♣♦r

  → X × X, τ : X × X

  X 7→

  (x, y) (y, x) ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ X✱ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳

  ❉❡ ❢❛t♦✱ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f : X → Y ❡ x, y ∈ X✱ t❡♠♦s (∆(f ) ◦ τ X )(x, y) = ((f × f ) ◦ τ X )(x, y)

  = (f × f )(τ X (x, y)) = (f × f )(y, x) = (f (y), f (x)) = τ Y (f (x), f (y)) = τ Y ((f × f )(x, y)) = (τ Y ◦ (f × f ))(x, y) = (τ Y ◦ ∆(f ))(x, y).

  X = τ Y ◦ ∆(f ) P♦rt❛♥t♦✱ ∆(f) ◦ τ ✳

  ❊①❡♠♣❧♦ ✶✳✹✳✺ ❆ ❝♦❧❡çã♦ ❞❛s ✐♥✈❡rsõ❡s ♥♦s ❣r✉♣♦s é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉❛❧✳ ❊①♣❧✐❝✐t❛♠❡♥t❡✱ s❡❥❛ µ : Id

  n

  ...r nσ (n)

  ) =

  X σ ∈S

  n

  f (sign(σ)r 1σ(1)

  ...r nσ (n)

  ) =

  X σ

  ∈S

  sign(σ)f (r 1σ(1)

  

n

  )...f (r nσ (n)

  ) = det S ((f (r ij )) i,j ) = det S (GL n (f )((r ij ) i,j )) = det S (GL n (f )(A)) = (det S ◦GL n (f ))(A).

  P♦rt❛♥t♦✱ f ×

  ◦ det R = det S ◦GL n (f ) ✳

  ❊①❡♠♣❧♦ ✶✳✹✳✼ ❆ ❝♦❧❡çã♦ ev : Id V ect

  k

  → D ◦ D ❞❛❞❛ ♣♦r ev

  V : V →

  V ∗∗ v

  7→ ev

  sign(σ)r 1σ(1)

  ∈S

  Grp → (−) op ❞❡✜♥✐❞❛ ♣♦r

  , A 7→ det R (A)

  µ G

  : G → G op

  , x 7→ x −1

  ♣❛r❛ ❝❛❞❛ ❣r✉♣♦ G✳ ❈❧❛r❛♠❡♥t❡✱ µ G

  é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ♣❛r❛ ❝❛❞❛ ❣r✉♣♦ G✱ ❡ ❝♦♠♦ f(x

  −1 ) = f (x)

  −1 ✱ ♣❛r❛ t♦❞♦ ♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s f

  ✱ s❡❣✉❡ q✉❡ µ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❊①❡♠♣❧♦ ✶✳✹✳✻ ❖ ❞❡t❡r♠✐♥❛♥t❡ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❊①♣❧✐✲ ❝✐t❛♠❡♥t❡✱ det : GL n

  → (−) ×

  ❞❡✜♥✐❞❛ ♣♦r det R : GL n (R) → R ×

  ♣❛r❛ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡✱ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❙❛❜❡♠♦s q✉❡ det

  X σ

  R (A) = P

  σ ∈S

  n

  sign(σ)r 1σ(1)

  ...r nσ (n) ✳

  ❉❡ ❢❛t♦✱ ♣❛r❛ f : R → S ✉♠ ♠♦r✜s♠♦ ❡♠ Cring ❡ A = (r ij ) i,j ∈

  GL n (R) ✱ t❡♠♦s

  (f ×

  ◦ det R )(A) = f ×

  (det R (A)) = f (det R (A)) = f (det R ((r ij ) i,j )) = f (

  V (v)

  é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✱ ❡♠ q✉❡ ♦ ♠♦r✜s♠♦ ∗ ev (v) : V → k

  V f 7→ f (v)

  é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ k✲❧✐♥❡❛r✳ ❉❡ ❢❛t♦✱ ♣❛r❛ T : V → W ✱ v ∈ V ❡ ∗ f ∈ W

  ✱ t❡♠♦s ∗∗

  (((D ◦ D)(T ) ◦ ev )(v))(f ) = ((T ◦ ev )(v))(f )

  V V ∗∗

  = (T (ev (v)))(f )

  V ∗

  = (ev (v) ◦ T )(f )

  V ∗

  = (ev (v))(T (f ))

  V = (ev (v))(f ◦ T )

  V = (f ◦ T )(v) = f (T (v)) = (ev (T (v)))(f )

  W = ((ev ◦ T )(v))(f )

  W = ((ev ◦ Id V ect (T ))(v))(f ). W k

  = ev ◦ Id V ect (T ) V ect P♦rt❛♥t♦✱ (D ◦ D)(T ) ◦ ev

  V W k ✳ ❙❡ Id k ❡ D ◦ D k ❢♦r❡♠ r❡str✐t♦s ❛ vect ✱ ❡♥tã♦ ev é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❉❡✜♥✐çã♦ ✶✳✹✳✽ ❙❡❥❛♠ C, D ❝❛t❡❣♦r✐❛s✳ ✭✐✮ C ❡ D sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st❡♠ ❢✉♥t♦r❡s F : C → D ❡ G : D → C

  C D t❛✐s q✉❡ G ◦ F ≃ Id ❡ F ◦ G ≃ Id ❡ ❞❡♥♦t❛✲s❡ C ≃ D✳ ✭✐✐✮ C ❡ D sã♦ ✐s♦♠♦r❢❛s s❡ ❡①✐st❡♠ ❢✉♥t♦r❡s F : C → D ❡ G : D → C

  C D t❛✐s q✉❡ G ◦ F = Id ❡ F ◦ G = Id ❡ ❞❡♥♦t❛✲s❡ C ∼ D✳

  ❚♦❞♦ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❝❛t❡❣♦r✐❛s é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛✱ ♠❛s ❛ r❡❝í✲ ♣r♦❝❛ ♥ã♦ é ✈❡r❞❛❞❡✐r❛✳

  P❛r❛ ✉♠❛ ❝❛t❡❣♦r✐❛ C✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❡♠ Ob(C) ❛ r❡❧❛çã♦ ∼ t❛❧ q✉❡ X ∼ Y s❡✱ ❡ s♦♠❡♥t❡ s❡✱ X ≃ Y ✳ ❊♥tã♦ ∼ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡ ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ✉♠ ♦❜❥❡t♦ é ❝❤❛♠❛❞❛ ❝❧❛ss❡ ❞❡ ✐s♦♠♦r✜s♠♦ ❞❡ss❡ ♦❜❥❡t♦✳ ◆❡ss❡ ❝❛s♦✱ ❞✉❛s ❝❛t❡❣♦r✐❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ ❛s s✉❛s ❝❧❛ss❡s ❞❡ ✐s♦♠♦r✜s♠♦✱ ❡♥q✉❛♥t♦ q✉❡ sã♦ ✐s♦♠♦r❢❛s s❡ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ s✉❛s ❝❧❛ss❡s ❞❡ ♦❜❥❡t♦s✳

  ❆❣♦r❛✱ ❛♣r❡s❡♥t❛♠♦s ❞♦✐s t✐♣♦s ❞❡ ❝♦♠♣♦s✐çõ❡s ❡♥tr❡ tr❛♥s❢♦r♠❛✲ çõ❡s ♥❛t✉r❛✐s✿ ❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧ ❡ ❝♦♠♣♦s✐çã♦ ❤♦r✐③♦♥t❛❧✳

  ❉❡✜♥✐çã♦ ✶✳✹✳✾ ❙❡❥❛♠ C, D ❝❛t❡❣♦r✐❛s✱ F, G, H : C → D ❢✉♥t♦r❡s ❡ µ : F → G

  ✱ ν : G → H tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s✳ ❆ ❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧ ❞❡ ν ❡ µ é ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ν ◦ µ : F → H ❞❛❞❛ ♣♦r

  (ν ◦ µ) = ν ◦ µ ,

  X X X ♣❛r❛ t♦❞♦ X ∈ C.

  ❉❡✜♥✐çã♦ ✶✳✹✳✶✵ ❙❡❥❛♠ C✱ D✱ E ❝❛t❡❣♦r✐❛s✱ F, G : C → D✱ J, H : D

  → E ❢✉♥t♦r❡s ❡ µ : F → G✱ ν : J → H tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s✳

  ❆ ❝♦♠♣♦s✐çã♦ ❤♦r✐③♦♥t❛❧ ❞❡ ν ❡ µ é ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ν ∗ µ : J ◦ F → H ◦ G

  ❞❛❞❛ ♣♦r (ν ∗ µ) = ν ◦ J(µ ),

  X G X ♣❛r❛ t♦❞♦ X ∈ C.

  (X) ◦ J(µ ) = H(µ ) ◦ ν

  ◆♦t❡♠♦s q✉❡ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ ν G F ✳ ❉❡ (X)

  X X (X) : F (X) →

  ❢❛t♦✱ ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ ν ❡ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ♠♦r✜s♠♦ µ

  X G(X) ✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦

  ν

  

(X)

F

J(F (X)) H(F (X))

  J H (µ ) (µ )

  X X

J(G(X)) H(G(X))

  ν

  

G (X)

  G ◦ J(µ ) = H(µ ) ◦ ν F ♦✉ s❡❥❛✱ ν (X)

  X X (X) ✳ ❈♦♠ r❡s♣❡✐t♦ às ❝♦♠♣♦s✐çõ❡s ✈❡rt✐❝❛✐s ❡ ❤♦r✐③♦♥t❛✐s✱ t❡♠♦s ♦ s❡✲

  ❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ Pr♦♣♦s✐çã♦ ✶✳✹✳✶✶ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ✈á❧✐❞❛s✿ ✭✐✮ ❛ ❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s é ✉♠❛ tr❛♥s❢♦r✲ ♠❛çã♦ ♥❛t✉r❛❧❀ ✭✐✐✮ ❛ ❝♦♠♣♦s✐çã♦ ❤♦r✐③♦♥t❛❧ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s é ✉♠❛ tr❛♥s✲ ❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙❡❥❛ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❈♦♠ ❛s ♥♦t❛çõ❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛

  F (X) H(X) F (Y ) H(Y )

  Y

  (ν∗µ)

  X H (G(f )) J (F (f ))

  (ν∗µ)

  (J ◦ F )(X) (H ◦ G)(X) (J ◦ F )(Y ) (H ◦ G)(Y )

  ✭✐✐✮ ❙❡❥❛ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❈♦♠ ❛s ♥♦t❛çõ❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠♣♦s✐çã♦ ❤♦r✐③♦♥t❛❧✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛

  = (ν ◦ µ) Y ◦ F (f ).

  Y ◦ F (f )

  Y ◦ µ

  X = ν

  Y ◦ G(f ) ◦ µ

  X = ν

  X ◦ µ

  X = H(f ) ◦ ν

  ❝♦♠✉t❛♠✳ ❆ss✐♠✱ H(f ) ◦ (ν ◦ µ)

  (f ) µ

  (ν◦µ)

  (f ) F

  X G

  µ

  F (X) G(X) F (Y ) G(Y )

  Y

  (f ) ν

  (f ) G

  X H

  G(Y ) H(Y ) ν

  ❝♦♠✉t❛✳ ❙❛❜❡♠♦s q✉❡ ♦s ❞✐❛❣r❛♠❛s ❡ G(X) H(X)

  Y

  (f ) (ν◦µ)

  (f ) F

  X H

  Y

  ❝♦♠✉t❛✳ ❆❧é♠ ❞✐ss♦✱ ♦s ❞✐❛❣r❛♠❛s ❡

  µ

  G (X)

  ν

  X G

J(G(X)) H(G(X))

  G (Y )

  X )

  ν ∗ µ

  J ◦ F H ◦ G

  µ ν C E .

  P❛r❛ ❛ ✏❝♦♠♣♦s✐çã♦ ❤♦r✐③♦♥t❛❧✑✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡♣r❡s❡♥t❛çã♦ D C E F G J H

  C D . F H ν ◦ µ

  G C D F H µ ν

  ❖❜s❡r✈❛çã♦ ✶✳✹✳✶✷ ❆ ❡①♣❧✐❝❛çã♦ s♦❜r❡ ♦ ♥♦♠❡ ✏❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧✑ s❡ ❞❡✈❡ à r❡♣r❡s❡♥t❛çã♦ ❞❡ss❛ ❝♦♠♣♦s✐çã♦ ❡♠ ❢♦r♠❛ ❞✐❛❣r❛♠át✐❝❛ ❝♦♠♦ ❛❜❛✐①♦

  = (ν ∗ µ) Y ◦ J(F (f )).

  Y ) ◦ J(F (f ))

  (Y ) ◦ J(µ

  = ν G

  Y ◦ F (f ))

  (Y ) ◦ J(µ

  = ν G

  (Y ) ◦ J(G(f ) ◦ µ

  F (X) G(X) F (Y ) G(Y )

  = ν G

  X )

  H (G(f ))

  = ν G

  X )

  ◦ J(µ

  G (X)

  X = H(G(f )) ◦ ν

  ❝♦♠✉t❛♠✳ ❆ss✐♠✱ H(G(f )) ◦ (ν ∗ µ)

  Y

  (f ) µ

  (f ) F

  J (G(f ))

  ν

  (Y ) ◦ J(G(f )) ◦ J(µ P❛r❛ ❢✉♥t♦r❡s F, G✱ ❞❡♥♦t❛♠♦s ♣♦r Nat(F, G) ❛ ❝♦❧❡çã♦ ❞❡ tr❛♥s✲ ❢♦r♠❛çõ❡s ♥❛t✉r❛✐s µ : F → G✳ P♦❞❡♠♦s ❛❣♦r❛ ❛♣r❡s❡♥t❛r ♠❛✐s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❝❛t❡❣♦r✐❛✳ ❊①❡♠♣❧♦ ✶✳✹✳✶✸ ❙❡❥❛♠ C✱ D ❝❛t❡❣♦r✐❛s✳ ❊♥tã♦ F un(C, D) ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠❛ ❝❛t❡❣♦r✐❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❢✉♥t♦r❡s F : C → D

  F un (F, G) = N at(F, G) ❡ Hom (C,D) ✳ ❖ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞♦ ❢✉♥✲

  : F → F t♦r F : C → D é ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ id F ✳ ❆ ❝♦♠♣♦s✐çã♦ é ❞❛❞❛ ♣❡❧❛ ❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s✳

  ❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ❝❛r❛❝t❡r✐③❛ ❢✉♥t♦r❡s q✉❡ ❞❡✜♥❡♠ ❡q✉✐✈❛❧ê♥❝✐❛s ❡♥tr❡ ❝❛t❡❣♦r✐❛s ❝♦♠♦ ❛q✉❡❧❡s q✉❡ sã♦ ✜é✐s✱ ♣❧❡♥♦s ❡ ❞❡♥s♦s✳ ❖ ❧❡✐t♦r ❛t❡♥t♦ ❞❡✈❡ ♣❡r❝❡❜❡r q✉❡ ♥❛ ♣❛rt❡ ✏s❡✑ é ✉s❛❞♦ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❢♦rt❡ ❞♦ ❛①✐♦♠❛ ❞❛ ❡s❝♦❧❤❛✳ ❚❡♦r❡♠❛ ✶✳✹✳✶✹ ❉✉❛s ❝❛t❡❣♦r✐❛s C ❡ D sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ ❢✉♥t♦r F : C → D ✜❡❧✱ ♣❧❡♥♦ ❡ ❞❡♥s♦✳ ❉❡♠♦♥str❛çã♦✿ (⇒) ❈♦♠♦ C ❡ D sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ ❡①✐st❡♠ ❢✉♥t♦r❡s F : C → D C → G ◦ F

  ✱ G : D → C ❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s µ : Id ❡ ν : Id D → F ◦ G

  ✳ ❘❡❢❡r✐♠♦✲♥♦s à ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ µ

  X G(F (X))

  X f G

  (F (f ))

G(F (Y ))

  Y µ

  Y

  ❝♦♠♦ s❡♥❞♦ ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ µ ♣❛r❛ ♦ ♠♦r✜s♠♦ f : X → Y ❡♠ C✳ ❆✜r♠❛çã♦ ✶✿ F é ✜❡❧✳

  C (X, Y ) → Hom D (F (X), F (Y )) ❉❡✈❡♠♦s ♠♦str❛r q✉❡ F : Hom é

  ′ ′ ∈ Hom C (X, Y ) )

  ❛♣❧✐❝❛çã♦ ✐♥❥❡t♦r❛✳ ❙❡❥❛♠ f, f t❛✐s q✉❡ F (f) = F (f ✳ ′

  )) ❊♥tã♦ G(F (f)) = G(F (f ❡ t❡♠♦s

  µ ◦ f = G(F (f )) ◦ µ Y

  X ′

  = G(F (f )) ◦ µ

  X ′

  = µ ◦ f , Y

  ❡♠ q✉❡ ♥❛ ♣r✐♠❡✐r❛ ❡ t❡r❝❡✐r❛ ✐❣✉❛❧❞❛❞❡s ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ µ ′ ′

  ◦ f = µ ◦ f ♣❛r❛ ♦s ♠♦r✜s♠♦s f ❡ f ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦rt❛♥t♦✱ µ Y Y

  ′ ❡ ❝♦♠♦ µ Y é ✉♠ ✐s♦♠♦r✜s♠♦✱ ♦❜t❡♠♦s f = f ✳

  P♦r ❛r❣✉♠❡♥t♦ ❛♥á❧♦❣♦✱ G : D → C t❛♠❜é♠ é ✜❡❧✳ ❊ss❡ ❢❛t♦ s❡rá ✉s❛❞♦ ♥❛ ♣ró①✐♠❛ ❛✜r♠❛çã♦✳

  ❆✜r♠❛çã♦ ✷✿ F é ♣❧❡♥♦✳ C (X, Y ) → Hom D (F (X), F (Y ))

  ❉❡✈❡♠♦s ♠♦str❛r q✉❡ F : Hom é ❛♣❧✐❝❛çã♦ s♦❜r❡❥❡t♦r❛✳ ❙❡❥❛ g : F (X) → F (Y ) ✉♠ ♠♦r✜s♠♦ ❡♠ D

  ✳ ❈♦♥s✐❞❡r❡♠♦s f : X → Y ♦ ♠♦r✜s♠♦ ❡♠ C ❞❡✜♥✐❞♦ ♣♦r f = −1

  µ ◦ G(g) ◦ µ Y X ✳ ❊♥tã♦✱ ♣❡❧❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ µ ♣❛r❛ f✱ t❡♠♦s

  ◦ f G(F (f )) ◦ µ = µ

  X Y −1

  ◦ µ ◦ G(g) ◦ µ = µ

  Y Y

  X = G(g) ◦ µ X .

  = G(g) ◦ µ P♦rt❛♥t♦✱ G(F (f)) ◦ µ ❡ ❝♦♠♦ µ é ✉♠ ✐s♦♠♦r✜s♠♦✱

  X X

  X ♦❜t❡♠♦s G(F (f)) = G(g)✳ ❙❡♥❞♦ G ✜❡❧✱ ❝♦♥❝❧✉í♠♦s q✉❡ F (f) = g✳

  ❆✜r♠❛çã♦ ✸✿ F é ❞❡♥s♦✳ ❙❡❥❛ Z ∈ D✳ ❊♥tã♦✱ G(Z) ∈ C ❡ F (G(Z)) ≃ Z✱ ✈✐❛ ♦ ✐s♦♠♦r✜s♠♦

  ν : Z → F (G(Z)) Z ✳

  (⇐) ❉❡✈❡♠♦s ❞❡✜♥✐r ✉♠ ❢✉♥t♦r G : D → C ❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s

  µ : Id C → G ◦ F D → F ◦ G ✱ ν : Id ✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ❞❡✜♥✐r ♦❜❥❡t♦s

  X Z ∈ C g ❡ ♠♦r✜s♠♦s f ❡♠ C✱ ❡♠ q✉❡ Z ∈ D ❡ g sã♦ ❝❡rt♦s ♠♦r✜s♠♦s

  ❡♠ D✳ ❈♦♠♦ F : C → D é ❞❡♥s♦✱ ❡♥tã♦✱ ♣❛r❛ ❝❛❞❛ Z ∈ D✱ ❡①✐st❡ ✉♠ ♦❜❥❡t♦

  X Z ∈ C : Z → F (X Z ) ❡ ✉♠ ✐s♦♠♦r✜s♠♦ ν Z ❡♠ D✳

  ❆❣♦r❛✱ ❝♦♠♦ F é ✜❡❧ ❡ ♣❧❡♥♦✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ ❛ ❛♣❧✐✲ C (X, Y ) → Hom D (F (X), F (Y ))

  ❝❛çã♦ F : Hom é ❜✐❥❡t♦r❛✳ ▲♦❣♦✱ s❡ g : F (X) → F (Y ) é ✉♠ ♠♦r✜s♠♦ ❡♠ D✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ f g : X → Y g ) = g

  ❡♠ C t❛❧ q✉❡ F (f ✳ ❱❛♠♦s ❛♣r❡s❡♥t❛r três ♣r♦♣r✐❡❞❛✲ ❞❡s ❞❡ss❡s ♠♦r✜s♠♦s✳

  F = h Pr♦♣r✐❡❞❛❞❡ ✶✿ f (h) ✱ ♣❛r❛ t♦❞♦ ♠♦r✜s♠♦ h : X → Y ❡♠ C✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ F (h) : F (X) → F (Y ) é ✉♠ ♠♦r✜s♠♦ ❡♠ D✱ ❡①✐st❡

  : X → Y ) = F (h) ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ f F ❡♠ C t❛❧ q✉❡ F (f F ✳

  (h) (h) = h

  ❈♦♠♦ F é ✜❡❧✱ f F (h) ✳

  ′ ′

  g = f g ◦ f g Pr♦♣r✐❡❞❛❞❡ ✷✿ f ◦g ✱ ♣❛r❛ q✉❛✐sq✉❡r ♠♦r✜s♠♦s g :

  ′ ′ ′ ′′ F (X) → F (X ) : F (X ) → F (X )

  ❡ g ❡♠ D✳

  ❈♦♠♦ g ′

  Z

  F (X)

  = f F (id

  X

  ) (∗∗)

  = id X , ❡♠ (∗) ✉s❛♠♦s ❛ Pr♦♣r✐❡❞❛❞❡ ✷ ❡ ❡♠ (∗∗) ✉s❛♠♦s ❛ Pr♦♣r✐❡❞❛❞❡ ✶ ♣❛r❛ ♦ ♠♦r✜s♠♦ id

  X ✳ ▲♦❣♦✱ f g

  −1

  ◦ f g = id

  X ❡ ❛♥❛❧♦❣❛♠❡♥t❡ ♠♦str❛✲s❡ f g ◦ f g

  −1

  = id Y ✱ ♣r♦✈❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡✳

  ❆❣♦r❛✱ s❡❥❛ g : Z → W ✉♠ ♠♦r✜s♠♦ ❡♠ D ❡ ❝♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛✲ ❣r❛♠❛

  Z F (X Z ) W

  F (X W ) ν

  g ν

  −1

  W

  ❉❡✜♥✐♠♦s ♦ ♠♦r✜s♠♦ ♣♦♥t✐❧❤❛❞♦ ♣♦r ν

  W ◦ g ◦ ν

  −1 Z : F (X Z ) → F (X W ).

  P❡❧♦ ❞✐t♦ ❛❝✐♠❛✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ f ν

  W

  ◦g◦ν

  −1 Z

  : X Z → X W t❛❧ q✉❡ F (f ν

  W

  ◦g◦ν

  −1 Z

  ) = ν W

  ◦ g ◦ ν −1 Z

  ◦g = f id

  = f g

  ◦g : F (X) → F (X ′′

  ) ◦ F (f g ) = F (f g

  ) é ✉♠ ♠♦r✜s♠♦ ❡♠ D✱ ❡①✐st❡ ✉♠ ú♥✐❝♦

  ♠♦r✜s♠♦ f g

  ′

  ◦g : X → X

  ′′ t❛❧ q✉❡ F (f g

  ′

  ◦g ) = g

  ′ ◦ g

  ✳ ❆❣♦r❛✱ F (f g

  ′

  ◦g ) = g

  ′ ◦ g

  = F (f g

  ′

  ′ ◦ f g ).

  ◦ f g (∗)

  ◦ f g ✳

  −1

  ❡♠ D✳ ❉❡ ❢❛t♦✱ t❡♠♦s f g

  ✱ ♣❛r❛ q✉❛❧q✉❡r ✐s♦♠♦r✜s♠♦ g : F (X) → F (Y )

  −1

  = f g

  Pr♦♣r✐❡❞❛❞❡ ✸✿ f −1 g

  ′

  P♦rt❛♥t♦✱ F (f g

  ◦g = f g

  ′

  ◦ f g ) ❡ ❝♦♠♦ F é ✜❡❧✱ t❡♠♦s f g

  ′

  ◦g ) = F (f g

  ′

  ✳ (∆)

  ❙❛❜❡♥❞♦ ❞✐ss♦✱ ❞❡✜♥✐♠♦s G : D → C ♣♦r

  −1

  G(Z) = X Z , ❡ G(g) = f

  ν ◦g◦ν

  W Z

  ♣❛r❛ Z ∈ D ❡ g : Z → W ✉♠ ♠♦r✜s♠♦ ❡♠ D✳ ▼♦str❡♠♦s q✉❡ G : D → C é ✉♠ ❢✉♥t♦r✳ ❉❡ ❢❛t♦✱ s❡❥❛ Z ∈ D✳ ❊♥tã♦

  −1

  G(id ) = f Z ν

  ◦id ◦ν

  Z Z Z −1

  = f ν ◦ν

  Z Z

  = f id

  F (XZ )

  = f F

  (id )

  XZ

  (∗) = id

  X Z = id ,

  G (Z)

  X ❡♠ (∗) ✉s❛♠♦s ❛ Pr♦♣r✐❡❞❛❞❡ ✶ ♣❛r❛ ♦ ♠♦r✜s♠♦ id ✳ ❆❣♦r❛✱ s❡ g :

  Z

  ′ Z → W : W → V

  ❡ g sã♦ ♠♦r✜s♠♦s ❡♠ D✱ t❡♠♦s ′

  −1

  G(g ◦ g) = f ′ ν

  ◦g ◦g◦ν

  V Z −1 −1 ′

  = f ν

  ◦g ◦ν ◦ν ◦g◦ν

  V W W Z

  (∗)

  −1 −1

  = f ′ ◦ f ν ν

  ◦g ◦ν ◦g◦ν

  V W W Z

  ′ = G(g ) ◦ G(g),

  ′ ) = id ◦g) =

  ❡♠ (∗) ✉s❛♠♦s ❛ Pr♦♣r✐❡❞❛❞❡ ✷✳ P♦rt❛♥t♦✱ G(id Z ❡ G(g G (Z)

  ′ G(g ) ◦ G(g)

  ❡ ❛ss✐♠✱ G é ✉♠ ❢✉♥t♦r✳ C → G ◦ F

  ❆❣♦r❛✱ q✉❡r❡♠♦s ❞❡✜♥✐r ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ µ : Id ✳ F : F (X) → F (X F )

  ❙❡❥❛ X ∈ C✳ ❚❡♠♦s q✉❡ ν (X) (X) é ✉♠ ✭✐s♦✮♠♦r✜s♠♦ ν : X → X F

  ❡♠ D✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ f (X) (X) ❡♠ C t❛❧

  F

  ν ) = ν F F = G(F (X)) ν : X → q✉❡ F (f F (X) (X) ✳ ❈♦♠♦ X (X) ✱ ❡♥tã♦ f F (X)

G(F (X))

  ✳ C → G◦F = f

  ❉❡✜♥✐♥❞♦ µ : Id ♣♦r µ X ν ✱ ♣❛r❛ X ∈ C✱ ♠♦str❡♠♦s

  F (X)

  q✉❡ µ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱ s❡❥❛ X ∈ C✳ ❯s❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✸ ♣❛r❛ ♦ ✐s♦♠♦r✜s♠♦

  ν = f F (X) ✱ s❡❣✉❡ q✉❡ µ ν é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ C✱ ❝♦♠ ✐♥✈❡rs❛

  X F (X) −1

  µ = f

  −1

  X ✳ ν

  (X) F

  C → G ◦ F P❛r❛ ✈❡r✐✜❝❛r♠♦s q✉❡ µ : Id é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉✲ r❛❧✱ ❝♦♥s✐❞❡r❡♠♦s h : X → Y ✉♠ ♠♦r✜s♠♦ q✉❛❧q✉❡r ❡♠ C✳ ❉❡✈❡♠♦s

  ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛ X (G ◦ F )(X)

  ν

  ❈♦♠♦ X Z = G(Z)

  ✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ν Z : Z → F (G(Z))

  ✳ ▼♦str❡♠♦s q✉❡ ν : Id

  D → F ◦ G é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❘❡st❛✲

  ♥♦s ♠♦str❛r ❛♣❡♥❛s ❛ ♥❛t✉r❛❧✐❞❛❞❡✳ ❙❡❥❛ g : Z → W ✉♠ ♠♦r✜s♠♦ ❡♠ D

  ✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛ Z

  (F ◦ G)(Z) W

  (F ◦ G)(W ) ν

  Z

  F (G(g)) g

  W

  é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ P❛r❛ ❝❛❞❛ Z ∈ D✱ ❡①✐st❡ ♦ ✐s♦♠♦r✜s♠♦ ν

  é ❝♦♠✉t❛t✐✈♦✳ ❚❡♠♦s F (G(g)) ◦ ν

  Z = F (f

  ν

  W

  ◦g◦ν

  −1 Z

  ) ◦ ν Z

  (∆) = ν

  W ◦ g ◦ ν

  −1 Z

  Z : Z → F (X Z ) ❡♠ D✳

  P♦rt❛♥t♦✱ µ : Id C → G ◦ F

  Y (G ◦ F )(Y )

  ◦ µ

  µ

  X G

  (F (h)) h µ

  Y

  é ❝♦♠✉t❛t✐✈♦✳ ❚❡♠♦s G(F (h)) ◦ µ

  X = f

  ν

  

F (Y )

  ◦F (h)◦ν

  −1 F (X)

  X = f ν

  X = µ Y ◦ h.

  

F

(Y )

  ◦ f F

  (h) ◦ f

  ν

  −1 F (X)

  ◦ µ

  X = µ Y ◦ h ◦ µ

  −1

  X ◦ µ

  X = µ Y ◦ h ◦ id

  ◦ ν Z

  = ν W

  ◦ g ◦ id Z

  = ν W

  ◦ g ❡ ✐ss♦ ♥♦s ❞✐③ q✉❡ ν é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳

  ❆ss✐♠✱ ❡♥❝♦♥tr❛♠♦s ✉♠ ❢✉♥t♦r G : D → C ❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s µ : Id C → G ◦ F

  ❡ ν : Id D → F ◦ G

  ✳ ▲♦❣♦✱ C ❡ D sã♦ ❝❛t❡❣♦r✐❛s ❡q✉✐✈❛❧❡♥t❡s✳

  ❈❛♣ít✉❧♦ ✷ ❈❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s

  ◆❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛♠♦s ❛s ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐✲ s❛♠♦s ❞❡✜♥✐r ♥ú❝❧❡♦s✱ ❝♦♥ú❝❧❡♦s ❡ ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✳ ❊st❛s ú❧t✐♠❛s sã♦ ❝❛t❡❣♦r✐❛s ❝♦♠ ❡str✉t✉r❛ ❛❞✐❝✐♦♥❛❧ ❞❛❞❛ ♣❡❧❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♦❜❥❡t♦ ③❡r♦✱ s♦♠❛ ❞❡ ♠♦r✜s♠♦s ❡ s♦♠❛ ❞✐r❡t❛ ❞❡ ♦❜❥❡t♦s✳ ❈♦♠♦ r❡❢❡rê♥❝✐❛ ❜ás✐❝❛✱ ❝✐t❛♠♦s

  ✷✳✶ ◆ú❝❧❡♦s ❡ ❝♦♥ú❝❧❡♦s

  ❉❡✜♥✐çã♦ ✷✳✶✳✶ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛✳ ❯♠ ♦❜❥❡t♦ Z ∈ C é ❞✐t♦ C (Z, X)

  ✭✐✮ ✐♥✐❝✐❛❧ s❡✱ ♣❛r❛ q✉❛❧q✉❡r X ∈ C✱ Hom é ✉♥✐tár✐♦❀ C (X, Z)

  ✭✐✐✮ ✜♥❛❧ s❡✱ ♣❛r❛ q✉❛❧q✉❡r X ∈ C✱ Hom é ✉♥✐tár✐♦✳ ◆♦t❡♠♦s q✉❡ ♦❜❥❡t♦s ✐♥✐❝✐❛❧ ❡ ✜♥❛❧ sã♦ ❝♦♥❝❡✐t♦s ❞✉❛✐s✳ ❙❡❣✉❡ ❞❛

  C (Z, Z) = {id Z } ❞❡✜♥✐çã♦ q✉❡ s❡ Z é ♦❜❥❡t♦ ✐♥✐❝✐❛❧ ♦✉ ✜♥❛❧✱ ❡♥tã♦ Hom ✳ ❯s❛r❡♠♦s ❡ss❡ ❢❛t♦ ♥♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✷ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛✳ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ✈á❧✐❞❛s✿ ✭✐✮ s❡ C t❡♠ ♦❜❥❡t♦ ✐♥✐❝✐❛❧✱ ❡st❡ é ú♥✐❝♦✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦❀ ✭✐✐✮ s❡ C t❡♠ ♦❜❥❡t♦ ✜♥❛❧✱ ❡st❡ é ú♥✐❝♦✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳

  ′ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙❡❥❛♠ Z ❡ Z ♦❜❥❡t♦s ✐♥✐❝✐❛✐s ❡♠ C✳ ❊♥tã♦ ❡①✐s✲

  ′ ′ → Z t❡♠ ú♥✐❝♦s ♠♦r✜s♠♦s f : Z → Z ❡ g : Z ❡ ♣♦rt❛♥t♦ g ◦ f ∈

  ′ ′

  ′

  Hom C (Z, Z) = {id Z } C (Z , Z ) = {id Z } ❡ f ◦ g ∈ Hom ✳ ▲♦❣♦✱ g ◦ f =

  ′

  ′

  id Z Z ❡ f ◦ g = id ✳ ❈♦♥❝❧✉í♠♦s q✉❡ Z ≃ Z ✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳

  ❉❡✜♥✐çã♦ ✷✳✶✳✸ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛✳ ❯♠ ♦❜❥❡t♦ Z ∈ C é ❞✐t♦ ♦❜❥❡t♦ ③❡r♦✱ ♦✉ ♦❜❥❡t♦ ♥✉❧♦✱ s❡ Z é ♦❜❥❡t♦ ✐♥✐❝✐❛❧ ❡ ✜♥❛❧✳

  ◆♦ ❝❛s♦ ❞❡ Z s❡r ♦❜❥❡t♦ ③❡r♦✱ ♣❛r❛ X ∈ C✱ ❞❡♥♦t❛♠♦s ♣♦r

  X : X → Z X : Z → X

  ❡ 0 C (X, Z) C (Z, X)

  ♦s ú♥✐❝♦s ♠♦r✜s♠♦s ❡♠ Hom ❡ Hom ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆♦t❡♠♦s q✉❡ ♦❜❥❡t♦ ③❡r♦ é ✉♠ ❝♦♥❝❡✐t♦ ❛✉t♦✲❞✉❛❧✳ ❈♦r♦❧ár✐♦ ✷✳✶✳✹ ❙❡ ✉♠❛ ❝❛t❡❣♦r✐❛ C ♣♦ss✉✐ ✉♠ ♦❜❥❡t♦ ③❡r♦✱ ❡st❡ é ú♥✐❝♦✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ❖❜s❡r✈❛çã♦ ✷✳✶✳✺ ❉❡✈✐❞♦ ❛♦ ❝♦r♦❧ár✐♦ ❛❝✐♠❛✱ r❡❢❡r✐♠♦✲♥♦s ❛ ✉♠ ♦❜✲ ❥❡t♦ ③❡r♦ ✭q✉❛♥❞♦ ♦ ♠❡s♠♦ ❡①✐st✐r✮ ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ C ❝♦♠♦ ♦ ♦❜❥❡t♦ ③❡r♦ ❞❡ C✳

  ◆♦s ❡①❡♠♣❧♦s ❛❜❛✐①♦ ✈❡r✐✜q✉❡♠♦s s❡ ❛s ❝❛t❡❣♦r✐❛s q✉❡ ❝♦♥❤❡❝❡♠♦s ♣♦ss✉❡♠ ♦❜❥❡t♦ ③❡r♦✳ ❙❡ ✉♠❛ ❝❛t❡❣♦r✐❛ ♥ã♦ t❡♠ ♦❜❥❡t♦ ③❡r♦✱ ❞✐③❡♠♦s q✉❡♠ sã♦ ♦s ♦❜❥❡t♦s ✐♥✐❝✐❛✐s ❡ ✜♥❛✐s✳ ❊①❡♠♣❧♦ ✷✳✶✳✻ ❆ ❝❛t❡❣♦r✐❛ Set ♥ã♦ ♣♦ss✉✐ ♦❜❥❡t♦ ③❡r♦✳

  ❉❡ ❢❛t♦✱ ♦ ♦❜❥❡t♦ ✐♥✐❝✐❛❧ ❡♠ Set é ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ∅ ❡ ♦s ♦❜❥❡t♦s ✜♥❛✐s sã♦ ♦s ❝♦♥❥✉♥t♦s ✉♥✐tár✐♦s✳ ❊①❡♠♣❧♦ ✷✳✶✳✼ ◆❛ ❝❛t❡❣♦r✐❛ Rel✱ ♦ ♦❜❥❡t♦ ③❡r♦ é ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ∅✳ ❊①❡♠♣❧♦ ✷✳✶✳✽ ◆❛s ❝❛t❡❣♦r✐❛s Grp✱ Ab ❡ Div✱ ♦ ♦❜❥❡t♦ ③❡r♦ é ♦ ❣r✉♣♦ tr✐✈✐❛❧ {e}✳ ❊①❡♠♣❧♦ ✷✳✶✳✾ ◆❛ ❝❛t❡❣♦r✐❛ Ring✱ ♦ ♦❜❥❡t♦ ③❡r♦ é ♦ ❛♥❡❧ tr✐✈✐❛❧ {0}✳ ❊①❡♠♣❧♦ ✷✳✶✳✶✵ ❆s ❝❛t❡❣♦r✐❛s ring ❡ Cring ♥ã♦ tê♠ ♦❜❥❡t♦ ③❡r♦✳

  ❉❡ ❢❛t♦✱ ❡♠ ❛♠❜❛s ❝❛t❡❣♦r✐❛s✱ ♦ ♦❜❥❡t♦ ✐♥✐❝✐❛❧ é ♦ ❛♥❡❧ ❞♦s ✐♥t❡✐✲ r♦s Z ❡ ♦ ♦❜❥❡t♦ ✜♥❛❧ é ♦ ❛♥❡❧ tr✐✈✐❛❧ {0}✳ ❇❛st❛ ♦❜s❡r✈❛r♠♦s q✉❡ ring Cring R } R R

  Hom (Z, R) = Hom (Z, R) = {f (z) = z1 ✱ ❡♠ q✉❡ f ✱ ♣❛r❛ t♦❞♦ z ∈ Z✳

  ❊①❡♠♣❧♦ ✷✳✶✳✶✶ ❙❡❥❛♠ R ✉♠ ❛♥❡❧✱ k ✉♠ ❝♦r♣♦ ❡ A ✉♠❛ k✲á❧❣❡❜r❛✳ M M m

  R k k A A ◆❛s ❝❛t❡❣♦r✐❛s ✱ V ect ✱ vect ✱ ❡ ✱ ♦ ♦❜❥❡t♦ ③❡r♦ é ♦ ♠ó❞✉❧♦ tr✐✈✐❛❧ {0}✳ k ❊①❡♠♣❧♦ ✷✳✶✳✶✷ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❆ ❝❛t❡❣♦r✐❛ Alg ♥ã♦ t❡♠ ♦❜❥❡t♦ ③❡r♦✳

  ❉❡ ❢❛t♦✱ ♦ ♦❜❥❡t♦ ✐♥✐❝✐❛❧ é ♦ ❝♦r♣♦ k ❡ ♦ ♦❜❥❡t♦ ✜♥❛❧ é ❛ k✲á❧❣❡❜r❛ tr✐✈✐❛❧ {0}✳ ◆❛s ❝❛t❡❣♦r✐❛s ❞❡ ❣r✉♣♦s✱ ❛♥é✐s ❡ ♠ó❞✉❧♦s✱ sã♦ ❝♦♠✉♥s ♦s ❤♦♠♦✲

  ♠♦r✜s♠♦s tr✐✈✐❛✐s✳ ❚❛✐s ❤♦♠♦♠♦r✜s♠♦s ♣♦❞❡♠ s❡r ❣❡♥❡r❛❧✐③❛❞♦s ♣❛r❛ ❝❛t❡❣♦r✐❛s ❝♦♠ ♦❜❥❡t♦ ③❡r♦✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✸ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C✳ P❛r❛

  X X, Y ∈ C : X → Y

  ✱ ♦ ♠♦r✜s♠♦ ♥✉❧♦ ❞❡ X ❡♠ Y ✱ ❞❡♥♦t❛❞♦ ♣♦r 0 Y ✱ é ❞❡✜♥✐❞♦ ♣❡❧♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦

  

X

Y

  X Y

  X Y

  Z ♦✉ s❡❥❛✱ ❡♠ t❡r♠♦s ❞❡ ❝♦♠♣♦s✐çã♦✱ t❡♠♦s

  X X ◦ 0 = 0 Y .

  Y

  X : X → Y

  Pr♦♣♦s✐çã♦ ✷✳✶✳✶✹ ❖ ♠♦r✜s♠♦ ♥✉❧♦ 0 Y ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ ♦❜❥❡t♦ ③❡r♦✳

  ′ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ Z, Z ♦❜❥❡t♦s ③❡r♦s✳ P❡❧♦ ❈♦r♦❧ár✐♦

  X ′ ′

  C (X, Z ) = {φ } ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ι : Z → Z ✳ ❙❡❥❛♠ Hom ❡

  X ′

  ′ −1 Hom C (Z , Y ) = {φ Y } ∈ Hom C (X, Z ) Y ◦ ι ∈

  ✳ ❚❡♠♦s q✉❡ ι ◦ 0 ❡ 0

  X X ′ −1

  Hom C (Z , Y ) = φ Y ◦ ι = φ Y ✳ ▲♦❣♦✱ ι◦0 ❡ 0 ✳ P♦❞❡♠♦s ♥♦s ❜❛s❡❛r

  ♥♦ ❞✐❛❣r❛♠❛

  

X

Y

  X Y

  X Y

  Z

  X

  φ φ

  Y

  ι ′

  Z ❝✉❥♦s tr✐â♥❣✉❧♦s ❝♦♠✉t❛♠✳ ❊♥tã♦

  X X = Y ◦ 0

  Y

  −1

  X = Y ◦ ι ◦ ι ◦ 0

  X = φ Y ◦ φ .

  ➱ ❞❡ s❡ ❡s♣❡r❛r q✉❡ ❛ ❝♦♠♣♦s✐çã♦ ❝♦♠ ♠♦r✜s♠♦s ♥✉❧♦s r❡s✉❧t❡♠ ❡♠ ♠♦r✜s♠♦s ♥✉❧♦s✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✺ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C✱ f : X → Y

  ✉♠ ♠♦r✜s♠♦ ❡♠ C ❡ W ∈ C✳ ❊♥tã♦✱ t❡♠♦s W W Y

  X f ◦ 0 = 0 ◦ f = 0 .

  X Y ❡ 0 W W ❉❡♠♦♥str❛çã♦✿ P♦❞❡♠♦s ♥♦s ❜❛s❡❛r ♥♦ ❞✐❛❣r❛♠❛

  W

  f

  X W

  X Y

  X W Y

  Z X ∈ Hom C (Z, Y ) =

  ❝✉❥♦s tr✐â♥❣✉❧♦s ❝♦♠✉t❛♠✳ ❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ f ◦0 {0 Y } X = 0 Y

  ✳ ▲♦❣♦✱ f ◦ 0 ✳ ❊♥tã♦ W W

  ◦ 0 f ◦ 0 = f ◦ 0

  X X W ◦ 0

  = Y W = . Y W W Y

  X = 0 ◦ f = 0

  P♦rt❛♥t♦✱ f ◦ 0 ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ 0 ✳

  X Y W W ❖❜s❡r✈❛çã♦ ✷✳✶✳✶✻ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ❡ Y ∈

  Z Y Y Z Z C = 0 Y = 0 = 0 Y ◦ 0 = 0 Y ◦ id Z = 0 Y

  ✳ ❊♥tã♦ 0 ❡ 0 ✳ ❉❡ ❢❛t♦✱ 0 ✳ Y Z Y Y Y

  = 0 ❆♥❛❧♦❣❛♠❡♥t❡✱ 0 ✳

  Z ❉❡✜♥✐çã♦ ✷✳✶✳✶✼ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ ❡ f : X → Y

  ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ✭✐✮ ❯♠ ♥ú❝❧❡♦ ❞❡ f é ✉♠ ♣❛r (K, k)✱ ❡♠ q✉❡ K ∈ C ❡ k : K → X é ✉♠

  K ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ f ◦ k = 0 ❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧

  Y ′ ′ ′ ′ ′

  , k ) ∈ C : K → X é s❛t✐s❢❡✐t❛✿ ♣❛r❛ q✉❛❧q✉❡r ♣❛r (K ✱ ❡♠ q✉❡ K ❡ k

  ′

  K ′

  = 0 é ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ f ◦ k Y ✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦

  ′ ′ u : K → K = k ◦ u t❛❧ q✉❡ k ✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛

  K′ Y

  ′ K

  ′

  k f u

  X Y k

  K K

  Y

  é ❝♦♠✉t❛t✐✈♦✳ ✭✐✐✮ ❯♠ ❝♦♥ú❝❧❡♦ ❞❡ f é ✉♠ ♣❛r (Q, q)✱ ❡♠ q✉❡ Q ∈ C ❡ q : Y → Q é ✉♠

  X ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ q ◦ f = 0 Q ❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧

  ′ ′ ′ ′ ′ , q ) ∈ C : Y → Q

  é s❛t✐s❢❡✐t❛✿ ♣❛r❛ q✉❛❧q✉❡r ♣❛r (Q ✱ ❡♠ q✉❡ Q ❡ q

  X ′

  ◦ f = 0 ′ é ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ q Q ✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦

  ′ ′ u : Q → Q = u ◦ q t❛❧ q✉❡ q ✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛

  X Q

  Q q f u

  X Y

  ′

  q ′

  X Q Q′

  é ❝♦♠✉t❛t✐✈♦✳ ◆♦t❡♠♦s q✉❡ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦ sã♦ ❝♦♥❝❡✐t♦s ❞✉❛✐s✳ ❊♠ ♦✉tr❛s ♣❛✲

  ❧❛✈r❛s✱ ✉♠ ♥ú❝❧❡♦ ❞❡ ✉♠ ♠♦r✜s♠♦ ❡♠ C é ✉♠ ❝♦♥ú❝❧❡♦ ❞❡ss❡ ♠♦r✜s♠♦ op

  ❡♠ C ✳ ❖❜s❡r✈❛çã♦ ✷✳✶✳✶✽ ❱❛❧❡ ♥♦t❛r♠♦s q✉❡ s❡ ❛♣❧✐❝❛r♠♦s ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♥ú❝❧❡♦ (K, k) ♣❛r❛ ♦ ♣❛r (K, k)✱ ♦ ú♥✐❝♦ ♠♦r✜s♠♦ q✉❡ s❡ ♦❜té♠ ❞❡ K

  K : K → K ♣❛r❛ K é ♦ ♠♦r✜s♠♦ id ✳ ❯s❛♠♦s ❡ss❡ ❢❛t♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✾ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ ❡ f ✉♠ ♠♦r✜s♠♦ ❡♠ C✳

  ′ ′ , k )

  ✭✐✮ ❙❡ (K, k) ❡ (K sã♦ ♥ú❝❧❡♦s ❞❡ f✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✲ ′ ′

  ◦ u ✜s♠♦ u : K → K t❛❧ q✉❡ k = k ✱ ✐st♦ é✱ ♥ú❝❧❡♦s ❞❡ f sã♦ ú♥✐❝♦s✱ s❛❧✈♦ ✐s♦♠♦r✜s♠♦✳

  ′ ′ , q )

  ✭✐✐✮ ❙❡ (Q, q) ❡ (Q sã♦ ❝♦♥ú❝❧❡♦s ❞❡ f✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦✲ ′ ′

  → Q ♠♦r✜s♠♦ u : Q t❛❧ q✉❡ q = u ◦ q ✱ ✐st♦ é✱ ❝♦♥ú❝❧❡♦s ❞❡ f sã♦ ú♥✐❝♦s✱ s❛❧✈♦ ✐s♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❆♣❧✐❝❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ♥ú❝❧❡♦s (K, k) ❡

  ′ ′ ′ ′ (K , k ) , k )

  ♣❛r❛ ♦s ♣❛r❡s (K ❡ (K, k)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡①✐st❡♠ ú♥✐❝♦s ′ ′ ′ ′

  → K ◦ u = k ◦ v ♠♦r✜s♠♦s v : K ❡ u : K → K t❛✐s q✉❡ k = k ❡ k ✳ ❖❜s❡r✈❛♠♦s q✉❡ ♦ ♠♦r✜s♠♦ v ◦ u : K → K s❛t✐s❢❛③

  ′ ◦ u = k. k ◦ (v ◦ u) = (k ◦ v) ◦ u = k

  K = k K ❈♦♠♦ k ◦ id ✱ s❡❣✉❡ ❞❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛ q✉❡ v ◦ u = id ✳

  ′

  ′

  K ❆♥❛❧♦❣❛♠❡♥t❡✱ u ◦ v = id ✳ ❆ss✐♠✱ u : K → K é ♦ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦

  ′ ◦ u t❛❧ q✉❡ k = k ✳ op op

  ′ ′ , q )

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ■st♦ é✱ ❡♠ C ✱ (Q, q) ❡ (Q sã♦ ♥ú❝❧❡♦s ′

  op

  C (Q, Q ) = ❞❡ f✳ P❡❧♦ ✐t❡♠ ✭✐✮✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ u ∈ Hom op

  ′ ′ ′ Hom C (Q , Q) ◦ u = u ◦ q t❛❧ q✉❡ q = q ✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦

  ′ ′ → Q

  ✐s♦♠♦r✜s♠♦ u : Q t❛❧ q✉❡ q = u ◦ q ✳ ❖❜s❡r✈❛çã♦ ✷✳✶✳✷✵ ❉❡✈✐❞♦ à ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ r❡❢❡r✐♠♦✲♥♦s ❛ ✉♠ ♥ú❝❧❡♦ ❡ ❛ ✉♠ ❝♦♥ú❝❧❡♦ ❞❡ f ❝♦♠♦ ♦ ♥ú❝❧❡♦ ❡ ♦ ❝♦♥ú❝❧❡♦ ❞❡ f✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡✳ ❯s❛♠♦s ❛s ♥♦t❛çõ❡s (Ker(f), k) ❡ (Cok(f), q) ♦✉ (Ker f, k) ❡ (Cok f, q) ♣❛r❛ ♦ ♥ú❝❧❡♦ ❡ ♦ ❝♦♥ú❝❧❡♦ ❞❡ f✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❛♠❜é♠ é ❝♦♠✉♠ r❡❢❡r✐r♠♦s ❛♦ ♥ú❝❧❡♦ ❡ ❛♦ ❝♦♥ú❝❧❡♦ ❞❡ f ❝♦♠♦ ♦s ♠♦r✜s♠♦s k ❡ q✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✷✶ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ ❡ f : X → Y

  ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ✭✐✮ ❙❡ f ♣♦ss✉✐ ♥ú❝❧❡♦ (K, k)✱ ❡♥tã♦ k : K → X é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ ✭✐✐✮ ❙❡ f ♣♦ss✉✐ ❝♦♥ú❝❧❡♦ (Q, q)✱ ❡♥tã♦ q : Y → Q é ✉♠ ❡♣✐♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙❡❥❛♠ g, h : Z → K ♠♦r✜s♠♦s ❡♠ C t❛✐s q✉❡ k ◦ g = k ◦ h

  ✳ ❖ ♠♦r✜s♠♦ k ◦ g : Z → X é t❛❧ q✉❡ K Z f ◦ (k ◦ g) = (f ◦ k) ◦ g = 0 ◦ g = 0 . Y Y

  ❆♣❧✐❝❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ♥ú❝❧❡♦ (K, k) ♣❛r❛ ♦ ♣❛r (Z, k◦

  g) ✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u : Z → K t❛❧ q✉❡ k ◦ g = k ◦ u✳ ❊♥tã♦ k ◦ g = k ◦ h = k ◦ u ✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❡ u✱ u = g = h✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ❖s ❡①❡♠♣❧♦s ❛❜❛✐①♦ ♠♦str❛♠ ❝❛t❡❣♦r✐❛s ❝✉❥♦s ♠♦r✜s♠♦s tê♠ ♥ú✲

  ❝❧❡♦s ❡ ❝♦♥ú❝❧❡♦s✳ ❊①❡♠♣❧♦ ✷✳✶✳✷✷ ❊♠ Grp✱ t♦❞♦ ♠♦r✜s♠♦ ♣♦ss✉✐ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦✳

  ❉❡ ❢❛t♦✱ s❡❥❛ f : G → H ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✳ ❙❛❜❡♠♦s }

  H q✉❡ K = {x ∈ G : f(x) = e é ✉♠ s✉❜❣r✉♣♦ ❞❡ G✳ ❖ ♣❛r

  (K, k), ❡♠ q✉❡ k : K → G

  é ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✱ é ♦ ♥ú❝❧❡♦ ❞❡ f✳ H

  ❆❣♦r❛✱ ♣❛r❛ K ✉♠ s✉❜❣r✉♣♦ ❞❡ H✱ ❞❡♥♦t❛♠♦s ♣♦r K ❛ ✐♥t❡rs❡çã♦ H

  ❞❡ t♦❞♦s ♦s s✉❜❣r✉♣♦s ♥♦r♠❛✐s ❡♠ H q✉❡ ❝♦♥tê♠ K✳ ▲♦❣♦✱ K é ♦ ♠❡♥♦r s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❡♠ H q✉❡ ❝♦♥té♠ K✳ ❖ ♣❛r

  H H (H/f (G) , q),

  ❡♠ q✉❡ q : H → H/f(G) é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✱ é ♦ ❝♦♥ú❝❧❡♦ ❞❡ f✳

  ′ ′ ′ : H → Q ◦ f

  ❉❡ ❢❛t♦✱ s❡❥❛ q ✉♠ ♠♦r✜s♠♦ ❡♠ Grp t❛❧ q✉❡ q ′ ′

  ) ) é ♦ ♠♦r✜s♠♦ tr✐✈✐❛❧✳ ❊♥tã♦ f(G) ⊆ Ker(q ✳ ❈♦♠♦ Ker(q é ✉♠

  H s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ H ❡ f(G) é ♦ ♠❡♥♦r s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❝♦♥t❡♥❞♦

  H ′ f (G) ⊆ Ker(q )

  ✱ f(G) ✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ q✉♦❝✐❡♥t❡✱ H

  ′ → Q

  ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s u : H/f(G) t❛❧ q✉❡ ′ q = u ◦ q

  ✳ ❊①❡♠♣❧♦ ✷✳✶✳✷✸ ❊♠ Ring✱ t♦❞♦ ♠♦r✜s♠♦ ♣♦ss✉✐ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦✳

  ❉❡ ❢❛t♦✱ s❡❥❛ f : R → S ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s✳ ❙❛❜❡♠♦s q✉❡ }

  I = {r ∈ R : f (r) = 0 S é ✉♠ ✐❞❡❛❧ ❞❡ R ❡ J = Sf(R)S é ♦ ✐❞❡❛❧ ❞❡ S

  ❣❡r❛❞♦ ♣♦r f(R)✳ ❖s ♣❛r❡s (I, k)

  ❡ (J, q) sã♦ ♦ ♥ú❝❧❡♦ ❡ ♦ ❝♦♥ú❝❧❡♦ ❞❡ f✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♠ q✉❡ k : I → R

  ❡ q : S → S/J sã♦ ❛ ✐♥❝❧✉sã♦ ❡ ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ′ ′

  : S → Q ❱❡r✐✜q✉❡♠♦s ♦ ❝♦♥ú❝❧❡♦✳ ❙❡❥❛ q ✉♠ ♠♦r✜s♠♦ ❡♠ Ring

  ′ ′ ′ ◦ f = 0 ) ) t❛❧ q✉❡ q ✳ ❊♥tã♦ f(R) ⊆ Ker(q ✳ ❈♦♠♦ Ker(q é ✉♠ ✐❞❡❛❧✱

  ′ ) s❡❣✉❡ q✉❡ J ⊆ Ker(q ✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ q✉♦❝✐❡♥t❡✱ ❡①✐st❡

  ′ ′ = u ◦ q

  ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s u : S/J → Q t❛❧ q✉❡ q ✳

  M R ❊①❡♠♣❧♦ ✷✳✶✳✷✹ ❊♠ ✱ t♦❞♦ ♠♦r✜s♠♦ ♣♦ss✉✐ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦✳

  ❉❡ ❢❛t♦✱ s❡❥❛ f : M → N ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s à ❡s✲ N } q✉❡r❞❛✳ ❙❛❜❡♠♦s q✉❡ P = {m ∈ M : f(m) = 0 ❡ f(M) sã♦ R✲ s✉❜♠ó❞✉❧♦s ❞❡ M ❡ N✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ♣❛r❡s

  (P, k) ❡ (N/f(M), q) sã♦ ♦ ♥ú❝❧❡♦ ❡ ♦ ❝♦♥ú❝❧❡♦ ❞❡ f✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♠ q✉❡ k : P → M

  ❡ q : N → N/f(M) sã♦ ❛ ✐♥❝❧✉sã♦ ❡ ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛♠ ♦s ♥ú❝❧❡♦s ❡ ❝♦♥ú❝❧❡♦s ❞❡ ♠♦✲

  ♥♦♠♦r✜s♠♦s✱ ❡♣✐♠♦r✜s♠♦s✱ ♠♦r✜s♠♦s ✐❞❡♥t✐❞❛❞❡ ❡ ♠♦r✜s♠♦s ♥✉❧♦s✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✷✺ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C ❡ f : X → Y

  ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ X )

  ✭✐✮ ❙❡ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ ❡♥tã♦ (Z, 0 é ♦ ♥ú❝❧❡♦ ❞❡ f✳ Y

  ) ✭✐✐✮ ❙❡ f é ✉♠ ❡♣✐♠♦r✜s♠♦✱ ❡♥tã♦ (Z, 0 é ♦ ❝♦♥ú❝❧❡♦ ❞❡ f✳

  Z X = 0 Y Y = 0

  ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❚❡♠♦s q✉❡ f ◦ 0 ❡ ❥á ✈✐♠♦s q✉❡ 0 Y ✳

  ′

  ′ ′ ′ K : K → X = 0

  ❙❡❥❛ k ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ f ◦ k Y ✳ ❊♥tã♦

  ′ ′

  K K

  ′ ′ f ◦ k = f ◦ 0 = 0

  X ❡ ❝♦♠♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦s k X ✱ ♦✉

  ′ ′

  K K ′ ′

  = 0 X ◦ 0 : K → Z s❡❥❛✱ k ✳ ❈♦♠♦ 0 é ♦ ú♥✐❝♦ ♠♦r✜s♠♦ ❡♠ ′

  Hom C (K , Z) X ) ✱ ❝♦♥❝❧✉í♠♦s q✉❡ (Z, 0 é ♦ ♥ú❝❧❡♦ ❞❡ f✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ❈♦r♦❧ár✐♦ ✷✳✶✳✷✻ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C ❡ X ∈ C X : X → X

  ✳ ❖ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ id t❡♠ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦

  X X ) ) ❞❛❞♦s ♣♦r (Z, 0 ❡ (Z, 0 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✷✼ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C ❡

  X X, Y ∈ C : X → Y ✳ ❖ ♠♦r✜s♠♦ ♥✉❧♦ 0 Y t❡♠ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦ ❞❛❞♦s

  X ) Y ) ♣♦r (X, id ❡ (Y, id ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  X X ′ ′

  ◦ id X = 0 : K → X ❉❡♠♦♥str❛çã♦✿ ❈❧❛r❛♠❡♥t❡✱ 0 ✳ ❙❡❥❛ k

  Y Y

  ′

  X K ′ ′ ′

  ◦ k = 0 = id X ◦ k ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ 0 ✳ ❊♥tã♦ k ✳ ❙❡

  Y Y ′ ′

  α : K → X = id X ◦ α é ♦✉tr♦ ♠♦r✜s♠♦ t❛❧ q✉❡ k ✱ ❡♥tã♦ α = k ✳ ▲♦❣♦✱

  X X (X, id X ) Y )

  é ♦ ♥ú❝❧❡♦ ❞❡ 0 ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ (Y, id é ♦ ❝♦♥ú❝❧❡♦ ❞❡ 0 ✳ Y Y

  ✷✳✷ ❈❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s

  ❉❡✜♥✐çã♦ ✷✳✷✳✶ ❯♠❛ ❝❛t❡❣♦r✐❛ C é ❞✐t❛ ♣ré✲❛❞✐t✐✈❛ s❡ ✭✐✮ C ♣♦ss✉✐ ♦❜❥❡t♦ ③❡r♦❀

  C (X, Y ) ✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ 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 ✱ t❡♠✲s❡ ′ ′ g ◦ (f + f ) = g ◦ f + g ◦ f

  ′ ′ (g + g ) ◦ f = g ◦ f + g ◦ f. op

  ◆♦t❡♠♦s q✉❡ s❡ C, D sã♦ ❝❛t❡❣♦r✐❛s ♣ré✲❛❞✐t✐✈❛s✱ ❡♥tã♦ C ❡ C × D t❛♠❜é♠ ♦ sã♦✱ ✐ss♦ s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❛s ♠❡s♠❛s✳ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❡ X, Y ∈ C✳ ❉❡♥♦t❛♥❞♦ ♣♦r e Hom

  C (X, Y )

  

C (X,Y ) ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞♦ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ Hom ✱ ❡♥✉♥✲

  ❝✐❛♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳

  X = 0

  Pr♦♣♦s✐çã♦ ✷✳✷✳✷ ❈♦♠ ❛ ♥♦t❛çã♦ ❛❝✐♠❛✱ e Hom C (X,Y ) ✳ Y

  C (X, Z) = ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s Z ♦ ♦❜❥❡t♦ ③❡r♦ ❞❡ C✳ ❊♥tã♦ Hom

  X X

  X X

  X X {0 } ∈ Hom }

  • 0 C (X, Z) = {0 + 0 = ❡ ♣♦rt❛♥t♦✱ 0 ✱ ♦✉ s❡❥❛✱ 0

  X ✳ ▲♦❣♦✱

  X X = Y ◦ 0

  Y

  X X = Y ◦ (0 + 0 )

  X X ◦ 0 ◦ 0

  = Y + 0 Y

  X X = + 0 . Y Y

  X X

  X X = 0 + 0 C (X, Y )

  ▲♦❣♦✱ 0 Y Y Y ✳ ❙♦♠❛♥❞♦ ♦ ♦♣♦st♦ ❞❡ 0 Y ❡♠ Hom ✱

  X Hom = 0 ♦❜t❡♠♦s e C (X,Y ) Y ✳

  ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C✳ ▼♦str❛♠♦s ♥❛ Pr♦✲ ♣♦s✐çã♦ q✉❡ s❡ f : X → Y é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C✱ ❡♥tã♦ f

  X ) t❡♠ ♥ú❝❧❡♦ (Z, 0 ✳ ❯♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♥t❡r❡ss❛♥t❡ ❞❡ ❝❛t❡❣♦r✐❛s ♣ré✲ ❛❞✐t✐✈❛s é q✉❡ ✈❛❧❡ ❛ r❡❝í♣r♦❝❛ ❞❡ss❡ r❡s✉❧t❛❞♦✳ P❛r❛ ✈❡r ✐ss♦✱ ♠♦str❛♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✸ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ✈á❧✐❞❛s✿

  ✭✐✮ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ ♠♦r✜s♠♦ g : W W

  W → X ❡♠ C ❝♦♠ f ◦ g = 0 Y ✐♠♣❧✐❝❛ g = 0 X ❀

  ✭✐✐✮ f é ✉♠ ❡♣✐♠♦r✜s♠♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ ♠♦r✜s♠♦ g : Y →

  X Y W ❝♦♠ g ◦ f = 0 W ✐♠♣❧✐❝❛ g = 0 W ✳

  W ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ✭⇒✮ ❙❡❥❛ g : W → X t❛❧ q✉❡ f ◦ g = 0 ✳ ❊♥tã♦

  Y W W f ◦ g = f ◦ 0

  ❡ ❝♦♠♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ g = 0 ✳

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

  W W ❊♥tã♦ f ◦ (g − h) = 0 Y ✳ P♦r ❤✐♣ót❡s❡✱ g − h = 0 X ❡ ♣♦rt❛♥t♦✱

  W g = h + 0 = h

  X ✳ ▲♦❣♦✱ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ op ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ❈♦r♦❧ár✐♦ ✷✳✷✳✹ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C

  ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ X )

  ✭✐✮ ❙❡ (Z, 0 é ♦ ♥ú❝❧❡♦ ❞❡ f✱ ❡♥tã♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ Y

  ) ✭✐✐✮ ❙❡ (Z, 0 é ♦ ❝♦♥ú❝❧❡♦ ❞❡ f✱ ❡♥tã♦ f é ✉♠ ❡♣✐♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙❡❥❛ g : W → X ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡

  W f ◦ g = 0 X )

  Y ✳ ❈♦♠♦ (Z, 0 é ♦ ♥ú❝❧❡♦ ❞❡ f✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ W

  ◦ k = 0 k : W → Z

  X t❛❧ q✉❡ g = 0 X ✳ ❉❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ s❡❣✉❡ q✉❡ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ❉❛❞❛ ❛ ❡str✉t✉r❛ ❛❞✐❝✐♦♥❛❧ q✉❡ ❛s ❝❛t❡❣♦r✐❛s ♣ré✲❛❞✐t✐✈❛s ♣♦ss✉❡♠✱

  ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r s✉❜❝❛t❡❣♦r✐❛s q✉❡ tê♠ t❛❧ ❡str✉t✉r❛ ❡ ❢✉♥t♦r❡s q✉❡ ♣r❡s❡r✈❛♠ ❛ ♠❡s♠❛✳ ❉❡✜♥✐çã♦ ✷✳✷✳✺ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C

  ✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ D ❞❡ C é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣ré✲ D (X, Y ) C (X, Y )

  ❛❞✐t✐✈❛ s❡ Z ∈ D ❡ Hom é ✉♠ s✉❜❣r✉♣♦ ❞❡ Hom ✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ D✳ ❉❡✜♥✐çã♦ ✷✳✷✳✻ ❙❡❥❛♠ C, D ❝❛t❡❣♦r✐❛s ♣ré✲❛❞✐t✐✈❛s✳ ❯♠ ❢✉♥t♦r F : C

  → D C (X, Y )

  é ❞✐t♦ ❛❞✐t✐✈♦ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C ❡ f, g ∈ Hom ✱ t❡♠✲s❡ F (f + g) = F (f ) + F (g),

  C (X, Y ) → Hom D (F (X), F (Y )) ♦✉ s❡❥❛✱ ❛ ❛♣❧✐❝❛çã♦ F : Hom é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✳

  ❙❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ❡ ❞❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r q✉❡✱ ♣❛r❛ q✉❛✐s✲ F

  X (X) ) = 0 q✉❡r X, Y ∈ C✱ F (0 Y F ✳

  (Y ) ❊①❡♠♣❧♦s ❞❡ ❢✉♥t♦r❡s ❛❞✐t✐✈♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ❛♣ós ❞❡✜♥✐r♠♦s ❛s

  ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞❛ ❞❡✜♥✐çã♦ ❞❡ s♦♠❛ ❞✐r❡t❛ ❞❡ ♦❜❥❡t♦s✳ ❉❡✜♥✐çã♦ ✷✳✷✳✼ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❡ X, Y ∈ C✳ ❯♠❛

  X , p Y , i X , i Y ) s♦♠❛ ❞✐r❡t❛ ❞❡ X ❡ Y é ✉♠❛ q✉í♥t✉♣❧❛ (S, p ✱ ❡♠ q✉❡ S ∈ C

  X : S → X Y : S → Y X : X → S Y : Y → S ❡ p ✱ p ✱ i ✱ i sã♦ ♠♦r✜s♠♦s ❡♠ C q✉❡ s❛t✐s❢❛③❡♠

  ◦ i ◦ i ◦ p ◦ p p

  X X = id X , p Y Y = id Y

  X X + i Y Y = id S .

  ❡ i ◦i ◦i

  X X = id

  X Y Y = id Y ◆♦t❡♠♦s q✉❡ ❛s r❡❧❛çõ❡s p ❡ p ✐♠♣❧✐❝❛♠ q✉❡ i

  X Y

  X Y ✱ i sã♦ ♠♦♥♦♠♦r✜s♠♦s ❡ p ✱ p sã♦ ❡♣✐♠♦r✜s♠♦s✳ ❚❛✐s ♠♦r✜s♠♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ✐♥❝❧✉sõ❡s ❡ ♣r♦❥❡çõ❡s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ➱ ✐♠❡❞✐❛t♦ op

  X , i Y , p X , p Y ) ✈❡r✐✜❝❛r q✉❡ (S, i é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ X ❡ Y ❡♠ C ✳ ❱❛♠♦s ✉s❛r ❡ss❡ ❢❛t♦ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✽ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❡ X, Y ∈ C t❛✐s

  X , p Y , i X , i Y ) q✉❡ ❡①✐st❡ ✉♠❛ s♦♠❛ ❞✐r❡t❛ (S, p ✳ ❊♥tã♦

  X Y p Y ◦ i X = 0 X ◦ i Y = 0 .

  Y ❡ p

  X ❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s p Y ◦ i

  X + p Y ◦ i X = p Y ◦ i X ◦ id X + id Y ◦ p Y ◦ i

  X = p Y ◦ i X ◦ p X ◦ i X + p Y ◦ i Y ◦ p Y ◦ i

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

  X = p Y ◦ id S ◦ i

  X = p Y ◦ i X .

  Y ◦ i X + p Y ◦ i X = p Y ◦ i

  X Y ◦ i

  X ▲♦❣♦✱ p ✳ ❙♦♠❛♥❞♦ ♦ ♦♣♦st♦ ❞❡ p ❡♠

  X Y Hom C (X, Y ) Y ◦ i X = 0 X ◦ i Y = 0

  ✱ ♦❜t❡♠♦s p Y ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ p X ✳ ❆ ✉♥✐❝✐❞❛❞❡ ❞❛ s♦♠❛ ❞✐r❡t❛✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✱ ❡stá ❣❛r❛♥✲ t✐❞❛ ♣❡❧❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✳ ◆❛ ✈❡r❞❛❞❡✱ ♠♦str❛♠♦s q✉❡ ❞❛❞❛s ❞✉❛s s♦♠❛s ❞✐r❡t❛s ❞❡ ♦❜❥❡t♦s ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ q✉❡ r❡s♣❡✐t❛ ♦s ♠♦r✜s♠♦s ❞❡ ✐♥❝❧✉sã♦ ❡ ♣r♦❥❡çã♦✳

  ✳

  ◦ µ = p Y é ♠♦str❛❞♦ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳

  ′ Y ✳

  ❆♥❛❧♦❣❛♠❡♥t❡✱ µ ◦ i Y = i

  X .

  ′

  X ◦ id X = i

  ◦ p Y ◦ i X = i ′

  ′ Y

  X ◦ p X ◦ i X + i

  X = i ′

  ▲♦❣♦✱ ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ❞❛ ♣r♦♣♦s✐çã♦ ❝♦♠✉t❛✳ ▼♦str❛♠♦s ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛✳ ❚❡♠♦s u ◦ i

  ′ Y

  ′

  X ❡ p

  X ◦ µ = p

  P♦rt❛♥t♦✱ p ′

  X = p X .

  = p X + 0 S

  X ◦ p Y

  Y

  ◦ p Y = id X ◦ p X + 0

  ′ Y

  X ◦ i

  ▼♦str❡♠♦s q✉❡ µ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❚❡♠♦s q✉❡ i X ◦ p

  X , i Y ◦ p

  X ◦ p X + p

  ) ◦ µ = i

  ′

  ❡ ❛♥❛❧♦❣❛♠❡♥t❡✱ µ ◦ ν = id S

  ❡♠ ✭∗✮ ✉s❛♠♦s ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛✳ P♦rt❛♥t♦✱ ν ◦ µ = id S

  = i X ◦ p X + i Y ◦ p Y = id S ,

  ◦ µ (∗)

  ′ Y

  Y ◦ p

  X ◦ µ + i

  ′

  X ◦ p

  ′ Y

  ′ Y

  X

  ′

  X ◦ p

  ν ◦ µ = (i

  ′ Y ✳ ❊♥tã♦

  X

  X ◦ p ′

  → S ♣♦r ν = i

  ✳ ❉❡✜♥✐♠♦s ν : S ′

  ′ , S)

  ∈ Hom C (S

  ′

  ′

  Pr♦♣♦s✐çã♦ ✷✳✷✳✾ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ ❡ X, Y ∈ C✳ ❙❡✲ ❥❛♠ (S, p

  X

  X

  X Y µ i

  S S ′

  ′ Y

  p

  X

  ′

  p

  Y

  p

  X Y µ p

  Y

  S S ′

  ′ t❛❧ q✉❡ ♦s ❞✐❛❣r❛♠❛s

  ) s♦♠❛s ❞✐r❡t❛s ❞❡ X ❡ Y ✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ µ : S → S

  ′ Y

  X , i

  , i ′

  ′ Y

  X , p

  ′

  ′ , p

  X , p Y , i X , i Y ) ❡ (S

  i

  i

  X ◦ i

  ′ Y

  ′

  ◦ p Y ) = p

  ′ Y

  X ◦ p X + i

  ′

  X ◦ (i

  ′

  X ◦ µ = p

  ′

  ◦ p Y ✳ ❊♥tã♦ p

  X ◦ p X + i

  ′

  ♣♦r µ = i ′

  µ : S → S ′

  ) ✳ ❉❡✜♥✐♠♦s

  ◦p Y ∈ Hom C (S, S ′

  ′ Y

  X ◦p X , i

  ′

  s❡❥❛♠ ❝♦♠✉t❛t✐✈♦s✳ ❚❛❧ ✐s♦♠♦r✜s♠♦ é ú♥✐❝♦ ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♣r✐✲ ♠❡✐r♦ ♦✉ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛s ❝♦♠✉t❛r❡♠✳ ❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ i

  ′ Y

  i

  X

  • i Y ◦ p
  • i Y ◦ p

  ❘❡st❛✲♥♦s ♠♦str❛r q✉❡ ♦ ♠♦r✜s♠♦ µ é ♦ ú♥✐❝♦ ❝♦♠✉t❛♥❞♦ ♦ ♣r✐♠❡✐r♦ ′

  ♦✉ ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛s✳ ❙❡❥❛ η : S → S t❛❧ q✉❡ ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ′ ′

  ◦ η = p X ◦ η = p Y ❝♦♠✉t❛ ❝♦♠ η ♥♦ ❧✉❣❛r ❞❡ µ✱ ✐st♦ é✱ p X ❡ p Y ✳ ❚❡♠♦s

  ′

  η = id S ◦ η ′ ′ ′ ′

  = (i ◦ p + i ◦ p ) ◦ η

  X X Y Y ′ ′ ′ ′

  = i ◦ p ◦ η + i ◦ p ◦ η

  X X Y Y ′ ′

  = i ◦ p X + i ◦ p Y

  X Y = µ.

  ❖ ❝❛s♦ ❡♠ q✉❡ η é t❛❧ q✉❡ ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ ❝♦♠ η ♥♦ ❧✉❣❛r ❞❡ µ é ❛♥á❧♦❣♦✳ ❖❜s❡r✈❛çã♦ ✷✳✷✳✶✵ ❉❡✈✐❞♦ à ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ r❡❢❡r✐♠♦✲♥♦s ❛ ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ X ❡ Y ❝♦♠♦ ❛ s♦♠❛ ❞✐r❡t❛ ❞❡ X ❡ Y ✳ ➱ ❝♦♠✉♠ ❛ ♥♦t❛çã♦ (X ⊕ Y, p X , p Y , i X , i Y )

  ✱ ❛ss✐♠ ❝♦♠♦ é ❝♦♠✉♠ r❡❢❡r✐r♠♦s à s♦♠❛ ❞✐r❡t❛ X , p Y , i X , i Y

  ❞❡ X ❡ Y ❝♦♠♦ ♦ ♦❜❥❡t♦ X ⊕ Y ✱ ✜❝❛♥❞♦ ♦s ♠♦r✜s♠♦s p s✉❜❡♥t❡♥❞✐❞♦s✳ ❉❡✜♥✐çã♦ ✷✳✷✳✶✶ ❯♠❛ ❝❛t❡❣♦r✐❛ C é ❞✐t❛ ❛❞✐t✐✈❛ s❡ ✭✐✮ C é ♣ré✲❛❞✐t✐✈❛❀ ✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ ❡①✐st❡ ❛ s♦♠❛ ❞✐r❡t❛ X ⊕ Y ∈ C✳ ❉❡✜♥✐çã♦ ✷✳✷✳✶✷ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ♣ré✲❛❞✐t✐✈❛ D ❞❡ C é ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ s❡ D é ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛✳

  ❆❜❛✐①♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✳ ❊s❝r❡✈❡♠♦s ❛ s♦♠❛ ❞✐r❡t❛ s✐♠♣❧❡s♠❡♥t❡ ♣❡❧♦ ♦❜❥❡t♦ X ⊕ Y ✱ ✜❝❛♥❞♦ s✉❜❡♥t❡♥❞✐❞❛s ❛s ✐♥✲ ❝❧✉sõ❡s ❡ ♣r♦❥❡çõ❡s✳ ❊①❡♠♣❧♦ ✷✳✷✳✶✸ ❆s ❝❛t❡❣♦r✐❛s Ab ❡ Div sã♦ ❛❞✐t✐✈❛s✳

  ❉❡ ❢❛t♦✱ ❥á ✈✐♠♦s q✉❡ ♦ ❣r✉♣♦ tr✐✈✐❛❧ {e} é ♦ ♦❜❥❡t♦ ③❡r♦ ❞❡ Ab✳ Ab (G, H)

  P❛r❛ G, H ∈ Ab✱ Hom t❡♠ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❞❛❞❛ ♣♦r f, g ∈ Hom Ab (G, H) : (f + g)(x) = f (x) + g(x), ∀x ∈ G.

  ❆ s♦♠❛ ❞✐r❡t❛ ❞❡ G, H ∈ Ab é ❞❛❞❛ ♣❡❧♦ ♣r♦❞✉t♦ ❞✐r❡t♦ G × H✳ ❆s ♠❡s♠❛s ❝♦♥s✐❞❡r❛çõ❡s ✈❛❧❡♠ ❡♠ Div✳

  ❊①❡♠♣❧♦ ✷✳✷✳✶✹ ❙❡❥❛♠ R ✉♠ ❛♥❡❧✱ k ✉♠ ❝♦r♣♦ ❡ A ✉♠❛ k✲á❧❣❡❜r❛✳ M M m

  R k A A ❆s ❝❛t❡❣♦r✐❛s ✱ V ect ✱ ❡ sã♦ ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✳

  ❉❡ ❢❛t♦✱ ❥á ✈✐♠♦s q✉❡ ♦ ♠ó❞✉❧♦ tr✐✈✐❛❧ {0} é ♦❜❥❡t♦ ③❡r♦ ♥❡ss❛s ❝❛t❡❣♦r✐❛s✳ ❆ s♦♠❛ ❞✐r❡t❛ ❞❡ ♠ó❞✉❧♦s M ❡ N é ❞❛❞❛ ♣❡❧❛ s♦♠❛ ❞✐r❡t❛ ❞❡ ♠ó❞✉❧♦s M ⊕ N✳

  ❙❛❜❡♥❞♦ ❞❡ss❡s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✱ ❛♣r❡s❡♥t❛♠♦s ❡①❡♠✲ ♣❧♦s ❞❡ ❢✉♥t♦r❡s ❛❞✐t✐✈♦s✳ ❊①❡♠♣❧♦ ✷✳✷✳✶✺ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C

  C : C → C ❡ D ✉♠❛ s✉❜❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❞❡ C✳ ❖s ❢✉♥t♦r❡s ✐❞❡♥t✐❞❛❞❡ Id ✱

  D : D → C Z : C → C ✐♥❝❧✉sã♦ I ❡ ❝♦♥st❛♥t❡ C sã♦ ❛❞✐t✐✈♦s✳ ❊①❡♠♣❧♦ ✷✳✷✳✶✻ ❙❡❥❛♠ C, D ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✳ ❖ ❢✉♥t♦r ♣r♦❥❡çã♦ P C : C × D → C

  é ❛❞✐t✐✈♦✳ ❊①❡♠♣❧♦ ✷✳✷✳✶✼ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ X ∈ C✳ ❖s ❢✉♥t♦r❡s L X , R X : C → Ab

  ✱ ❛♥á❧♦❣♦s ❛♦s ❢✉♥t♦r❡s ❞♦s ❊①❡♠♣❧♦s sã♦ ❛❞✐t✐✈♦s✳

  M R → Ab

  ❊①❡♠♣❧♦ ✷✳✷✳✶✽ ❖ ❢✉♥t♦r ❞❡ ❡sq✉❡❝✐♠❡♥t♦ U : ❞♦ ❊①❡♠♣❧♦ é ❛❞✐t✐✈♦✳

  M Z − : Ab → R

  ❊①❡♠♣❧♦ ✷✳✷✳✶✾ ❖ ❢✉♥t♦r R ⊗ ❞♦ ❊①❡♠♣❧♦ é ❛❞✐t✐✈♦✳ k → V ect k

  ❊①❡♠♣❧♦ ✷✳✷✳✷✵ ❖ ❢✉♥t♦r D : V ect ❞♦ ❊①❡♠♣❧♦ é ❛❞✐t✐✈♦✳

  P❛r❛ ♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✱ ♣r❡❝✐s❛♠♦s ❞❡✜♥✐r ❛ s♦♠❛ ❞✐r❡t❛ ❞❡ ♠♦r✲ ✜s♠♦s✳

  ′ :

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

  ′ ′

  X → Y , p X , p X , i X , i X ) ♠♦r✜s♠♦s ❡♠ C✳ ❙❡❥❛♠ (X ⊕ X ❡ (Y ⊕

  ′ ′ ′

  ′ ′

  Y , p Y , p Y , i Y , i Y ) ❛s s♦♠❛s ❞✐r❡t❛s ❞❡ X, X ❡ Y, Y ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ′ ′ ′ ′ : X ⊕ X → Y ⊕ Y

  ❆ s♦♠❛ ❞✐r❡t❛ ❞❡ f ❡ f é ♦ ♠♦r✜s♠♦ f ⊕ f ❡♠ C ❞❛❞♦ ♣♦r

  ′ ′

  ′ ′

  f ⊕ f = i Y ◦ f ◦ p X + i Y ◦ f ◦ p X . Pr♦♣♦s✐çã♦ ✷✳✷✳✷✷ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ f : X → Y ✱

  ′ ′ ′ ′ ′ ′ f : X → Y : X ⊕ X → Y ⊕ Y

  ♠♦r✜s♠♦s ❡♠ C✳ ❊♥tã♦ f ⊕ f é t❛❧ q✉❡ ♦s ❞✐❛❣r❛♠❛s p p

  

X X′

  ′ ′

  X X ⊕ X

  X

  ′ ′

  f f f ⊕f

  ′ ′ Y Y ⊕ Y Y p p

  Y Y ′

  i i

  

X X′

  ′ ′

  X X ⊕ X

  X

  ′ ′

  f f f ⊕f

  ′ ′ Y Y ⊕ Y Y i i

  Y Y ′

  s❡❥❛♠ ❝♦♠✉t❛t✐✈♦s✳ ❚❛❧ ♠♦r✜s♠♦ é ú♥✐❝♦ ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♣r✐✲ X ⊕ id Y =

  ♠❡✐r♦ ♦✉ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛s ❝♦♠✉t❛r❡♠✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ id id

  X ⊕Y ✳

  ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ t❡♠♦s ′ ′

  ′ ′

  p Y ◦ (f ⊕ f ) = p Y ◦ (i Y ◦ f ◦ p X + i Y ◦ f ◦ p X ) ′

  ′ ′

  ◦ i ◦ f ◦ p ◦ i ◦ f ◦ p = p Y Y X + p Y Y

  X

  ′

  Y ′

  ′

  ◦ f ◦ p ◦ f ◦ p = id Y X + 0

  X Y

  ′

  X ⊕X

  = f ◦ p X + 0 Y

  = f ◦ p X .

  ′ ′ ′

  ′

  Y ◦ (f ⊕ f ) = f ◦ p

  X Y ◦ (f ⊕ f ) = f ◦ P♦rt❛♥t♦✱ p ❡ ❛♥❛❧♦❣❛♠❡♥t❡✱ p

  ′

  p

  X ✳ ❆ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ s❡❣✉❡ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ op

  ❞♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ♣❛r❛ C ✳ ′

  ❘❡st❛✲♥♦s ♠♦str❛r q✉❡ ♦ ♠♦r✜s♠♦ f ⊕f é ♦ ú♥✐❝♦ t❛❧ q✉❡ ♦ ♣r✐♠❡✐r♦ ′ ′

  → Y ⊕ Y ♦✉ ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠✳ ❙❡❥❛ g : X ⊕ X ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ ❝♦♠ g ♥♦ ❧✉❣❛r ❞❡

  ′ ′

  ′ ′

  ◦ g = f ◦ p ◦ g = f ◦ p f ⊕ f Y

  X Y

  X ✳ ❊♥tã♦ p ❡ p ✳ ❚❡♠♦s

  ′

  g = id Y ◦ g ⊕Y

  ′ ′

  = (i Y ◦ p Y + i Y ◦ p Y ) ◦ g

  = i Y ◦ p Y ◦ g + i Y

  ′ p

  Y ′

  Z

  p

  ′ g

  g ⊕g

  Y

  ′ Z Z ⊕ Z

  Y ′

  é ❝♦♠✉t❛t✐✈♦✳ P❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ s❛❜❡♠♦s q✉❡ ♦s ❞✐❛❣r❛♠❛s Y Y ⊕ Y

  Z′

  p

  ′

  ◦f

  ′

  g

  Z ′ p

  g

  Z ′ p

  Y

  p

  ′

  f

  X′

  Y ′ p

  X ′

  f p

  ′

  ′

  f ⊕f

  X

  Y Y ⊕ Y ′ p

  X X ⊕ X ′

  Z′

  p

  X′

  X ′

  ′

  = f ⊕ f ′ .

  Pr♦♣♦s✐çã♦ ✷✳✷✳✷✸ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ f : X → Y ✱ f ′

  X ⊕Y ✳

  X ⊕ id Y = id

  ′ é ❛♥á❧♦❣♦✳ P♦r ❞❡✜♥✐çã♦✱ id

  ✳ ❖ ❝❛s♦ ❡♠ q✉❡ g é t❛❧ q✉❡ ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ ❞❛ ♣r♦♣♦s✐çã♦ ❝♦♠✉t❛ ❝♦♠ g ♥♦ ❧✉❣❛r ❞❡ f ⊕f

  P♦rt❛♥t♦✱ g = f ⊕f ′

  ′

  → Y ′

  X

  ◦ p

  ◦ f ′

  ′

  ◦ g = i Y ◦ f ◦ p X + i Y

  ′

  ◦ p Y

  : X ′

  ✱ g : Y → Z✱ g ′

  Z

  Z Z ⊕ Z ′ p

  ) g ◦f p

  ′

  )◦(f ⊕f

  ′

  (g⊕g

  X

  X X ⊕ X ′

  : Y ′

  ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛

  ′ ).

  ′ ) ◦ (f ⊕ f

  ′ ) = (g ⊕ g

  ′ ◦ f

  ♠♦r✜s♠♦s ❡♠ C✳ ❊♥tã♦ (g ◦ f ) ⊕ (g

  → Z ′

  Y ′ sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❊♥tã♦ ′ ′ ′ p Z ◦ (g ⊕ g ) ◦ (f ⊕ f ) = g ◦ p Y ◦ (f ⊕ f )

  = g ◦ f ◦ p X .

  ′ ′ Z ◦ (g ⊕ g ) ◦ (f ⊕ f ) = g ◦ f ◦ p

  X P♦rt❛♥t♦✱ p ❡✱ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ′ ′ ′ ′

  ′ ′

  p Z ◦ (g ⊕ g ) ◦ (f ⊕ f ) = g ◦ f ◦ p

  X ✳ P❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ s❡❣✉❡

  ′ ′ ′ ′ ◦ f ) = (g ⊕ g ) ◦ (f ⊕ f ) q✉❡ (g ◦ f) ⊕ (g ✳

  Pr♦♣♦s✐çã♦ ✷✳✷✳✷✹ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ f, g : X → Y ✱ ′ ′ ′ ′

  → Y f , g : X ♠♦r✜s♠♦s ❡♠ C✳ ❊♥tã♦

  ′ ′ ′ ′ (f + g) ⊕ (f + g ) = (f ⊕ f ) + (g ⊕ g ).

  ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛ p p

  

X X′

  ′ ′

  X X ⊕ X

  X

  ′ ′ ′ ′

  f f

  • g (f ⊕f )+(g⊕g ) +g

  ′ ′ Y Y ⊕ Y Y p p

  Y Y ′

  é ❝♦♠✉t❛t✐✈♦✳ P❡❧❛ Pr♦♣♦s✐çã♦ s❛❜❡♠♦s q✉❡ ♦s ❞✐❛❣r❛♠❛s p p

  X X′

  ′ ′

  X X ⊕ X

  X

  ′ ′

  f f f ⊕f

  ′ ′

  Y Y ⊕ Y Y p p

  

Y Y ′

  p p

  

X X′

  ′ ′

  X X ⊕ X

  X

  ′ ′

  g g g ⊕g

  ′ ′ Y Y ⊕ Y Y p p

  Y Y ′ sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❊♥tã♦ ′ ′ ′ ′ p Y ◦ ((f ⊕ f ) + (g ⊕ g )) = p Y ◦ (f ⊕ f ) + p Y ◦ (g ⊕ g )

  = f ◦ p X + g ◦ p

  X = (f + g) ◦ p X .

  ′ ′ Y ◦ ((f ⊕ f ) + (g ⊕ g )) = (f + g) ◦ p

  X P♦rt❛♥t♦✱ p ❡ ❛♥❛❧♦❣❛♠❡♥t❡✱ ′ ′ ′ ′

  ′ ′

  ◦ ((f ⊕ f p Y ) + (g ⊕ g )) = (f + g ) ◦ p

  X ✳ P❡❧❛ Pr♦♣♦s✐çã♦

  ′ ′ ′ ′

  • g ) = (f ⊕ f ) + (g ⊕ g ) s❡❣✉❡ q✉❡ (f + g) ⊕ (f ✳

  ❈♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ s♦♠❛ ❞✐r❡t❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ s❡❣✉✐♥t❡ ❢✉♥t♦r✳ ❊①❡♠♣❧♦ ✷✳✷✳✷✺ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛✳ ❖ ❢✉♥t♦r − ⊕ − : C

  × C → C ❞❡✜♥✐❞♦ ♣♦r

  ′ ′ ′ ′ (− ⊕ −)(X, X ) = X ⊕ X ) = f ⊕ f ,

  ❡ (− ⊕ −)(f, f ′

  ♣❛r❛ X, Y ∈ C ❡ f, f ♠♦r✜s♠♦s ❡♠ C✱ é ❛❞✐t✐✈♦✳ ❉❡ ❢❛t♦✱ − ⊕ − é ✉♠ ❢✉♥t♦r ♣❡❧❛s ♣r♦♣♦s✐çõ❡s ✐st♦ é✱

  (− ⊕ −)(id X , id Y ) = id X ⊕ id Y = id X ⊕Y = id

  (−⊕−)(X,Y ) ❡

  ′ ′ ′ ′ (− ⊕ −)(g, g ) ◦ (− ⊕ −)(f, f ) = (g ⊕ g ) ◦ (f ⊕ f )

  ′ ′ = (g ◦ f ) ⊕ (g ◦ f )

  ′ ′ = (− ⊕ −)(g ◦ f, g ◦ f ).

  ➱ ❛❞✐t✐✈♦ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✷✻ ❙❡❥❛♠ C, D ❝❛t❡❣♦r✐❛s ❛❞✐t✐✈❛s✱ F : C → D ✉♠

  X , p Y , i X , i Y ) ❢✉♥t♦r ❛❞✐t✐✈♦ ❡ X, Y ∈ C✳ ❙❡ (X ⊕ Y, p é ❛ s♦♠❛ ❞✐r❡t❛

  X ), F (p Y ), F (i X ), F (i Y )) ❞❡ X ❡ Y ✱ ❡♥tã♦ (F (X ⊕ Y ), F (p é ❛ s♦♠❛ ❞✐r❡t❛ ❞❡ F (X) ❡ F (Y )✳ ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ t❡♠♦s

  F (p X ) ◦ F (i X ) = F (p X ◦ i X ) = F (id X )

  X Y

  2 X )

  1 Y

  2 Y ) ◦ (f ⊕ g) ◦ ∆

  X = (p

  1 Y ◦ (f ⊕ g) + p

  2 Y ◦ (f ⊕ g)) ◦ ∆

  X = (f ◦ p

  1 X

  2 X ) ◦ ∆

  X = (f ◦ p

  1 X

  2 X ) ◦ (i

  1 X

  = f ◦ p

  2 Y .

  1 X ◦ i

  1 X

  1 X ◦ i

  2 X

  2 X ◦ i

  1 X

  2 X ◦ i

  2 X = f ◦ id X + f ◦ 0

  X X

  X X

  X = f + 0

  X Y

  ❊♥tã♦ δ Y ◦ (f ⊕ g) ◦ ∆ X = (p

  1 Y

  = id F (X) .

  1 X , p

  ▲♦❣♦✱ F (p X ) ◦ F (i X ) = id F

  (X) ❡ ❛♥❛❧♦❣❛♠❡♥t❡✱ F (p Y ) ◦ F (i Y ) = id F

  (Y ) ✳ ❆❧é♠ ❞✐ss♦✱ ♣♦r s❡r F ✉♠ ❢✉♥t♦r ❛❞✐t✐✈♦✱ t❡♠♦s F (i X ) ◦ F (p X ) + F (i Y ) ◦ F (p Y ) = F (i X ◦ p X ) + F (i Y ◦ p Y )

  = F (i X ◦ p X + i Y ◦ p Y ) = F (id

  X ⊕Y

  ) = id

  F (X⊕Y ) .

  ❈♦♠♦ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ❞❡ss❛ s❡çã♦✱ ♠♦str❛♠♦s ❝♦♠♦ ❛ s♦♠❛ ❡ ❛ s♦♠❛ ❞✐r❡t❛ ❞❡ ♠♦r✜s♠♦s ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✷✼ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ ❡ f, g : X → Y ♠♦r✜s♠♦s ❡♠ C✳ ❊♥tã♦✱ ❡①✐st❡♠ ♠♦r✜s♠♦s ∆

  X : X → X ⊕ X ❡

  δ Y : Y ⊕ Y → Y ❡♠ C t❛✐s q✉❡ f + g = δ Y ◦ (f ⊕ g) ◦ ∆

  X .

  ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ (X⊕X, p

  2 X , i

  2 X ❡ δ Y = p

  1 X , i

  2 X )

  ❡ (Y ⊕Y, p

  1 Y , p

  2 Y , i

  1 Y , i

  2 Y )

  ❛s s♦♠❛s ❞✐r❡t❛s ❞❡ X, X ❡ Y, Y ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✜♥✐♠♦s ∆ X : X →

  X ⊕ X ❡ δ

  Y : Y ⊕ Y → Y ♣♦r

  ∆ X = i

  1 X

  • i
  • p
  • p
  • g ◦ p
  • g ◦ p
  • i
  • f ◦ p
  • g ◦ p
  • g ◦ p
  • g ◦ 0
  • g ◦ id
  • 0
  • g = f + g.

  ✷✳✸ ❈❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s

  ❉❡✜♥✐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✜s♠♦ f ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✱ ♣♦❞❡♠♦s ❝♦♥✲ n , k n )} n , k ) s✐❞❡r❛r ❛ s❡q✉ê♥❝✐❛ {(K ∈N ✱ ❡♠ q✉❡ (K

  1 1 é ♦ ♥ú❝❧❡♦ ❞❡ f✱ (K , k ) , k )

  2n 2n é ♦ ❝♦♥ú❝❧❡♦ ❞❡ k 2n−1 ❡ (K 2n+1 2n+1 é ♦ ♥ú❝❧❡♦ ❞❡ k 2n ✱ ♣❛r❛ t♦❞♦ n ∈ N✳ ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♠♦str❛ q✉❡✱ ♣❛r❛ n í♠♣❛r✱ (K n , k n ) n , k n )

  é ♦ ♥ú❝❧❡♦ ❞❡ f ❡ ♣❛r❛ n ♣❛r✱ (K é ♦ ❝♦♥ú❝❧❡♦ ❞♦ ♥ú❝❧❡♦ ❞❡ f✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✷ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C ❡ f : X → Y

  ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ✭✐✮ ❙❡ (Ker(f), k) é ♦ ♥ú❝❧❡♦ ❞❡ f ❡ (Cok(k), π) é ♦ ❝♦♥ú❝❧❡♦ ❞❡ k✱ ❡♥tã♦ (Ker(f), k) é ♦ ♥ú❝❧❡♦ ❞❡ π✳ ✭✐✐✮ ❙❡ (Cok(f), q) é ♦ ❝♦♥ú❝❧❡♦ ❞❡ f ❡ (Ker(q), ι) é ♦ ♥ú❝❧❡♦ ❞❡ q✱ ❡♥tã♦ (Cok(f ), q)

  é ♦ ❝♦♥ú❝❧❡♦ ❞❡ ι✳ Ker

  (f ) ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❏á t❡♠♦s π◦k = 0 ✱ q✉❡ é ❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦

  Cok (k)

  ♣❛r❛ (Ker(f), k) s❡r ♦ ♥ú❝❧❡♦ ❞❡ π✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛ ′

  K u

  ′

  k f Ker(f ) X Y. k

  π v Cok(k)

  Ker (f ) ❈♦♠♦ f ◦ k = 0 s❡❣✉❡✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ❝♦✲

  Y ♥ú❝❧❡♦ (Cok(k), π) ♣❛r❛ ♦ ♣❛r (Y, f)✱ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ v : Cok(k) → Y t❛❧ q✉❡ f = v ◦ π✳

  ′

  K ′ ′ ′

  : K → X = 0 ❆❣♦r❛✱ s❡❥❛ k ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ π ◦k ✳

  Cok (k)

  ❊♥tã♦

  ′ ′

  K K ′ ′ f ◦ k = v ◦ π ◦ k = v ◦ 0 = 0 .

  Cok Y (k)

  ′ ′ , k )

  P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ♥ú❝❧❡♦ (Ker(f), k) ♣❛r❛ ♦ ♣❛r (K ✱ ′ ′

  → Ker(f ) = k ◦ u ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u : K t❛❧ q✉❡ k ✳ P♦r✲ t❛♥t♦✱ (Ker(f), k) é ♦ ♥ú❝❧❡♦ ❞❡ π✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ◆❡ss❡ ❝❛s♦✱ t❡rí❛♠♦s ♦ ❞✐❛❣r❛♠❛ Ker(q) v

  ι q Cok(f )

  X Y f

  ′

  q u ′

  Q Pr♦♣♦s✐çã♦ ✷✳✸✳✸ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✳ ❆s s❡❣✉✐♥t❡s ❛✜r✲ ♠❛çã♦❡s sã♦ ✈á❧✐❞❛s✿ ✭✐✮ t♦❞♦ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C é ♦ ♥ú❝❧❡♦ ❞♦ s❡✉ ❝♦♥ú❝❧❡♦❀ ✭✐✐✮ t♦❞♦ ❡♣✐♠♦r✜s♠♦ ❡♠ C é ♦ ❝♦♥ú❝❧❡♦ ❞♦ s❡✉ ♥ú❝❧❡♦✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙❡❥❛ k : K → X ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C ❡ (Cok(k), π) s❡✉ ❝♦♥ú❝❧❡♦✳ P♦r ✭✐✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ❡①✐st❡ ✉♠ ♠♦r✲ ✜s♠♦ f ❡♠ C t❛❧ q✉❡ (K, k) é ♦ ♥ú❝❧❡♦ ❞❡ f✳ P❡❧❛ Pr♦♣♦s✐çã♦ (K, k)

  é ♦ ♥ú❝❧❡♦ ❞♦ s❡✉ ❝♦♥ú❝❧❡♦✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ C✱ t♦❞♦ ✐s♦♠♦r✜s♠♦ é ✉♠

  ♠♦♥♦♠♦r✜s♠♦ ❡ ✉♠ ❡♣✐♠♦r✜s♠♦✳ ❊♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✱ ✈❛❧❡ ❛ r❡❝í♣r♦❝❛✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✹ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦ Z ∈ C

  ✳ ❙❡ f : X → Y é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡ ✉♠ ❡♣✐♠♦r✜s♠♦ ❡♠ C✱ ❡♥tã♦ f é ✉♠ ✐s♦♠♦r✜s♠♦✳

  ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ f é ✉♠ ❡♣✐♠♦r✜s♠♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ Y

  (Z, 0 ) Y )

  é ♦ ❝♦♥ú❝❧❡♦ ❞❡ f✳ P❡❧❛ Pr♦♣♦s✐çã♦ (Y, id é ♦ ♥ú❝❧❡♦ Y

  ❞❡ 0 ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ f é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ Y

  ❛♥t❡r✐♦r✱ (X, f) é ♦ ♥ú❝❧❡♦ ❞❡ 0 ✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♥ú❝❧❡♦ ❞❛❞❛ ♣❡❧❛ Pr♦♣♦s✐çã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ u : X → Y t❛❧ q✉❡ f = id Y ◦ u

  ✳ ▲♦❣♦✱ f = u é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❆❜❛✐①♦ ❛♣r❡s❡♥t❛♠♦s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s ❡ ✉♠ ❡①❡♠✲

  ♣❧♦ ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❞✐t✐✈❛ q✉❡ ♥ã♦ é ❛❜❡❧✐❛♥❛✳ ❊①❡♠♣❧♦ ✷✳✸✳✺ ❆ ❝❛t❡❣♦r✐❛ Ab é ❛❜❡❧✐❛♥❛✳

  ❉❡ ❢❛t♦✱ ❥á ✈✐♠♦s ♥♦ ❊①❡♠♣❧♦ ♣r♦✈❛✲s❡ q✉❡ t♦❞♦ ♠♦r✜s♠♦ ❡♠ Ab ♣♦ss✉✐ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦✳ ❆❣♦r❛✱ s❡ f : G → H é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠ Ab✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ (X, f) é ♦ ♥ú❝❧❡♦ ❞❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛ π : H → H/f(G)✳ P♦rt❛♥t♦✱ t♦❞♦ ♠♦♥♦♠♦r✜s♠♦ é ✉♠ ♥ú❝❧❡♦✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ t♦❞♦ ❡♣✐♠♦r✜s♠♦ é ✉♠ ❝♦♥ú❝❧❡♦✳

  M R ❊①❡♠♣❧♦ ✷✳✸✳✻ ❆ ❝❛t❡❣♦r✐❛ é ❛❜❡❧✐❛♥❛✳

  M R ❉❡ ❢❛t♦✱ ❛ ♣r♦✈❛ é ❛♥á❧♦❣❛ ❛ ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳ ❆ ❝❛t❡❣♦r✐❛

  M R é ❛❞✐t✐✈❛ ♣❡❧♦ ❊①❡♠♣❧♦ ❡ t♦❞♦ ♠♦r✜s♠♦ ❡♠ ♣♦ss✉✐ ♥ú❝❧❡♦s ❡ ❝♦♥ú❝❧❡♦s ♣❡❧♦ ❊①❡♠♣❧♦ ❊①❡♠♣❧♦ ✷✳✸✳✼ ❆ ❝❛t❡❣♦r✐❛ Div é ❛❞✐t✐✈❛✱ ♠❛s ♥ã♦ é ❛❜❡❧✐❛♥❛✳

  ❉❡ ❢❛t♦✱ ❛ ❝❛t❡❣♦r✐❛ Div é ❛❞✐t✐✈❛ ♣❡❧♦ ❊①❡♠♣❧♦ ◆♦ ❡♥t❛♥t♦✱ ♦ ♠♦r✜s♠♦ π : Q → Q/Z ❡♠ Div é ♠♦♥♦♠♦r✜s♠♦ ♣❡❧♦ ❊①❡♠♣❧♦ ❡ é ❡♣✐♠♦r✜s♠♦✱ ♣♦✐s é s♦❜r❡❥❡t♦r✳ P♦ré♠✱ π ♥ã♦ é ✐s♦♠♦r✜s♠♦ ❡♠ Div

  ✳ ❯s❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ Div ♥ã♦ ♣♦❞❡ s❡r ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✳

  ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♠♦str❛ q✉❡✱ ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛✱ t♦❞♦ ♠♦r✜s♠♦ ❛❞♠✐t❡ ❞✉❛s ❞❡❝♦♠♣♦s✐çõ❡s✱ ✉♠❛ ❡♥✈♦❧✈❡♥❞♦ ✉♠ ♠♦♥♦♠♦r✲ ✜s♠♦ ❝❛♥ô♥✐❝♦ ❡ ♦✉tr❛ ❡♥✈♦❧✈❡♥❞♦ ✉♠ ❡♣✐♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✽ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛ ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ✭✐✮ ❙❡❥❛♠ (Cok(f), q) ♦ ❝♦♥ú❝❧❡♦ ❞❡ f ❡ (Ker(q), ι) ♦ ♥ú❝❧❡♦ ❞❡ q✳ ❊♥✲ tã♦ ❡①✐st❡ u : X → Ker(q) t❛❧ q✉❡ f = ι ◦ u✳

  ✭✐✐✮ ❙❡❥❛♠ (Ker(f), k) ♦ ♥ú❝❧❡♦ ❞❡ f ❡ (Cok(k), π) ♦ ❝♦♥ú❝❧❡♦ ❞❡ k✳ ❊♥tã♦ ❡①✐st❡ v : Cok(k) → Y t❛❧ q✉❡ f = v ◦ π✳

  X ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❈♦♠♦ q ◦ f = 0 ✱ s❡❣✉❡✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡

  Cok (f ) ✉♥✐✈❡rs❛❧ ❞♦ ♥ú❝❧❡♦ (Ker(q), ι) ♣❛r❛ ♦ ♣❛r (X, f)✱ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u : X → Ker(q) t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛

  X Cok (f )

  X f q u

  Cok(f ) Y

  ι Ker(q)

  Ker (q) Cok (f )

  é ❝♦♠✉t❛t✐✈♦✳ P♦rt❛♥t♦✱ f = ι ◦ u✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ❱❛♠♦s ♠♦str❛r q✉❡ ❛s ❞❡❝♦♠♣♦s✐çõ❡s ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r sã♦ ❛

  ❝♦♠♣♦s✐çã♦ ❞❡ ✉♠ ❡♣✐♠♦r✜s♠♦ ❡ ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡ ❡st❛ ❞❡❝♦♠♣♦s✐✲ çã♦ é ú♥✐❝❛✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞♦ s❡❣✉✐♥t❡ ❧❡♠❛✳ ▲❡♠❛ ✷✳✸✳✾ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛ ❡ f : X → Y ✉♠ ♠♦r✲ ✜s♠♦ ❡♠ C✳ ❙❡❥❛♠ f = ι ◦ u ❡ f = v ◦ π ❛s ❞❡❝♦♠♣♦s✐çõ❡s ❞❡ f ❝♦♠♦ ♥♦s ✐t❡♥s ✭✐✮ ❡ ✭✐✐✮ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳

  ′ ′ ′ ′ : X → K : K → Y ◦u

  ✭✐✮ ❙❡ u ❡ ι sã♦ ♠♦r✜s♠♦s ❡♠ C t❛✐s q✉❡ f = ι ❡ ′

  ι é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ψ : Ker(q) →

  K t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛ u

  Ker(q)

  X ψ

  ′

  u ι K Y

  

  ι

  ′ é ❝♦♠✉t❛t✐✈♦✳ ❆❧é♠ ❞✐ss♦✱ s❡ u ❡ u sã♦ ❡♣✐♠♦r✜s♠♦s✱ ❡♥tã♦ ψ é ✉♠ ✐s♦♠♦r✜s♠♦✳

  ′ ′ ′ ′ : X → Q : Q → Y ◦π

  ✭✐✐✮ ❙❡ π ❡ v sã♦ ♠♦r✜s♠♦s ❡♠ C t❛✐s q✉❡ f = v ′

  ❡ π é ✉♠ ❡♣✐♠♦r✜s♠♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ψ : Q → Cok(k) t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛

  

  π Q

  X ψ

  π ′ v Cok(k)

  Y v

  ′ é ❝♦♠✉t❛t✐✈♦✳ ❆❧é♠ ❞✐ss♦✱ s❡ v ❡ v sã♦ ♠♦♥♦♠♦r✜s♠♦s✱ ❡♥tã♦ ψ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ u

  Ker(q)

  X ψ

  ′

  u ι

  ′

  q ′ Cok(ι ).

  K Y

  ′

  ι q

  ′

  ψ Cok(f )

  ′ ′ ′ ), q )

  ❙❡❥❛ (Cok(ι ♦ ❝♦♥ú❝❧❡♦ ❞❡ ι ✳ ❚❡♠♦s ′ ′ ′ ′ K ′

  X q ◦ f = q ◦ ι ◦ u = 0 ′ ◦ u = 0 ′ . Cok Cok

  (ι ) (ι )

  ′ ′ ), q )

  P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ❝♦♥ú❝❧❡♦ (Cok(f), q) ♣❛r❛ ♦ ♣❛r (Cok(ι ✱ ′ ′

  : Cok(f ) → Cok(ι ) ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ψ t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛

  X Cok (f )

  Cok(f ) q f

  ′

  ψ

  X Y

  ′

  q ′

  

X Cok(ι )

Cok (ι′ )

  ′ ′ = ψ ◦ q

  é ❝♦♠✉t❛t✐✈♦✳ P♦rt❛♥t♦✱ q ✳ ′

  ′ )

  ❆❣♦r❛✱ ι é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ P❡❧❛ Pr♦♣♦s✐çã♦ (K, ι é ♦ ′

  ♥ú❝❧❡♦ ❞❡ q ✳ ❚❡♠♦s Ker Ker

  (q) (q) ′ ′ ′ q ◦ ι = ψ ◦ q ◦ ι = ψ ◦ 0 = 0 ′ .

  Cok Cok (f ) (ι )

  ′ )

  P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ♥ú❝❧❡♦ (K, ι ♣❛r❛ ♦ ♣❛r (Ker(q), ι)✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ψ : Ker(q) → K t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛

  Ker (q) (ι′ ) Cok

  Ker(q) ι

  ′

  q ′

  ψ Cok(ι )

  Y

  ′

  ι K K

  Cok (ι′ )

  ′ ′ ◦ ψ = ψ ◦ u s❡❥❛ ❝♦♠✉t❛t✐✈♦✳ P♦rt❛♥t♦✱ ι = ι ✳ ❆❧é♠ ❞✐ss♦✱ u ✳ ❉❡ ❢❛t♦✱

  ′ ′ ′ ι ◦ u = f = ι ◦ u = ι ◦ ψ ◦ u. ′ ′ ′ ′

  ◦ u ◦ ψ ◦ u = ι

  ▲♦❣♦✱ ι ❡ ❝♦♠♦ ι é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ ′ ′ ′

  ◦ ψ u = ψ ◦ u = ψ ◦ u ✳ ❊♥tã♦ ι = ι ❡ u ✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛ ❞❛

  ♣r♦♣♦s✐çã♦ é ❝♦♠✉t❛t✐✈♦✳ ′ ′

  ◦ ψ = ι ❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡ u ❡ u s❡❥❛♠ ❡♣✐♠♦r✜s♠♦s✳ ❈♦♠♦ ι

  ′ ❡ ψ ◦ u = u ❡♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ ψ é ✉♠ ♠♦♥♦✲ ♠♦r✜s♠♦ ❡ ✉♠ ❡♣✐♠♦r✜s♠♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ ψ é ✉♠ ✐s♦♠♦r✜s♠♦✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳

  ❈♦r♦❧ár✐♦ ✷✳✸✳✶✵ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛ ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❙❡❥❛♠ f = ι ◦ u ❡ f = v ◦ π ❛s ❞❡❝♦♠♣♦s✐çõ❡s ❞❡ f ❝♦♠♦ ♥♦s ✐t❡♥s ✭✐✮ ❡ ✭✐✐✮ ❞❛ Pr♦♣♦s✐çã♦ ❊♥tã♦ ✭✐✮ u é ✉♠ ❡♣✐♠♦r✜s♠♦✳ ✭✐✐✮ v é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ▼♦str❡♠♦s q✉❡ u é ✉♠ ❡♣✐♠♦r✜s♠♦ ✉s❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ❙❡❥❛ h : Ker(q) → W ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡

  X h ◦ u = 0

  W ✳ ′

  ) ❙❡❥❛ (Ker(h), k ♦ ♥ú❝❧❡♦ ❞❡ h✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ♥ú✲

  ′ ′

  ) : X →

  ❝❧❡♦ (Ker(h), k ♣❛r❛ ♦ ♣❛r (X, u)✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ u Ker(h) t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛

  X W

  X u h

  ′

  u Ker(q)

  W

  ′

  k

  (h)

  Ker(h) Ker

  W

  ′ ′ ′ ′ ◦ u ◦ u

  é ❝♦♠✉t❛t✐✈♦✳ P♦rt❛♥t♦✱ u = k ✳ ▲♦❣♦✱ f = ι ◦ u = ι ◦ k ✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛ u

  Ker(q)

  X ψ

  ′

  u ι

  Ker(h) Y

  ′

  ι ◦k ′

  é ❝♦♠✉t❛t✐✈♦✳ ❆❣♦r❛✱ ❝♦♠♦ ι ❡ k sã♦ ♠♦♥♦♠♦r✜s♠♦s ❡♥tã♦✱ ♣❡❧❛ Pr♦✲ ′

  ♣♦s✐çã♦ ι ◦ k é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❡①✐st❡ ′

  ◦ ψ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ψ : Ker(q) → Ker(h) t❛❧ q✉❡ ι = ι ◦ k ❡ ❝♦♠♦

  ′ ι Ker = k ◦ ψ

  é ✉♠ ♠♦♥♦♠♦r✜s♠♦✱ t❡♠♦s id (q) ✳ ▲♦❣♦✱ Ker (h) Ker (q)

  ′ h = h ◦ id = h ◦ k ◦ ψ = 0 ◦ ψ = 0 . Ker (q)

  W W Ker (q)

  X P♦rt❛♥t♦✱ h ◦ u = 0 ✐♠♣❧✐❝❛ h = 0 ✳ ▲♦❣♦✱ u é ✉♠ ❡♣✐♠♦r✲ W W

  ✜s♠♦✳ op ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳

  ❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❝❤❛♠❛❞♦ ❚❡♦r❡♠❛ ❞♦ ■s♦♠♦r✜s♠♦ ♣❛r❛ ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s✳ ❚❡♦r❡♠❛ ✷✳✸✳✶✶ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛ ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C ❝♦♠ ♥ú❝❧❡♦ (Ker(f), k) ❡ ❝♦♥ú❝❧❡♦ (Cok(f), q)✳ ❙❡❥❛♠ (Cok(k), π)

  ♦ ❝♦♥ú❝❧❡♦ ❞❡ k ❡ (Ker(q), ι) ♦ ♥ú❝❧❡♦ ❞❡ q✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ φ : Cok(k) → Ker(q) t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛ f

  X Y π ι

  Cok(k) Ker(q) φ

  é ❝♦♠✉t❛t✐✈♦✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ❛s ❞❡❝♦♠♣♦s✐çõ❡s f = ι ◦ u ❡ f = v ◦ π ❞♦s ✐t❡♥s ✭✐✮ ❡ ✭✐✐✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❛ Pr♦♣♦s✐çã♦ P❡❧♦ ❝♦r♦❧ár✐♦ ❛♥t❡r✐♦r✱ u é ✉♠ ❡♣✐♠♦r✜s♠♦ ❡ v é ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ P❡❧♦ ▲❡♠❛ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ ψ : Ker(q) → Cok(k) t❛❧ q✉❡ ♦ ❞✐❛❣r❛♠❛ u

  Ker(q)

  X ψ

  π ι

  Cok(k) Y v

  é ❝♦♠✉t❛t✐✈♦✳ P♦rt❛♥t♦✱ π = ψ ◦ u ❡ ι = v ◦ ψ✳ P❛r❛ φ : Cok(k) → −1

  Ker(q) ✱ φ = ψ ✱ t❡♠♦s φ ◦ π = u ❡ ι ◦ φ = v✳ ❊♥tã♦ ι ◦ φ ◦ π = ι ◦ u = f.

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

  ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ♥ú❝❧❡♦ ❡ ❝♦♥ú❝❧❡♦✱ q✉❡ ❞❡✈❡ s❡r ❢❡✐t❛ ♣❛r❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ❝♦♠ ♦❜❥❡t♦ ③❡r♦✳ ◆♦ ❡♥t❛♥t♦✱ ❡s❝♦❧❤❡♠♦s ❛♣r❡s❡♥tá✲❧❛ ♥❡ss❛ s❡çã♦✱ ♣♦✐s ✈❛♠♦s ♠♦str❛r q✉❡ ❡♠ ❝❛t❡❣♦r✐❛s ❛❜❡❧✐❛♥❛s t♦❞♦ ♠♦r✜s♠♦ t❡♠ ✐♠❛❣❡♠ ❡ ❝♦✐♠❛❣❡♠✳ ❉❡✜♥✐çã♦ ✷✳✸✳✶✷ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ✭✐✮ ❯♠❛ ✐♠❛❣❡♠ ❞❡ f é ✉♠ ♣❛r (I, ι)✱ ❡♠ q✉❡ I ∈ C ❡ ι : I → Y é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ f = ι ◦ g✱ ♣❛r❛ ❛❧❣✉♠ ♠♦r✜s♠♦ g : X → I✱ t❛❧ q✉❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ é s❛t✐s❢❡✐t❛✿ ♣❛r❛ q✉❛❧q✉❡r ♣❛r

  ′ ′ ′ ′ ′ ∈ C → Y

  (I , ι ) : I ✱ ❡♠ q✉❡ I ❡ ι é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡

  ′ ′ ′ ′ ◦ g f = ι : X → I

  ✱ ♣❛r❛ ❛❧❣✉♠ ♠♦r✜s♠♦ g ✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ′ ′

  ◦ u u : I → I t❛❧ q✉❡ ι = ι ✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛

  ι

  I g f u

  X Y

  ′

  g ′

  ′

  I ι é ❝♦♠✉t❛t✐✈♦✳ ✭✐✐✮ ❯♠❛ ❝♦✐♠❛❣❡♠ ❞❡ f é ✉♠ ♣❛r (C, π)✱ ❡♠ q✉❡ C ∈ C ❡ π : X → C é ✉♠ ❡♣✐♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ f = g ◦ π✱ ♣❛r❛ ❛❧❣✉♠ ♠♦r✜s♠♦ g : C → Y

  ✱ t❛❧ q✉❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ é s❛t✐s❢❡✐t❛✿ ♣❛r❛ q✉❛❧q✉❡r ′ ′ ′ ′ ′

  ∈ C , π ) : X → C

  ♣❛r (C ✱ ❡♠ q✉❡ C ❡ π é ✉♠ ❡♣✐♠♦r✜s♠♦ ❡♠ C ′ ′ ′ ′

  ◦ π → Y : C

  ❝♦♠ f = g ✱ ♣❛r❛ ❛❧❣✉♠ ♠♦r✜s♠♦ g ✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ′ ′

  → C ♠♦r✜s♠♦ u : C t❛❧ q✉❡ π = u ◦ π ✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛

  ′

  π ′

  C

  ′

  g f u

  X Y g

  π C

  é ❝♦♠✉t❛t✐✈♦✳

  ◆♦t❡♠♦s q✉❡ ✐♠❛❣❡♠ ❡ ❝♦✐♠❛❣❡♠ sã♦ ❝♦♥❝❡✐t♦s ❞✉❛✐s✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠❛ ✐♠❛❣❡♠ ❞❡ ✉♠ ♠♦r✜s♠♦ ❡♠ C é ✉♠❛ ❝♦✐♠❛❣❡♠ ❞❡ss❡ op

  ♠♦r✜s♠♦ ❡♠ C ✳ ❱❛❧❡ ♥♦t❛r♠♦s q✉❡ s❡ ❛♣❧✐❝❛r♠♦s ❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛ ✐♠❛❣❡♠

  (I, ι) I : I → I

  ♣❛r❛ ♦ ♣❛r (I, ι)✱ ♦ ú♥✐❝♦ ♠♦r✜s♠♦ q✉❡ s❡ ♦❜té♠ é id ✳ ❯s❛♠♦s ❡ss❡ ❢❛t♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✶✸ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳

  ′ ′ , ι )

  ✭✐✮ ❙❡ (I, ι) ❡ (I sã♦ ✐♠❛❣❡♥s ❞❡ f✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✲ ′ ′

  ◦ u ✜s♠♦ u : I → I t❛❧ q✉❡ ι = ι ✳

  ′ ′ , π )

  ✭✐✐✮ ❙❡ (C, π) ❡ (C sã♦ ❝♦✐♠❛❣❡♥s ❞❡ f✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ′ ′

  → C ✐s♦♠♦r✜s♠♦ u : C t❛❧ q✉❡ π = u ◦ π ✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❆♣❧✐❝❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛s ✐♠❛❣❡♥s (I, ι) ❡

  ′ ′ ′ ′ (I , ι ) , ι )

  ♣❛r❛ ♦s ♣❛r❡s (I ❡ (I, ι)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡①✐st❡♠ ♠♦r✜s✲ ′ ′ ′ ′

  → I ◦ u = ι ◦ v

  ♠♦s u : I → I ❡ v : I t❛✐s q✉❡ ι = ι ❡ ι ✳ ◆♦t❡♠♦s q✉❡ ♦ ♠♦r✜s♠♦ v ◦ u : I → I s❛t✐s❢❛③

  ′ ι ◦ (v ◦ u) = (ι ◦ v) ◦ u = ι ◦ u = ι.

  ❙❡❣✉❡ q✉❡ v ◦ u : I → I é ♦ ú♥✐❝♦ ♠♦r✜s♠♦ q✉❡ s❡ ♦❜té♠ ❛♦ ❛♣❧✐❝❛r

  I ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ✐♠❛❣❡♠ (I, ι) ♣❛r❛ ♦ ♣❛r (I, ι)✳ ▲♦❣♦✱ v ◦ u = id ✳

  ′

  ′

  I ❆♥❛❧♦❣❛♠❡♥t❡✱ u ◦ v = id ✳ ❆ss✐♠✱ u : I → I é ✉♠ ✐s♦♠♦r✜s♠♦ t❛❧

  ′ ◦ u q✉❡ ι = ι ✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✶✹ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❛❜❡❧✐❛♥❛ ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ✭✐✮ ❙❡❥❛♠ (Cok(f), q) ♦ ❝♦♥ú❝❧❡♦ ❞❡ f ❡ (Ker(q), ι) ♦ ♥ú❝❧❡♦ ❞❡ q✳ ❊♥✲ tã♦ (Ker(q), ι) é ❛ ✐♠❛❣❡♠ ❞❡ f✳ ✭✐✐✮ ❙❡❥❛♠ (Ker(f), k) ♦ ♥ú❝❧❡♦ ❞❡ f ❡ (Cok(k), π) ♦ ❝♦♥ú❝❧❡♦ ❞❡ k✳ ❊♥tã♦ (Cok(k), π) é ❛ ❝♦✐♠❛❣❡♠ ❞❡ f✳ ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❉❛ Pr♦♣♦s✐çã♦ ❡①✐st❡ u : X → Ker(q) t❛❧ q✉❡ f = ι ◦ u✱ q✉❡ é ❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ ♣❛r❛ (Ker(q), ι) s❡r ❛ ✐♠❛❣❡♠ ❞❡ f✳

  ′ ′ ′ ′ ′ , ι ) ∈ C : I → Y

  ❙❡❥❛ (I ✱ ❡♠ q✉❡ I ❡ ι é ✉♠ ♠♦♥♦♠♦r✜s♠♦ ′ ′ ′ ′

  : X → I ◦ g ❡♠ C t❛❧ q✉❡ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ g ❝♦♠ f = ι ✳ P❡❧♦

  ′ ′ ◦ψ

  ▲❡♠❛ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠♦r✜s♠♦ ψ : Ker(q) → I t❛❧ q✉❡ ι = ι ✳ op

  ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧✱ M ❡ N R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ❡ f : M → N ✉♠

  M R ♠♦r✜s♠♦ ❡♠ ✳ ❱✐♠♦s ♥♦ ❊①❡♠♣❧♦ q✉❡ ♦ ♥ú❝❧❡♦ ❡ ♦ ❝♦♥ú❝❧❡♦ ❞❡ f sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♣❛r❡s

  (P, k) ❡ (N/f(M), q),

  } N

  ❡♠ q✉❡ P = {m ∈ M : f(m) = 0 ✱ k : P → M é ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛ ❡ q : N → N/f(M) é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳ ❊♥tã♦✱ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ❊①❡♠♣❧♦ ❡ ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ❛ ✐♠❛❣❡♠ ❡ ❛ ❝♦✐♠❛❣❡♠ ❞❡ f sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♣❛r❡s

  (f (M ), ι) ❡ (M/P, π)

  ❡♠ q✉❡ ι : f(M) → Y é ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛ ❡ π : M → M/P é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳ ◆❡ss❡ ❝❛s♦✱ ♦ ❚❡♦r❡♠❛ ♠♦str❛ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ φ : M/P → f(M) t❛❧ q✉❡ f = ι ◦ φ ◦ π✳

  ❈❛♣ít✉❧♦ ✸ ❈❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s

  ❖s ♠♦♥ó✐❞❡s sã♦ ✉♠❛ ❞❛s ❡str✉t✉r❛s ♠❛✐s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ á❧❣❡✲ ❜r❛ ♦r❞✐♥ár✐❛✳ P♦r ❡①❡♠♣❧♦✱ ❣r✉♣♦s sã♦ ♠♦♥ó✐❞❡s ❡♠ q✉❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s sã♦ ✐♥✈❡rtí✈❡✐s✱ ❛♥é✐s sã♦ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ❝♦♠ ❛ ❛❞✐çã♦ ❡ ♠♦♥ó✐❞❡s ❝♦♠ ♦ ♣r♦❞✉t♦✱ ♠ó❞✉❧♦s sã♦ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ❝♦♠ ❛ s♦♠❛✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡♥❞♦♠♦r✜s♠♦s ❞❡ ♦❜❥❡t♦s ❛❧❣é❜r✐❝♦s sã♦ ♠♦♥ó✐❞❡s ❝♦♠ ❛ ❝♦♠♣♦s✐çã♦✳ P♦rt❛♥t♦✱ ❛s ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✱ s❡♥❞♦ ❛ ❝❛t❡❣♦r✐✜❝❛çã♦ ❞♦s ♠♦♥ó✐❞❡s✱ sã♦ ✉♠❛ ❞❛s ❡str✉t✉r❛s ♠❛✐s ❢✉♥❞❛♠❡♥t❛✐s ❡♠ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s✳

  ◆♦çõ❡s ✐♠♣♦rt❛♥t❡s q✉❡ ♣♦❞❡♠ s❡r ❞❡✜♥✐❞❛s ❡♠ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐✲ ❞❛✐s sã♦ ♦❜❥❡t♦s ♠♦♥ó✐❞❡s ❡ ❝❛t❡❣♦r✐❛s ❡♥r✐q✉❡❝✐❞❛s✳ ▼♦♥ó✐❞❡s✱ ❛♥é✐s✱ á❧❣❡❜r❛s✱ ❜✐á❧❣❡❜r❛s ❡ s✉❛s ✈❡rsõ❡s ❝♦♠✉t❛t✐✈❛s ♦✉ t♦♣♦❧ó❣✐❝❛s sã♦ ♦❜✲ ❥❡t♦s ♠♦♥ó✐❞❡s ❡♠ ❞❡t❡r♠✐♥❛❞❛s ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✳ ❊♠ ❛s á❧❣❡❜r❛s ❞❡ ◆✐❝❤♦❧s sã♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♦❜❥❡t♦s ♠♦♥ó✐❞❡s ♥❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❞♦s ♠ó❞✉❧♦s ❞❡ ❨❡tt❡r✲❉r✐♥❢❡❧❞ s♦❜r❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ q✉❡ ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ ♣♦♥t✉❛✲ ❞❛s✳ ❯♠❛ ❝❛t❡❣♦r✐❛ ❡♥r✐q✉❡❝✐❞❛ s♦❜r❡ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ C é ✉♠❛

  D (X, Y ) ∈ C ❝❛t❡❣♦r✐❛ D t❛❧ q✉❡✱ ♣❛r❛ X, Y ∈ D✱ Hom ❡ ❡①✐st❡ ✉♠❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛s ❡str✉t✉r❛s ❞❡ C ❡ D ✭✈❡❥❛ ❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ❡ss❡s ❝❛t❡❣♦r✐❛s s❡ ❞❡✈❡ ❛♦ ❢❛t♦ ❞❡ q✉❡✱ ❡♠ ♠✉✐t❛s s✐t✉❛çõ❡s✱ ♦s ♠♦r✜s♠♦s ❢r❡q✉❡♥t❡♠❡♥t❡ tê♠ ❡str✉t✉r❛ ❛❞✐❝✐♦♥❛❧ q✉❡ ❞❡✈❡ s❡r r❡s♣❡✐✲ t❛❞❛✳ ❈❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s t❛♠❜é♠ tê♠ ❛♣❧✐❝❛çõ❡s ❡♠ ❧ó❣✐❝❛ ❝❛t❡❣ó✲ r✐❝❛ ✭✈❡r ❡ ♥♦ ❡st✉❞♦ ❞♦ ❣r✉♣♦ ❞❡ ❚❤♦♠♣s♦♥ ✭✈❡r

  ◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛s ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✳ ❉❡✜♥✐♠♦s t❛✐s ❝❛t❡❣♦r✐❛s ❡ ♦s ❝❤❛♠❛❞♦s ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s✱ q✉❡ sã♦ ❢✉♥t♦r❡s q✉❡ ♣r❡s❡r✈❛♠ ❛ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧✳ ❆ s❡❣✉✐r✱ ❞❡✜♥✐♠♦s ❡q✉✐✈❛❧ê♥❝✐❛s ♠♦✲

  ♥♦✐❞❛✐s✳ ❖ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡ ❞❡st❡ ❝❛♣ít✉❧♦ é ♦ ❢❛t♦ ❞❡ q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦✲ ♥♦✐❞❛❧ ❡sq✉❡❧ét✐❝❛✱ ❡ ❡st❛ ú❧t✐♠❛ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❝❛t❡❣♦r✐❛ ♠❛✐s s✐♠♣❧❡s✳ ◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ♠♦str❛♠♦s q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐✲ ❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✱ q✉❡ t❡♠ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧ ♠❛✐s s✐♠♣❧❡s✳ ❈♦♠♦ r❡❢❡rê♥❝✐❛ ❜ás✐❝❛✱ ❝✐t❛♠♦s

  ✸✳✶ ❈❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s

  ❯♠ ♠♦♥ó✐❞❡ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ t❡r♥❛ (M, ∗, 1)✱ ❡♠ q✉❡ M

  é ✉♠ ❝♦♥❥✉♥t♦✱ ∗ : M × M → M é ✉♠❛ ❢✉♥çã♦✱ 1 ∈ M ❡✱ ♣❛r❛ q✉❛✐sq✉❡r x, y, z ∈ M✱ ✈❛❧❡♠ (x ∗ y) ∗ z = x ∗ (y ∗ z), 1 ∗ x = x ❡ x ∗ 1 = x. ❇❛s❡❛❞♦s ♥❡ss❛ ❞❡✜♥✐çã♦ ❞❡ ♠♦♥ó✐❞❡✱ ❞❡✜♥✐♠♦s ❝❛t❡❣♦r✐❛s ♠♦♥♦✐✲

  ❞❛✐s✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡♥♦t❛♠♦s ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s g ❡ f ♣♦r gf ✐♥✈és ❞❡ g ◦ f✳

  ❉❡✜♥✐çã♦ ✸✳✶✳✶ ❯♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ✉♠❛ sê①t✉♣❧❛ (C, ⊗, 1, a, l, r) ❡♠ q✉❡ ✭✐✮ C é ✉♠❛ ❝❛t❡❣♦r✐❛✳ ✭✐✐✮ ⊗ : C × C → C é ✉♠ ❢✉♥t♦r ❞❛❞♦ ♣♦r

  ⊗(X, Y ) = X ⊗ Y ❡ ⊗ (f, g) = f ⊗ g,

  ′ ′ → Y

  ♣❛r❛ X, Y ∈ C ❡ f : X → Y ✱ g : X ♠♦r✜s♠♦s ❡♠ C✱ ❡♠ q✉❡ ′ ′ f ⊗ g : X ⊗ X → Y ⊗ Y

  ✳ ✭✐✐✐✮ 1 ∈ C é ❝❤❛♠❛❞♦ ♦❜❥❡t♦ ✉♥✐❞❛❞❡✳

  X,Y,Z : (X ⊗Y )⊗Z → X ⊗(Y ⊗Z) : X, Y, Z ∈ C}

  X ✭✐✈✮ {a

  ✱ {l : 1⊗X → X : X ∈ C}

  X ❡ {r : X ⊗ 1 → X : X ∈ C} sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s t❛✐s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z, W ∈ C✱ ♦s ❞✐❛❣r❛♠❛s

  ((X ⊗ Y ) ⊗ Z) ⊗ W a ⊗id a

  X,Y,Z W X ⊗Y,Z,W

  (X ⊗ (Y ⊗ Z)) ⊗ W (X ⊗ Y ) ⊗ (Z ⊗ W ) a a

  ⊗Z,W ⊗W X,Y

X,Y,Z

  X ⊗ ((Y ⊗ Z) ⊗ W ) X ⊗ (Y ⊗ (Z ⊗ W )) id ⊗a

  

X Y,Z,W

  a

X,1,Y

  (X ⊗ 1) ⊗ Y X ⊗ (1 ⊗ Y ) r id

  X ⊗id Y X ⊗l Y

  X ⊗ Y ❝♦♠✉t❛♠✱ ♦✉ s❡❥❛✱ a X,Y,Z a

  X = (id X ⊗ a Y,Z,W )a X,Y (a X,Y,Z ⊗ id W ) ⊗W ⊗Y,Z,W ⊗Z,W

  X ⊗ id Y = (id X ⊗ l Y )a .

  ❡ r X,1,Y ❆s ❝♦♠✉t❛t✐✈✐❞❛❞❡s ❞♦ ♣r✐♠❡✐r♦ ❡ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛s ❞❛ ❞❡✜♥✐çã♦

  ❛♥t❡r✐♦r sã♦ ❝❤❛♠❛❞❛s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡✳

  ❈❤❛♠❛♠♦s ♦ ❢✉♥t♦r ⊗ ❞❡ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✱ ❛♣❡s❛r ❞❡ ♥ã♦ s❡r s❡♠✲ ♣r❡ ✉♠ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ♦r❞✐♥ár✐♦✳ ❈❤❛♠❛♠♦s a ❡ l, r ❞❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ❞❡ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ✉♥✐❞❛❞❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆♥❛❧♦❣❛♠❡♥t❡✱

  X,Y,Z X , r

  X ♣❛r❛ X, Y, Z ∈ C✱ ❝❤❛♠❛♠♦s a ❡ l ❞❡ ✐s♦♠♦r✜s♠♦s ❞❡ ❛ss♦✲ ❝✐❛t✐✈✐❞❛❞❡ ❡ ❞❡ ✉♥✐❞❛❞❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ❘❡❢❡r✐♠♦✲♥♦s ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ♣♦r C q✉❛♥❞♦ ♦ ❢✉♥t♦r ♣r♦✲ ❞✉t♦ t❡♥s♦r✐❛❧✱ ♦ ♦❜❥❡t♦ ✉♥✐❞❛❞❡ ❡ ♦s ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ❞❡ ❛ss♦❝✐❛✲ t✐✈✐❞❛❞❡ ❡ ✉♥✐❞❛❞❡ ❡stã♦ s✉❜❡♥t❡♥❞✐❞♦s✳

  ❊①♣❧✐❝❛♠♦s ❛❧❣✉♥s ❞❡t❛❧❤❡s ❞❛ ❉❡✜♥✐çã♦ ❙❡♥❞♦ ⊗ : C×C → C ′ ′ ′ ′ ′ ′

  : X → Y : Y → Z ✉♠ ❢✉♥t♦r✱ ♣❛r❛ f : X → Y ✱ f ✱ g : Y → Z ❡ g ♠♦r✜s♠♦s ❡♠ C✱ t❡♠♦s

  ′ ′ ′ ′ gf ⊗ g f = (g ⊗ g )(f ⊗ f ).

  ′ ′ → Y

  ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ f : X → Y ❡ g : X sã♦ ✐s♦♠♦r✜s♠♦s ❡♠ C✱ −1 −1 ′ ′

  : Y → X : Y → X ❡♥tã♦ ❡①✐st❡♠ f ❡ g ❡ ❝♦♠♦ ⊗ é ✉♠ ❢✉♥t♦r✱ t❡♠♦s

  −1 −1 −1 −1 (f ⊗ g )(f ⊗ g) = f f ⊗ g g

  ′

  = id X ⊗ id

  X

  ′

  = ⊗(id X , id X )

  ′

  = ⊗(id ) (X,X )

  ′

  = id ⊗(X,X )

  ′

  = id X .

  ⊗X −1 −1 −1

  ′

  ⊗ g ) = id Y ⊗Y =

  ❆♥❛❧♦❣❛♠❡♥t❡✱ (f ⊗ g)(f ✳ ▲♦❣♦✱ (f ⊗ g) −1 −1

  ⊗ g f ✳

  ′ ′ ′ ′ X ⊗ g f = (id X ⊗ g )(id X ⊗ f )

  ❆❧é♠ ❞✐ss♦✱ id ✳ ❆ss✐♠✱ ♣❛r❛ X ∈ C✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❢✉♥t♦r X ⊗ − : C → C ❞❡✜♥✐❞♦ ♣♦r

  (X ⊗ −)(Y ) = X ⊗ Y X ⊗ f, ❡ (X ⊗ −)(f) = id

  ♣❛r❛ t♦❞♦ Y ∈ C ❡ f ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣❛r❛ Y ∈ C✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❢✉♥t♦r − ⊗ Y : C → C ❞❡✜♥✐❞♦ ♣♦r

  (− ⊗ Y )(X) = X ⊗ Y Y , ❡ (− ⊗ Y )(f) = f ⊗ id

  ♣❛r❛ t♦❞♦ X ∈ C ❡ f ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ P❛r❛ ♦ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❞❡ ❛ss♦❝✐❛t✐✈✐❞❛❞❡✱ ♦s ❢✉♥t♦r❡s ❡♥✈♦❧✈✐✲

  ❞♦s sã♦ a : ⊗ ◦ (⊗ × Id C ) → ⊗ ◦ (Id C × ⊗), ❛♠❜♦s ❞❡ C × C × C ♣❛r❛ C ❡ ✐st♦ q✉❡r ❞✐③❡r q✉❡

  ⊗(⊗(X, Y ), Z) = ⊗(X ⊗ Y, Z) = (X ⊗ Y ) ⊗ Z ❡ ⊗(X, ⊗(Y, Z)) = ⊗(X, Y ⊗ Z) = X ⊗ (Y ⊗ Z).

  P❛r❛ ♦s ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ❞❡ ✉♥✐❞❛❞❡✱ t❡♠♦s C C , l : 1 ⊗ − → Id ❡ r : − ⊗ 1 → Id

  ❛♠❜♦s ❞❡ C ♣❛r❛ C✳

  ′ ′ ′ ❙❡❥❛♠ f : X → X ✱ g : Y → Y ❡ h : Z → Z ♠♦r✜s♠♦s ❡♠ C✱ ♦s

  ❞✐❛❣r❛♠❛s ❞❡ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a✱ l ❡ r sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ a

X,Y,Z

  (X ⊗ Y ) ⊗ Z X ⊗ (Y ⊗ Z) (f ⊗g)⊗h f ⊗(g⊗h)

  ′ ′ ′ ′ ′ ′ (X ⊗ Y ) ⊗ Z X ⊗ (Y ⊗ Z ) a

  X′ ,Y ′ ,Z′

  l r

  X X

  1 ⊗ X

  X X ⊗ 1

  X id f f f

  ⊗f ⊗id

  1

  1 1 ⊗ Y Y Y ⊗ 1 Y.

  r l Y

  Y

  ◆♦t❡♠♦s q✉❡ s❡ (C, ⊗, 1, a, l, r) é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✱ ❡♥tã♦ op

  −1 −1 −1 (C , l , r )

  , ⊗, 1, a é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❖✉tr❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❛ss♦❝✐❛❞❛ ❛ C é ❞❡✜♥✐❞❛ ❛❜❛✐①♦✳ ❉❡✜♥✐çã♦ ✸✳✶✳✷ ❙❡❥❛ (C, ⊗, 1, a, l, r) ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❆ ❝❛t❡✲ rev rev rev rev rev rev

  , ⊗ , a , l , r ) ❣♦r✐❛ r❡✈❡rs❛ ❛ C é ❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ (C , 1 ❡♠ q✉❡ rev

  = C ✭✐✮ C ❀ rev

  : C × C → C ✭✐✐✮ ⊗ é ♦ ❢✉♥t♦r ❞❡✜♥✐❞♦ ♣♦r rev rev

  X ⊗ Y = Y ⊗ X g = g ⊗ f, ❡ f ⊗

  ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C ❡ f, g ♠♦r✜s♠♦s ❡♠ C❀ rev

  ✭✐✐✐✮ 1 = 1❀ ✭✐✈✮ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ C✱ rev rev rev

  −1 a = a , l = r X = l X .

  X,Y,Z Z,Y,X X ❡ r

  X Pr♦♣♦s✐çã♦ ✸✳✶✳✸ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡ f, g : X → Y ⊗ f = id ⊗ g

  ♠♦r✜s♠♦s ❡♠ C t❛✐s q✉❡ id ✳ ❊♥tã♦ f = g✳

  1

  1 ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ✉s❛♥❞♦ ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ l ♣❛r❛ ❡ss❡s ♠♦r✲ ✜s♠♦s✱ t❡♠♦s

  ⊗ f ) = l ⊗ g) = gl f l X = l Y (id Y (id X .

  1

  1 X = gl

  X X P♦rt❛♥t♦✱ fl ❡ ❝♦♠♦ l é ✉♠ ✐s♦♠♦r✜s♠♦✱ t❡♠♦s f = g✳

  = g ⊗ id ❆♥❛❧♦❣❛♠❡♥t❡✱ f ⊗ id ✐♠♣❧✐❝❛ f = g✳

  1

  1 ❖s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s tr❛t❛♠ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦s ✐s♦♠♦r✜s♠♦s ❞❡

  ✉♥✐❞❛❞❡✳ ❚❛✐s r❡s✉❧t❛❞♦s sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❈❛♣ít✉❧♦ ✹✱ ♥♦ ❡♥t❛♥t♦✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ♥❡ss❡ ❝❛♣ít✉❧♦ ❡①❛t❛♠❡♥t❡ ♣♦r ❡st❛r♠♦s tr❛t❛♥❞♦ ❞❡ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✳ Pr♦♣♦s✐çã♦ ✸✳✶✳✹ ❙❡❥❛♠ (C, ⊗, 1, a, l, r) ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡ X, Y ∈ C

  ✳ ❊♥tã♦ ♦s ❞✐❛❣r❛♠❛s a

  1,X,Y

  (1 ⊗ X) ⊗ Y 1 ⊗ (X ⊗ Y ) l l

  ⊗id ⊗Y

  X Y

  X X ⊗ Y

  a

X,Y,1

  (X ⊗ Y ) ⊗ 1 X ⊗ (Y ⊗ 1) r

  X ⊗Y

  id ⊗r

  X Y

  X ⊗ Y sã♦ ❝♦♠✉t❛t✐✈♦s✳

  ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛ a ⊗id

  

1,1,X Y

  ✭✶✮ (r ⊗id )⊗id (id ⊗l )⊗id

1 X Y

  1 X Y

  ((1 ⊗ 1) ⊗ X) ⊗ Y (1 ⊗ X) ⊗ Y (1 ⊗ (1 ⊗ X)) ⊗ Y a a a

  

1⊗1,X,Y 1,X,Y 1,1⊗X,Y

  ✭✷✮ ✭✸✮ (1 ⊗ 1) ⊗ (X ⊗ Y ) 1 ⊗ (X ⊗ Y ) 1 ⊗ ((1 ⊗ X) ⊗ Y ) r id

  ⊗id

  X ⊗Y ⊗(l ⊗id )

  1

  1 X Y

  id ⊗l

1 X ⊗Y

  ✭✹✮ ✭∗✮ a

  1,1,X⊗Y id

  ⊗a

  1 1,X,Y

  1 ⊗ (1 ⊗ (X ⊗ Y )) ✭✺✮

  ❡♠ q✉❡ ✭✺✮ é ♦ ❞✐❛❣r❛♠❛ ❢♦r♠❛❞♦ ♣❡❧❛s ✢❡❝❤❛s ❞❛ ❜♦r❞❛✳ ❆❜❛✐①♦ ❡①✲ ♣❧✐❝❛♠♦s ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ ❝❛❞❛ ❞✐❛❣r❛♠❛✳

  ❉✐❛❣r❛♠❛ ✭✶✮✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦ ♣❛r❛ ♦s ♦❜❥❡t♦s ⊗ l X )a = r ⊗ id

  X 1✱ 1 ❡ X✱ ♦✉ s❡❥❛✱ (id 1 1,1,X 1 ❡ ✉s❛♥❞♦ ♦ ❢✉♥t♦r − ⊗ Y ✱ t❡♠♦s

  ((id ⊗ l X ) ⊗ id Y )(a ⊗ id Y ) = (id ⊗ l X )a ⊗ id Y 1 1,1,X 1 1,1,X

  = (− ⊗ Y )((id ⊗ l X )a ) 1 1,1,X

  = (− ⊗ Y )(r ⊗ id X )

  1 ⊗ id

  = (r X ) ⊗ id Y .

  1 ❉✐❛❣r❛♠❛ ✭✷✮✱ s❡❣✉❡ ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a ♣❛r❛ ♦s ♦❜❥❡t♦s 1 ⊗ 1✱ X

  X Y ❡ Y ❡ ♦s ♠♦r✜s♠♦s r ✱ id ❡ id ✳

  1 ❉✐❛❣r❛♠❛ ✭✸✮✱ s❡❣✉❡ ❞❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a ♣❛r❛ ♦s ♦❜❥❡t♦s 1✱ X ❡ Y

  X Y ❡ ♣❛r❛ ♦s ♠♦r✜s♠♦s id 1 ✱ l ❡ id ✳

  ❉✐❛❣r❛♠❛ ✭✹✮✱ s❡❣✉❡ ❞♦ ❛①✐♦♠❛ ❞♦ tr✐â❣✉❧♦ ♣❛r❛ ♦s ♦❜❥❡t♦s 1✱ 1 ❡ X ⊗ Y

  ✳

  ❉✐❛❣r❛♠❛ ✭✺✮✱ s❡❣✉❡ ❞♦ ❛①✐♦♠❛ ❞♦ ♣❡♥tá❣♦♥♦ ♣❛r❛ ♦s ♦❜❥❡t♦s 1✱ 1✱

  X ❡ Y ✳

  ▼♦str❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛ ✭∗✮ é ❝♦♠✉t❛t✐✈♦✳ ❚❡♠♦s ⊗ (l ⊗ id ⊗ id

  (id

  X Y ))a (a Y ) 1 1,1⊗X,Y 1,1,X (3)

  = a ((id ⊗ l X ) ⊗ id Y )(a ⊗ id Y ) 1,X,Y 1 1,1,X

  (1) = a ((r ⊗ id X ) ⊗ id Y )

  1,X,Y

  1 (2)

  = (r ⊗ id X )a ⊗Y 1 1⊗1,X,Y

  (4) = (id ⊗ l X )a a

  1 ⊗Y 1,1,X⊗Y 1⊗1,X,Y (5)

  ⊗ l ⊗ a ⊗ id = (id X ⊗Y )(id )a (a Y ).

  1 1 1,X,Y 1,1⊗X,Y 1,1,X ⊗ id Y ⊗

  ❈♦♠♦ a 1,1⊗X,Y ❡ a 1,1,X sã♦ ✐s♦♠♦r✜s♠♦s✱ s❡❣✉❡ q✉❡ id

  1 (l X ⊗ id Y ) = (id ⊗ l X )(id ⊗ a )

  1 ⊗Y 1 1,X,Y ✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛ ✭∗✮ ❝♦✲ ⊗ (l X ⊗ id Y ) = id ⊗ l X a

  ♠✉t❛✳ ❘❡❡s❝r❡✈❡♥❞♦✱ t❡♠♦s id

  1 1 ⊗Y 1,X,Y ✳ P❡❧❛ X ⊗ id Y = l X a

  Pr♦♣♦s✐çã♦ t❡♠♦s l ⊗Y 1,X,Y ✳ ▲♦❣♦✱ ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛✳ ❆ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ s❡❣✉❡ ❞❛ rev

  ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ♣❛r❛ C ✳ ❉❡ ❢❛t♦✱ ♦ ♣r✐♠❡✐r♦ rev

  ❞✐❛❣r❛♠❛ ♣❛r❛ Y ❡ X ❡♠ C é

  

rev

  a

  

1,Y,X

  rev rev rev rev Y ) ⊗ X (Y ⊗ X)

  (1 ⊗ 1 ⊗

  rev rev rev

  l id l ⊗

  X Y Y ⊗rev X

  rev Y ⊗

  X rev rev q✉❡✱ ❝♦♠♦ ♠♦str❛♠♦s✱ é ❝♦♠✉t❛t✐✈♦✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ⊗ ✱ a rev

  ❡ l ✱ ♦ ❞✐❛❣r❛♠❛ ❛♥t❡r✐♦r t♦r♥❛✲s❡

  

−1

  a

X,Y,1

  X ⊗ (Y ⊗ 1) (X ⊗ Y ) ⊗ 1 r id

  X ⊗Y

  ⊗r

  X Y

  X ⊗ Y q✉❡ é ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ ❞❛ ♣r♦♣♦s✐çã♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ✐♥✈❡rs❛ ❞❡ a

X,Y,1 ✳

  Pr♦♣♦s✐çã♦ ✸✳✶✳✺ ❙❡❥❛ (C, ⊗, 1, a, l, r) ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❊♥✲ tã♦

  = id ⊗ l

  X ✭✐✮ l ✱ ♣❛r❛ t♦❞♦ X ∈ C❀

  1⊗X

  1 X = r X ⊗ id ✭✐✐✮ r ✱ ♣❛r❛ t♦❞♦ X ∈ C❀

  ⊗1

  1 = r

  ✭✐✐✐✮ l ✳

  1

1 X

  ❉❡♠♦♥str❛çã♦✿ ✭✐✮ P❛r❛ X ∈ C✱ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ l ❡ t❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛ l

  1⊗X

  1 ⊗ (1 ⊗ X) 1 ⊗ X id l

  ⊗l

  X X

  1

  1 ⊗ X

  X l

  X X l = l

  X (id ⊗ l X )

  X é ❝♦♠✉t❛t✐✈♦✱ ♦✉ s❡❥❛✱ l ✳ ❈♦♠♦ l é ✉♠ ✐s♦♠♦r✲

  1⊗X

  1 = id ⊗ l

  X ✜s♠♦✱ s❡❣✉❡ q✉❡ l ✳

  1⊗X

  1 rev ✭✐✐✮ ➱ ♦ ✐t❡♠ ✭✐✮ ♣❛r❛ C ✳ ✭✐✐✐✮ ❖s ❞✐❛❣r❛♠❛s a

  

1,1,1

  (1 ⊗ 1) ⊗ 1 1 ⊗ (1 ⊗ 1) l ⊗id l

  1

1 1⊗1

  1 ⊗ 1 a

  

1,1,1

  (1 ⊗ 1) ⊗ 1 1 ⊗ (1 ⊗ 1) r id

  ⊗id ⊗l

  1

  1

  1

  1

  1 ⊗ 1 sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❉❡ ❢❛t♦✱ ♦ ♣r✐♠❡✐r♦ tr✐â♥❣✉❧♦ ❝♦♠✉t❛ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♣❛r❛ X = Y = 1 ❡ ♦ s❡❣✉♥❞♦ ♣❡❧♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦ ♣❛r❛ X = Y = 1✳ ❈♦♠ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ss❡s ❞✐❛❣r❛♠❛s ❡ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ✐t❡♠ ✭✐✮ ❞❡ss❛ ♣r♦♣♦s✐çã♦ ♣❛r❛ X = 1✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s l ⊗ id = l a

  ✭✶✮

  1 1 1⊗1 1,1,1 r ⊗ id = (id ⊗ l )a

  ✭✷✮

  1

  1

  1 1 1,1,1 l = id ⊗ l .

  ✭✐✮ 1⊗1

  1

  1 ❊♥tã♦

  (1) ⊗ id l = l a

  1 1 1⊗1 1,1,1 (i) = (id ⊗ l )a

  1 1 1,1,1 (2) = r ⊗ id .

  1

  1 ⊗ id = r ⊗ id = r

  P♦rt❛♥t♦✱ l ❡✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ l ✳

  1

  1

  1

  1

  1

  1 ▼♦str❛r❡♠♦s ❛❣♦r❛ q✉❡ ♦ ♦❜❥❡t♦ ✉♥✐❞❛❞❡ é ú♥✐❝♦✱ ❛ ♠❡♥♦s ❞❡ ✐s♦✲

  ′ ′ ′ , l , r )

  ♠♦r✜s♠♦✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ♣❛r❛ t❡r♥❛s (1, l, r) ❡ (1 ✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ ι q✉❡ r❡s♣❡✐t❛ ♦s ✐s♦♠♦r✜s♠♦s ❞❡ ✉♥✐❞❛❞❡✳

  ′ ′ ′ , a, l , r )

  Pr♦♣♦s✐çã♦ ✸✳✶✳✻ ❙❡❥❛♠ (C, ⊗, 1, a, l, r)✱ (C, ⊗, 1 ❝❛t❡❣♦r✐❛s ′

  ♠♦♥♦✐❞❛✐s ❡ X ∈ C✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ ι : 1 → 1 t❛❧ q✉❡ ♦s ❞✐❛❣r❛♠❛s ι

  ⊗id

  X

  ′ 1 ⊗ X ⊗ X

  1

  ′

  l l

  X X

  X id

  ⊗ι

  

X

  ′ X ⊗ 1

  X ⊗ 1 r ′

  X

  r

  X X

  ❝♦♠✉t❛♠✳ ❚❛❧ ✐s♦♠♦r✜s♠♦ é ú♥✐❝♦ ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♣r✐♠❡✐r♦ ♦✉ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛s ❝♦♠✉t❛r❡♠✳ ′ ′ −1

  ′

  (r ) ❉❡♠♦♥str❛çã♦✿ ❉❡✜♥✐♠♦s ι : 1 → 1 ♣♦r ι = l ✳ ❈❧❛r❛♠❡♥t❡✱ ι

  1

  1 é ✉♠ ✐s♦♠♦r✜s♠♦✱ ♣♦✐s é ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ✐s♦♠♦r✜s♠♦s✳ ❆❣♦r❛✱ ✈❡❥❛♠♦s q✉❡ ♦s ❞✐❛❣r❛♠❛s a

  

1,1′,X

  ′ ′ ) ⊗ X ⊗ X)

  (1 ⊗ 1 1 ⊗ (1 l l

  ⊗id

  X

1′ 1′⊗X

  ′ ⊗ X

  1 a

  

1,1′,X

  ′ ′ ⊗ X)

  ) ⊗ X (1 ⊗ 1 1 ⊗ (1

  

′ ′

  r id ⊗id ⊗l

  X

  1

1 X

  1 ⊗ X l

  1′⊗X

  ′ ′

  ⊗ X) 1 ⊗ (1 ⊗ X

  1

  

′ ′

  id l ⊗l

1 X

  X X

  1 ⊗ X l

  

X

  sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❉❡ ❢❛t♦✱ ♦ ♣r✐♠❡✐r♦ tr✐â♥❣✉❧♦ ❝♦♠✉t❛ ♣❡❧❛ Pr♦♣♦s✐çã♦ ′

  ♣❛r❛ ♦s ♦❜❥❡t♦s 1✱ 1 ❡ X✱ s❡♥❞♦ 1 ❛ ✉♥✐❞❛❞❡✳ ❖ s❡❣✉♥❞♦ tr✐â♥❣✉❧♦ ′ ′

  ❝♦♠✉t❛ ♣❡❧♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦ ♣❛r❛ ♦s ♦❜❥❡t♦s 1✱ 1 ❡ X✱ s❡♥❞♦ 1 ❛ ✉♥✐❞❛❞❡✳ ❖ q✉❛❞r❛❞♦ ❝♦♠✉t❛ ♣❡❧❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ l ♣❛r❛ ♦ ♠♦r✜s♠♦

  ′ l

  X ✳ ▲♦❣♦✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s

  ′ ′ ′

  l ⊗ id X = l ⊗X a ,X

  1 1 1,1 ✭✶✮ ′ ′

  ′

  ⊗ id ⊗ l r X = (id )a ,X

1 X 1,1 ✭✷✮

  1 ′ ′

  ′

  ⊗ l l l ⊗X = l X (id ).

  X

  1

  1 X ✭✸✮ ❊♥tã♦

  (2) ′ ′

  ′

  l X (r ⊗ id X ) = l X (id ⊗ l )a ,X

  1 X 1,1

  1 (3)

  ′

  ′ ′

  = l l a ,X X ⊗X 1 1,1

  (1) ′

  ′

  = l (l ⊗ id X )

  X

  1 (∗)

  ′ ′ = l (ιr ⊗ id X )

  X

  1 (∗∗)

  ′ ′ = l (ι ⊗ id X )(r ⊗ id X ),

  X

  1

  ❡♠ ✭∗✮ ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ι ❡ ❡♠ ✭∗∗✮ ♦ ❢❛t♦ ❞❡ ⊗ s❡r ✉♠ ❢✉♥t♦r✳ ′ ′ ′ ′

  X (r ⊗ id X ) = l (ι ⊗ id X )(r ⊗ id X ) ⊗ id

  1

  1

  1 ′

  X P♦rt❛♥t♦✱ l X ❡ ❝♦♠♦ r é ✉♠

  X = l (ι ⊗ id X ) ✐s♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ l X ✳ ▲♦❣♦✱ ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛✳ ❆ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ é ❛♥á❧♦❣❛✳

  ❘❡st❛✲♥♦s ♠♦str❛r q✉❡ ♦ ♠♦r✜s♠♦ ι é ú♥✐❝♦ t❛❧ q✉❡ ✉♠ ❞♦s ❞✐❛❣r❛✲ ′ ′

  ♠❛s ❝♦♠✉t❛✳ ❙❡❥❛ ι : 1 → 1 ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ ♦ ♣r✐♠❡✐r♦ ′

  ❞✐❛❣r❛♠❛ ❝♦♠✉t❛✳ ❈♦♥s✐❞❡r❛♥❞♦ X = 1 ✱ t❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛

  ′

  ι ⊗id

  1′

  ′ ′ ′ 1 ⊗ 1 1 ⊗ 1

  ′

  l l

  1′ 1′

  ′

  1 é ❝♦♠✉t❛t✐✈♦✳ ❉❛í✱ ♣❡❧♦ ✐t❡♠ ✭✐✐✐✮ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s

  ′ ′

  

′ ′

  l = l (ι ⊗ id )

  1

  1

  1 ′ ′

  ′

  = r (ι ⊗ id )

  1

  1 ′ ′

  = ι r ,

  1 ′

  ❡♠ q✉❡ ♥❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ r ♣❛r❛ ♦ ♠♦r✲ ′ ′ ′ ′ ′ −1

  ′ ′

  = ι r = l (r ) = ι ✜s♠♦ ι ✳ P♦rt❛♥t♦✱ l 1 ✳ ▲♦❣♦✱ ι 1 ✳

  1

  1 ❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r♠♦s ❡①❡♠♣❧♦s ❞❡ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✱ ❞❡✜♥✐✲

  ♠♦s ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❡str✐t❛s✳ ❚❛✐s ❝❛t❡❣♦r✐❛s ❛♣r❡s❡♥t❛♠ ✉♠❛ ❡str✉t✉r❛ ♠❛✐s s✐♠♣❧❡s ❞♦ q✉❡ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s q✉❛✐sq✉❡r✳ ❉❡✜♥✐çã♦ ✸✳✶✳✼ ❯♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛ é ✉♠❛ t❡r♥❛ (C, ⊗, 1)✱ ❡♠ q✉❡ C é ✉♠❛ ❝❛t❡❣♦r✐❛✱ ⊗ : C × C → C é ✉♠ ❢✉♥t♦r ❡ 1 ∈ C t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ C✱

  (X ⊗ Y ) ⊗ Z = X ⊗ (Y ⊗ Z), 1 ⊗ X = X = X ⊗ 1 ❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s a✱ l ❡ r sã♦ ❛s r❡s♣❡❝t✐✈❛s tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s ✐❞❡♥t✐❞❛❞❡✳ ❊①❡♠♣❧♦ ✸✳✶✳✽ (Set, ×, {∗}, a, l, r) é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✱ ❡♠ q✉❡ × : Set × Set → Set

  é ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦✱ {∗} é q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ✉♥✐tár✐♦ ❡✱ ♣❛r❛ X, Y, Z ❝♦♥❥✉♥t♦s✱ a X,Y,Z : (X × Y ) × Z → X × (Y × Z),

  7→ ((x, y), z) (x, (y, z))

  {∗} × X → X → X l X : e r X : X × {∗}

  7→ 7→ (∗, x) x (x, ∗) x

  ❉❡ ❢❛t♦✱ ♠♦str❡♠♦s q✉❡ a, l ❡ r sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s q✉❡ s❛✲ ′ t✐s❢❛③❡♠ ♦s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦✳ ❙❡❥❛♠ f : X → X ✱

  ′ ′ g : Y → Y X,Y,Z , l

  X X ❡ h : Z → Z ❢✉♥çõ❡s✳ ➱ ❝❧❛r♦ q✉❡ a ❡ r sã♦

  ❜✐❥❡çõ❡s✱ ❧♦❣♦ ✐s♦♠♦r✜s♠♦s ❡♠ Set✳ ❆❣♦r❛✱ ✈❡❥❛♠♦s q✉❡ ♦s ❞✐❛❣r❛♠❛s a

X,Y,Z

  (X × Y ) × Z X × (Y × Z) f

  (f ×g)×h ×(g×h) ′ ′ ′ ′ ′ ′

  (X × Y ) × Z X × (Y × Z ) a

  

X′ ,Y ′ ,Z′

  l r

  X X

  {∗} × X X × {∗}

  X X id f

  ×f f ×id f

  {∗} {∗}

  ′ ′ ′ ′

  {∗} × X X × {∗}

  X X r l

  X′ X′

  ❝♦♠✉t❛♠✳ ❙❡❥❛♠ x ∈ X, y ∈ Y ❡ z ∈ Z✳ ❊♥tã♦ (f × (g × h))a X,Y,Z ((x, y), z) = (f × (g × h))(x, (y, z))

  = (f (x), (g(y), h(z)))

  ′ ′ ′

  = a X ,Y ,Z ((f (x), g(y)), h(z))

  ′ ′ ′

  = a X ,Y ,Z ((f × g) × h)((x, y), z) ❡ f l

  X (∗, x) = f (x)

  

  = l X (∗, f (x))

  

  = l X (id × f )(∗, x).

  {∗} ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡ r é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ▼♦str❡✲

  ♠♦s ♦s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦✳ ❙❡❥❛♠ X, Y, Z, W ❝♦♥✲ ❥✉♥t♦s ❡ x ∈ X, y ∈ Y, z ∈ Z, w ∈ W ✳ ❊♥tã♦ a X,Y,Z a

  X (((x, y), z), w) ×W ×Y,Z,W

  = a X,Y,Z ((x, y), (z, w)) ×W

  = (x, (y, (z, w))) = (id X × a Y,Z,W )(x, ((y, z), w)) = (id X × a Y,Z,W )a X,Y ((x, (y, z)), w)

  ×Z,W = (id X × a Y,Z,W )a X,Y (a X,Y,Z × id W )(((x, y), z), w)

  ×Z,W ❡

  (r X × id Y )((x, ∗), y) = (x, y) × l

  = (id

  X Y )(x, (∗, y)) × l

  = (id X Y )a X, ((x, ∗), y).

  {∗},Y k , ⊗ k , k, a, l, r) ❊①❡♠♣❧♦ ✸✳✶✳✾ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❊♥tã♦ (V ect é ✉♠❛

  × V ect → V ect k : V ect k k k

  ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✱ ❡♠ q✉❡ ⊗ é ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ s♦❜r❡ k ❡✱ ♣❛r❛ X, Y, Z ❝♦♥❥✉♥t♦s✱ a X,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z),

  7→ (x ⊗ y) ⊗ z x ⊗ (y ⊗ z)

  → X → X l X : k ⊗ X e r X : X ⊗ k

  7→ 7→ 1 ⊗ x x x ⊗ 1 x k , ⊗ k , k, a, l, r)

  ❆♥❛❧♦❣❛♠❡♥t❡✱ (vect é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❊①❡♠♣❧♦ ✸✳✶✳✶✵ ❙❡❥❛♠ k ✉♠ ❝♦r♣♦✱ G ✉♠ ❣r✉♣♦ ❡ ω ✉♠ ✸✲❝♦❝✐❝❧♦✱

  × × = k − {0}

  ♦✉ s❡❥❛✱ ω : G × G × G → k ✱ ❡♠ q✉❡ k ✱ é ✉♠❛ ❢✉♥çã♦ t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r a, b, c, d ∈ G✱ ω(a, b, c)ω(a, bc, d)ω(b, c, d) = ω(ab, c, d)ω(a, b, cd). ❆ ❝❛t❡❣♦r✐❛ C(G, ω) é ❛q✉❡❧❛ ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s

  G ✲❣r❛❞✉❛❞♦s✱ ♦✉ s❡❥❛✱ X ∈ C(G, ω) s❡ X é ✉♠ k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡

  L X = X g g g ✱ ❡♠ q✉❡ X sã♦ k✲s✉❜❡s♣❛ç♦s ❞❡ X✳

  ∈G L L

  X g Y g ∈ C(G, ω) P❛r❛ X = g ✱ Y = g ✱ ✉♠ ♠♦r✜s♠♦ ❡♠

  ∈G ∈G C

  (G, ω) g ) ⊆ Y g

  é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ k✲❧✐♥❡❛r f : X → Y t❛❧ q✉❡ f(X ✱ ♣❛r❛ t♦❞♦ g ∈ G✳

  L L X g , Y = Y g

  ❙❡ X, Y ∈ C(G, ω) ❝♦♠ ❣r❛❞✉❛çõ❡s X = g g ✱ ∈G ∈G

  ❡♥tã♦ X ⊗ Y ∈ C(G, ω) ❝♦♠ ❣r❛❞✉❛çã♦ M M

  ⊗ Y X ⊗ Y = (X ⊗ Y ) g , g = X a b .

  ❡♠ q✉❡ (X ⊗ Y ) g ab

  ∈G =g L

  δ k ❖ ♦❜❥❡t♦ 1 ∈ C(G, ω) é ♦ ❝♦r♣♦ k ❝♦♠ ❣r❛❞✉❛çã♦ k = 1,g ✱ g

  ∈G k ❡♠ q✉❡ δ 1,g é ♦ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r ❡ ⊗ = ⊗ ✳ L X g , Y =

  ❙❡❥❛♠ X, Y, Z ∈ C(G, ω) ❝♦♠ ❣r❛❞✉❛çõ❡s X = g ∈G

  L L Y g , Z = Z g a , y ∈

  ✳ P❛r❛ q✉❛✐sq✉❡r a, b, c ∈ G✱ x ∈ X g ∈G g ∈G

  Y b , z ∈ Z c ✱ ❝♦♥s✐❞❡r❡♠♦s ❛s s❡❣✉✐♥t❡s tr❛♥s❢♦r♠❛çõ❡s k✲❧✐♥❡❛r❡s a X,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z),

  (x ⊗ y) ⊗ z 7→ ω(a, b, c)x ⊗ (y ⊗ z) l X : k ⊗ X →

  X X : X ⊗ k → X.

  ❡ r −1

  1 ⊗ x 7→ ω(1, 1, a) x x ⊗ 1 7→ ω(a, 1, 1)x ❊♥tã♦ (C(G, ω), ⊗, 1, a, l, r) é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❉❡ ❢❛t♦✱

  X,Y,Z

  X ♠♦str❡♠♦s q✉❡✱ ♣❛r❛ X, Y, Z ∈ C(G, ω)✱ a ❡ l sã♦ ♠♦r✜s♠♦s ❡♠ C(G, ω)✳ ◆♦t❡♠♦s q✉❡✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ t❡♠♦s

  M ((X ⊗ Y ) ⊗ Z) g = (X ⊗ Y ) e ⊗ Z c ec

  =g M M

  = (X a ⊗ Y b ) ⊗ Z c ec

  =g ab =e M = (X a ⊗ Y b ) ⊗ Z c . abc

  =g L g = X a ⊗ (Y b ⊗ Z c )

  ❆♥❛❧♦❣❛♠❡♥t❡✱ (X ⊗ (Y ⊗ Z)) ✳ ❚❛♠❜é♠✱ abc

  =g M (k ⊗ X) g = δ k ⊗ X b = k ⊗ X g .

  1,a ab =g a b c

  ❆❣♦r❛✱ ♣❛r❛ g, a, b, c ∈ G✱ g = abc✱ x ∈ X ✱ y ∈ Y ✱ z ∈ Z ✱ t❡♠♦s ⊗ (Y ⊗ Z a((x ⊗ y) ⊗ z) = ω(a, b, c)x ⊗ (y ⊗ z) ∈ X a b c ),

  X (1 ⊗ x) = x ∈ X a .

  ❡ l X,Y,Z ((X a ⊗ Y b ) ⊗ Z c ) ⊆ X a ⊗ (Y b ⊗ Z c ) X (k ⊗ X a ) ⊆ X a

  ▲♦❣♦✱ a ❡ l X,Y,Z (((X ⊗ Y ) ⊗ Z) g ) ⊆ (X ⊗ (Y ⊗ Z)) g

  ❡ ✐ss♦ ♥♦s ❞✐③ q✉❡ a ❡ q✉❡ l X ((k ⊗ X) g ) ⊆ X g X,Y,Z

  X ✳ P♦rt❛♥t♦✱ a ❡ l sã♦ ♠♦r✜s♠♦s ❡♠ C(G, ω) ❡

  X ❛♥❛❧♦❣❛♠❡♥t❡✱ r é ✉♠ ♠♦r✜s♠♦ ❡♠ C(G, ω)✳

  ❘❡st❛✲♥♦s ♠♦str❛r ♦ ❛①✐♦♠❛ ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦✳ ❙❡❥❛♠ X, Y, Z, W ∈ C(G, ω) a b c d

  ✱ a, b, c, d ∈ G ❡ x ∈ X ✱ y ∈ Y ✱ z ∈ Z ✱ w ∈ W ✳ ❊♥tã♦ a X,Y,Z a

  X (((x ⊗ y) ⊗ z) ⊗ w) ⊗W ⊗Y,Z,W

  (∗) = a X,Y,Z (ω(ab, c, d)(x ⊗ y) ⊗ (z ⊗ w))

  ⊗W = ω(ab, c, d)ω(a, b, cd)x ⊗ (y ⊗ (z ⊗ w))

  (∗∗) = ω(a, b, c)ω(a, bc, d)ω(b, c, d)x ⊗ (y ⊗ (z ⊗ w)) = (id X ⊗ a Y,Z,W )(ω(a, b, c)ω(a, bc, d)x ⊗ ((y ⊗ z) ⊗ w)) = (id X ⊗ a Y,Z,W )a X,Y (ω(a, b, c)(x ⊗ (y ⊗ z)) ⊗ w)

  ⊗Z,W = (id X ⊗ a Y,Z,W )a X,Y (a X,Y,Z ⊗ id W )(((x ⊗ y) ⊗ z) ⊗ w),

  ⊗Z,W a b ❡♠ (∗) ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ x ∈ X ❡ y ∈ Y ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ x ⊗ y ∈ X a ⊗ Y b ⊆ (X ⊗ Y ) ab

  ❡ ❡♠ (∗∗) ✉s❛♠♦s ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ✸✲❝♦❝✐❝❧♦ ω✳ ❆❣♦r❛✱ ♦ tr✐â♥❣✉❧♦ (r X ⊗ id Y )((x ⊗ 1) ⊗ y) = ω(a, 1, 1)x ⊗ y

  −1 = ω(a, 1, 1)ω(1, 1, b)ω(1, 1, b) x ⊗ y = ω(a, 1, 1)ω(1, 1, b)(id X ⊗ l Y )(x ⊗ (1 ⊗ y))

  (∗∗∗) = ω(a, 1, b)(id X ⊗ l Y )(x ⊗ (1 ⊗ y)) = (id X ⊗ l Y )a (x ⊗ (1 ⊗ y)),

X,1,Y

  ❡♠ (∗ ∗ ∗) ✉s❛♠♦s ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ✸✲❝♦❝✐❝❧♦ ω✳ ❉❡ ❢❛t♦✱ t❡♠♦s ω(a, 1, 1)ω(a, 1, b)ω(1, 1, b)

  = ω(a, 1, 1)ω(a, 1 · 1, b)ω(1, 1, b) = ω(a1, 1, b)ω(a, 1, 1b) = ω(a, 1, b)ω(a, 1, b),

  ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ ω(a, 1, 1)ω(1, 1, b) = ω(a, 1, b)✳ ❊①❡♠♣❧♦ ✸✳✶✳✶✶ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛✳ ❉♦ ❊①❡♠♣❧♦ ♣❛r❛ C = D

  C ) ✱ End(C) = F un(C, C) é ✉♠❛ ❝❛t❡❣♦r✐❛✳ ▼♦str❡♠♦s q✉❡ (End(C), ⊗, Id

  é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✱ ❡♠ q✉❡ ⊗ : End(C) × End(C) → End(C)

  é ❞❡✜♥✐❞♦ ♣♦r ⊗(G, F ) = G ◦ F

  ❡ ⊗ (ν, µ) = ν ∗ µ, ♣❛r❛ q✉❛✐sq✉❡r F, G ∈ End(C) ❡ ν, µ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s✳ ❉❡ ❢❛t♦✱

  ′ ′ : G → H : R → S s❡❥❛♠ ν : F → G✱ ν ✱ µ : J → R✱ µ tr❛♥s❢♦r♠❛çõ❡s

  ′ ◦ ν : F → H

  ♥❛t✉r❛✐s✱ ❡♠ q✉❡ F, G, H, J, R, S ∈ End(C)✳ ❊♥tã♦ ν ✱ ′

  ′ ′ µ ◦µ : J → S ∗µ : G◦R → H◦S

  ✳ ◆♦t❡♠♦s q✉❡✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ν ❡ ν ∗ µ : F ◦ J → G ◦ R sã♦ ❞❛❞❛s ♣♦r

  ′ ′ ′ ′ (ν ∗µ ) X = ν G(µ ) X = ν R F (µ X ),

  S X ❡ (ν∗µ) (X) ♣❛r❛ t♦❞♦ X ∈ C.

  (X) ▼♦str❡♠♦s q✉❡ ⊗ é ✉♠ ❢✉♥t♦r✳ ❉❡ ❢❛t♦✱ ♣❛r❛ ❝❛❞❛ X ∈ C✱ t❡♠♦s

  ⊗(id ) X = ⊗(id F , id G )

  X (F,G)

  = (id F ∗ id G )

  ′ ∗ µ

  X )F (µ X )

  (∗) = ν

  ′ S

  (X) G(µ

  ′

  X )ν R

  (X) F (µ X )

  = (ν ′

  ∗ µ ′

  ) X (ν ∗ µ)

  X = ((ν

  ′ ) ◦ (ν ∗ µ))

  (X) F (µ

  X = (⊗(ν

  ′ , µ

  ′ ) ◦ ⊗(ν, µ)) X .

  ❡♠ q✉❡ (∗) s❡❣✉❡ ❞❛ ✐❣✉❛❧❞❛❞❡ ❞❛❞❛ ♥♦ ♣❛rá❣r❛❢♦ ❛♣ós ❛ ❉❡✜♥✐çã♦ P♦rt❛♥t♦✱ ⊗(id (F,G)

  ) = id ⊗(F,G) ❡ ⊗((ν

  ′ , µ

  ′ )◦(ν, µ)) = ⊗(ν

  ′ , µ

  ′ )◦

  ⊗(ν, µ) ✳

  ❈♦♠♦ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s ❛❞✐❝✐♦♥❛✐s✱ ♣♦❞❡✲ ♠♦s ❞❡✜♥✐r ♦s ❢✉♥t♦r❡s q✉❡ ♣r❡s❡r✈❛♠ t❛✐s ♣r♦♣r✐❡❞❛❞❡s✳ ❉❡✜♥✐çã♦ ✸✳✶✳✶✷ ❙❡❥❛♠ C, D ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✳ ❯♠ ❢✉♥t♦r ♠♦✲ ♥♦✐❞❛❧ ❡♥tr❡ C ❡ D é ✉♠❛ t❡r♥❛ (F, ζ, φ)✱ ❡♠ q✉❡ ✭✐✮ F : C → D é ✉♠ ❢✉♥t♦r❀ ✭✐✐✮ ζ : ⊗ ◦ (F × F ) → F ◦ ⊗ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✱ ✐st♦ é✱ ζ X,Y : F (X) ⊗ F (Y ) → F (X ⊗ Y )

  ′

  (X) ν S

  X = (id F ) G

  ◦ ν, µ ′

  (X) F ((id G ) X )

  = id F (G(X))

  = id (F ◦G)(X)

  = (id F ◦G

  )

  X = (id

  ⊗(F,G) )

  X ❡

  ⊗((ν ′

  , µ ′

  ) ◦ (ν, µ)) X = ⊗(ν ′

  ◦ µ)

  = ν ′ S

  X = ((ν

  ′ ◦ ν) ∗ (µ

  ′ ◦ µ))

  X = (ν

  ′ ◦ ν)

  S (X) F ((µ

  ′ ◦ µ) X )

  = ν ′ S

  (X) ν

  S (X) F (µ

  ′

  X µ X )

  ✱ ♣❛r❛ X, Y ∈ C❀ ✭✐✐✐✮ φ : 1 → F (1) é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ D❀ ❛❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ C✱ ♦s ❞✐❛❣r❛♠❛s

  (F (X) ⊗ F (Y )) ⊗ F (Z) F (X ⊗ Y ) ⊗ F (Z) F (X) ⊗ (F (Y ) ⊗ F (Z))

X,Y

Y,Z

X,Y,Z

  F (X) = F (r

  F (r

  X

  ) sã♦ ❝♦♠✉t❛t✐✈♦s✱ ♦✉ s❡❥❛✱

  ζ X,Y ⊗Z (id F (X)

  ⊗ ζ Y,Z )a F (X),F (Y ),F (Z)

  = F (a X,Y,Z )ζ X⊗Y,Z (ζ X,Y ⊗ id F

  (Z) ), l

  F (X) = F (l

  X )ζ 1,X (φ ⊗ id F (X) )

❡ r

  P❛r❛ ❡①♣❧✐❝✐t❛r✱ s❡ f : X → X ′

  X )ζ X,1 (id F (X) ⊗

  φ ).

  ⊗φ ζ

  ✱ g : Y → Y ′ sã♦ ♠♦r✜s♠♦s ❡♠ C✱ ♦

  ❞✐❛❣r❛♠❛ ❞❡ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ ζ é F (X) ⊗ F (Y ) F (X ⊗ Y )

  F (X ′

  ) ⊗ F (Y ′

  ) F (X ′

  ⊗ Y ′ ). ζ

  F (f ⊗g) F (f )⊗F (g) ζ

  X′ ,Y ′

  X,1

  F (X)

  F ((X ⊗ Y ) ⊗ Z) F (X) ⊗ F (Y ⊗ Z) F (X ⊗ (Y ⊗ Z)) 1 ⊗ F (X) F (X)

  ⊗ζ

  F (1) ⊗ F (X) F (1 ⊗ X) F (X) ⊗ 1

  F (X) F (X) ⊗ F (1) F (X ⊗ 1)

  ζ

  ⊗id

  F (Z)

  a

  F (X),F (Y ),F (Z)

  ζ

  X ⊗Y,Z

  id

  F (X)

  F (a

  id

  ) ζ

  X,Y ⊗Z

  l

  F (X)

  φ ⊗id

  F (X)

  ζ

  1,X

  F (l

  X

  ) r

  F (X)

X,Y

  ❊①❡♠♣❧♦ ✸✳✶✳✶✸ ❙❡❥❛ (C, ⊗, 1, a, l, r) ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❆ t❡r♥❛ Id

  Id Id C

  C C

  (Id C , ζ , φ ) = id

  X é ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧✱ ❡♠ q✉❡ ζ ⊗Y ✱ ♣❛r❛

  X,Y Id C

  X, Y ∈ C = id ❡ φ 1 ✳

  F F , φ ) : C →

  ❉❡✜♥✐çã♦ ✸✳✶✳✶✹ ❙❡❥❛♠ C✱DE ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✱ (F, ζ G G

  G G D , φ ) : D → E , φ )

  ❡ (G, ζ ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s✳ ❆ ❝♦♠♣♦s✐çã♦ ❞❡ (G, ζ F F G ◦F G ◦F

  , φ ) , φ ) ❡ (F, ζ é ❛ t❡r♥❛ (G ◦ F, ζ ✱ ❡♠ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r

  G ◦F G ◦F X, Y ∈ C

  ✱ ζ X,Y ❡ φ sã♦ ❛s ❝♦♠♣♦s✐çõ❡s

  G ◦F

  ζ

X,Y

  G(F (X)) ⊗ G(F (Y )) G(F (X ⊗ Y ))

  G F

  ζ G (ζ )

  

F (X),F (Y ) X,Y

  G(F (X) ⊗ F (Y ))

  

◦F

G

  φ

G(F (1))

1 G

  φ

  F

  G (φ )

  G(1) ♦✉ s❡❥❛✱

  G ◦F F G G ◦F F G ζ = G(ζ )ζ = G(φ )φ . X,Y X,Y ❡ φ

  F (X),F (Y ) Pr♦♣♦s✐çã♦ ✸✳✶✳✶✺ ❆ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s é ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♠ ❛s ♥♦t❛çõ❡s ❞❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r✱ ❞❡✈❡♠♦s ♠♦s✲ tr❛r q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ C✱

  G G ◦F ◦F

  ζ (id ⊗ ζ )a X,Y (G◦F )(X) Y,Z (G◦F )(X),(G◦F )(Y ),(G◦F )(Z)

  ⊗Z G G

  ◦F ◦F ⊗ id

  = (G ◦ F )(a X,Y,Z )ζ (ζ ), X ⊗Y,Z X,Y (G◦F )(Z)

  G ◦F G ◦F l = (G ◦ F )(l X )ζ (φ ⊗ id )

  (G◦F )(X) (G◦F )(X) ❡ 1,X

  G G ◦F ◦F r = (G ◦ F )(r X )ζ (id ⊗ φ ).

  (G◦F )(X) (G◦F )(X) X,1

  ❉❡ ❢❛t♦✱ t❡♠♦s G G

  ◦F ◦F ⊗ ζ

  ζ (id )a X,Y ⊗Z (G◦F )(X) Y,Z (G◦F )(X),(G◦F )(Y ),(G◦F )(Z)

  F G F G = G(ζ )ζ (id ⊗ G(ζ )ζ )

  X,Y F G (F (X)) Y,Z F ⊗Z (X),F (Y ⊗Z) (Y ),F (Z) a G (F (X)),G(F (Y )),G(F (Z))

  F G F = G(ζ )ζ (G(id F ) ⊗ G(ζ ))

  X,Y ⊗Z F (X) Y,Z (X),F (Y ⊗Z)

  G (id G ⊗ ζ )a G

  (F (X)) F (F (X)),G(F (Y )),G(F (Z)) (Y ),F (Z)

  (1) F F G

  = G(ζ )G(id F ⊗ ζ )ζ X,Y (X) Y,Z F

  ⊗Z (X),F (Y )⊗F (Z) G

  (id ⊗ ζ )a G (F (X)) G (F (X)),G(F (Y )),G(F (Z))

  F (Y ),F (Z) (2)

  F F = G(ζ )G(id F ) ⊗ ζ )G(a F )

  X,Y ⊗Z (X) Y,Z (X),F (Y ),F (Z) G G

  ζ (ζ ⊗ id ) F F G (F (Z))

  (X)⊗F (Y ),F (Z) (X),F (Y ) (3)

  F F = G(F (a X,Y,Z ))G(ζ )G(ζ ⊗ id )

  X X,Y F (Z) ⊗Y,Z

  G G ζ (ζ ⊗ id G )

  F F (F (Z)) (X)⊗F (Y ),F (Z) (X),F (Y )

  (4) F G

  = G(F (a X,Y,Z ))G(ζ )ζ

  X F ⊗Y,Z (X⊗Y ),F (Z)

  F G ⊗ id

  (G(ζ ) ⊗ G(id F ))(ζ G ) X,Y (Z) F (F (Z))

  (X),F (Y ) G G

  ◦F ◦F = (G ◦ F )(a X,Y,Z )ζ (ζ ⊗ id ),

  X X,Y (G◦F )(Z) ⊗Y,Z

  G ❡♠ ✭✶✮ ❡ ✭✹✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ ζ ✱ ✈✐❞❡ ❞✐❛❣r❛♠❛s

  G

  ζ

  F (X),F (Y )⊗F (Z)

  G(F (X)) ⊗ G(F (Y ) ⊗ F (Z)) G(F (X) ⊗ (F (Y ) ⊗ F (Z)))

  F F

  G G

  (id )⊗G(ζ ) (id ⊗ζ )

  F (X) F (X) Y,Z

Y,Z

  G(F (X)) ⊗ G(F (Y ⊗ Z)) G(F (X) ⊗ F (Y ⊗ Z))

  G

  ζ

  (X),F (Y ⊗Z) F G

  ζ

  F (X)⊗F (Y ),F (Z)

  G(F (X) ⊗ F (Y )) ⊗ G(F (Z)) G((F (X) ⊗ F (Y )) ⊗ F (Z))

  F F

  G G

  (ζ )⊗G(id ) (ζ ⊗id )

  F (Z) F (Z) X,Y

  X,Y G(F (X ⊗ Y )) ⊗ G(F (Z)) G(F (X ⊗ Y ) ⊗ F (Z)). G

  ζ

  F (X⊗Y ),F (Z)

  ❊♠ ✭✷✮ ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ G s❡r ❢✉♥t♦r ♠♦♥♦✐❞❛❧ ❡ ❡♠ ✭✸✮ ✉s❛♠♦s ♦

  ❢❛t♦ ❞❡ F s❡r ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳ ❆❣♦r❛✱ ✈❡❥❛♠♦s ♦ ♣ró①✐♠♦ ❞✐❛❣r❛♠❛ l = l (G◦F )(X) G (F (X))

  (∗) G G

  ⊗ id = G(l F )ζ (φ G )

  (X) (F (X)) 1,F (X)

  (∗∗) F F G G

  = G(F (l X ))G(ζ )G(φ ⊗ id F )ζ (φ ⊗ id G ) (X) (F (X))

  1,X 1,F (X) F G F G

  = G(F (l X ))G(ζ )ζ (G(φ ) ⊗ G(id F ))(φ ⊗ id G ) F (X) (F (X))

  1,X (1),F (X)

  G G ◦F ◦F

  = (G ◦ F )(l X )ζ (φ ⊗ id ), (G◦F )(X)

  1,X ❡♠ (∗) ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ G s❡r ❢✉♥t♦r ♠♦♥♦✐❞❛❧ ❡ ❡♠ (∗∗) ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ F s❡r ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳ ❆ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛ ❡♥✈♦❧✈❡♥❞♦ r

  X é ♣r♦✈❛❞❛ ❛♥❛❧♦❣❛♠❡♥t❡✳

  F F , φ )

  ❉❡✜♥✐çã♦ ✸✳✶✳✶✻ ❙❡❥❛♠ C, D ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❡ (F, ζ ✱ G G

  (G, ζ , φ ) : C → D ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧

  F F G G , φ ) , φ )

  ♠♦♥♦✐❞❛❧ ❡♥tr❡ (F, ζ ❡ (G, ζ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ µ : F → G t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ ♦s ❞✐❛❣r❛♠❛s

  µ

  X ⊗µ Y

  F (X) ⊗ F (Y ) G(X) ⊗ G(Y )

  F G

  ζ ζ

X,Y X,Y

  F (X ⊗ Y ) G(X ⊗ Y ) µ

  

⊗Y

  X

1 F G

  φ φ F (1) G(1)

  µ

  

1

  sã♦ ❝♦♠✉t❛t✐✈♦s✱ ♦✉ s❡❥❛✱ G F G F

  ζ (µ X ⊗ µ Y ) = µ X ζ = µ φ .

  X,Y ⊗Y X,Y ❡ φ

  1 ❯♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧

  ♠♦♥♦✐❞❛❧ q✉❡ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❉✐③❡♠♦s q✉❡ C ❡ D sã♦ ♠♦✲ F F

  , φ ) : ♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st✐r❡♠ ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s (F, ζ

  G G C → D , φ ) : D → C

  ✱ (G, ζ ❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ♠♦♥♦✐❞❛✐s µ : G ◦ F → Id C D

  ❡ ν : F ◦ G → Id ✳ P❛r❛ ♠♦str❛r♠♦s ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ ♣r❡❝✐s❛♠♦s

  ❞❡✜♥✐r ❝❛t❡❣♦r✐❛ ❡sq✉❡❧ét✐❝❛ ❡ ❡sq✉❡❧❡t♦ ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛✳

  ❉❡✜♥✐çã♦ ✸✳✶✳✶✼ ❯♠❛ ❝❛t❡❣♦r✐❛ C é ❞✐t❛ ❡sq✉❡❧ét✐❝❛ s❡✱ ♣❛r❛ q✉❛✐s✲ q✉❡r X, Y ∈ C t❛✐s q✉❡ X ≃ Y ✐♠♣❧✐❝❛r X = Y ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠❛ ❝❛t❡❣♦r✐❛ ❡sq✉❡❧ét✐❝❛ C ♥ã♦ ♣♦ss✉✐ ♦❜❥❡t♦s

  ❞✐st✐♥t♦s q✉❡ s❡❥❛♠ ✐s♦♠♦r❢♦s✳ ❉❡✜♥✐çã♦ ✸✳✶✳✶✽ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛✳ ❖ ❡sq✉❡❧❡t♦ ❞❡ C é ❛ s✉❜❝❛✲ t❡❣♦r✐❛ ♣❧❡♥❛ Sk(C) ❞❡ C q✉❡ ❝♦♥s✐st❡ ❡♠ ❝♦♥s✐❞❡r❛r ✉♠ só ♦❜❥❡t♦ ❡♠ ❝❛❞❛ ❝❧❛ss❡ ❞❡ ✐s♦♠♦r✜s♠♦ ❞❡ ♦❜❥❡t♦s ❞❡ C✳

  ➱ ❝❧❛r♦ q✉❡ Sk(C) é ✉♠❛ ❝❛t❡❣♦r✐❛ ❡sq✉❡❧ét✐❝❛✳ k

  ❊①❡♠♣❧♦ ✸✳✶✳✶✾ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❉❡♥♦t❛♠♦s ♣♦r Matr ❛ ❝❛t❡❣♦r✐❛ o = {0, 1, 2, · · · }

  ❝✉❥♦s ♦❜❥❡t♦s sã♦ ♦s ❡❧❡♠❡♥t♦s ❞❡ N ✳ P❛r❛ q✉❛✐sq✉❡r n, m ∈ N o ✱ ✉♠ ♠♦r✜s♠♦ ❞❡ n ♣❛r❛ m é ✉♠❛ ♠❛tr✐③ m × n ❝♦♠ ❡♥✲

  M atr (n, m) tr❛❞❛s ♥♦ ❝♦r♣♦ k✳ ◆♦t❛♠♦s Hom ♣❡❧♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❛s

  k

  m (k) ♠❛tr✐③❡s m × n s♦❜r❡ k✱ ✐st♦ é✱ M ×n ✳

  ❚❛❧ ❝❛t❡❣♦r✐❛ é ♠♦♥♦✐❞❛❧✱ ♣❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡❥❛ ❬✼❪✳ ▼♦str❡✲ k M atr (n, m)

  ♠♦s q✉❡ Matr é ❡sq✉❡❧ét✐❝❛✳ ❉❡ ❢❛t♦✱ s❡❥❛ A ∈ Hom ✉♠

  k

  M atr (m, n) n ✐s♦♠♦r✜s♠♦✳ ❊♥tã♦ ❡①✐st❡ B ∈ Hom k t❛❧ q✉❡ AB = I ❡ BA = I m

  ✳ ❆ss✐♠✱ n = tr(AB) = tr(BA) = m✳ ▲♦❣♦✱ ❞♦✐s ♦❜❥❡t♦s k k

  ✐s♦♠♦r❢♦s ❡♠ Matr sã♦ ✐❣✉❛✐s ❡ ✐ss♦ ♥♦s ❞✐③ q✉❡ Matr é ❡sq✉❡❧ét✐❝❛✳ ❙❡❥❛ (C, ⊗, 1, a, l, r) ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ●♦st❛rí❛♠♦s ❞❡ ❛♣r❡✲ s❡♥t❛r ✉♠❛ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧ ♣❛r❛ ❛ ❝❛t❡❣♦r✐❛ Sk(C)✳ ❋❛③❡♠♦s ✐ss♦

  ❛tr❛✈és ❞♦s ♣ró①✐♠♦s ❞♦✐s ❧❡♠❛s✳ P❛r❛ ❝❛❞❛ X ∈ C✱ ❞❡♥♦t❛♠♦s ♣♦r X ∈ Sk(C) ♦ ú♥✐❝♦ ♦❜❥❡t♦ ❞❡

  Sk(C) : X → X t❛❧ q✉❡ X ≃ X ❡ ✜①❛♠♦s ✉♠ ✐s♦♠♦r✜s♠♦ σ

  X ❡♠ C✳ ❉❡✜♥✐♠♦s ⊙ : Sk(C) × Sk(C) → Sk(C) ♣♦r

  −1 ⊙(X, Y ) = X ⊙ Y = X ⊗ Y ′ ′ (f ⊗ g)σ ,

  ❡ ⊙(f, g) = f ⊙g = σ

  X X ⊗Y ⊗Y ′ ′

  ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ Sk(C) ❡ f : X → X ✱ g : Y → Y ♠♦r✜s♠♦s ❡♠ Sk(C)

  ✳ P❛r❛ ♠❡❧❤♦r ✈✐s✉❛❧✐③❛çã♦✱ ❡①♣❧✐❝✐t❛♠♦s ❛ ❝♦♠♣♦st❛

  −1

  σ σ f

  

X ⊗Y ⊗g X′ ⊗Y ′

  ′ ′ ′ ′ ′ ′

  X ⊗ Y X ⊗ Y X ⊙ Y = X ⊗ Y X ⊗ Y = X ⊙ Y .

  ▲❡♠❛ ✸✳✶✳✷✵ ❈♦♠ ❛ ♥♦t❛çã♦ ❛❝✐♠❛✱ ⊙ : Sk(C) × Sk(C) → Sk(C) é ✉♠ ❢✉♥t♦r✳

  ′ ′ ′ → Y

  : X , g : Y → ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ♣❛r❛ f : X → Y, f

  ′ ′ ′ Z, g : Y → Z

  ♠♦r✜s♠♦s ❡♠ Sk(C)✱ t❡♠♦s ⊙(id ) = ⊙(id X , id Y )

  (X,Y )

  = σ −1 X ⊗Y

  X sã♦ ✐s♦♠♦r✜s♠♦s✱ ♣❛r❛ q✉❛✐sq✉❡r

  ′

  ⊗(Y

  ′

  X

  −1

  (f ⊙ (g ⊙ h))a X,Y,Z = σ

  ′ ♠♦r✜s♠♦s ❡♠ Sk(C)✱ t❡♠♦s

  ′ ✱ h : Z → Z

  ′ ✱ g : Y → Y

  X, Y, Z ∈ Sk(C) ✱ ♣♦✐s sã♦ ❝♦♠♣♦s✐çõ❡s ❞❡ ✐s♦♠♦r✜s♠♦s✳ ❆❣♦r❛✱ ♣❛r❛ f : X → X

  X ❡ r

  ′

  X,Y,Z ✱ l

  ▲❡♠❛ ✸✳✶✳✷✶ ❆ sê①t✉♣❧❛ (Sk(C), ⊙, 1, a, l, r) é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐✲ ❞❛❧✳ ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❡❥❛♠♦s q✉❡ a✱ l ❡ r sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s✳ ➱ ❝❧❛r♦ q✉❡ a

  X ⊗1 .

  1 )σ

  X ⊗ σ

  ❡ r X = r X (id

  X )σ 1⊗X

  1 ⊗ id

  , l X = l X (σ

  Z )σ (X⊙Y )⊗Z

  X ⊗Y ⊗ id

  ⊙Z

  ) (f ⊗ (g ⊙ h))σ

  −1 Y

  = σ −1

  X ⊗Y

  ⊗Z ) a X,Y,Z (σ

  −1 Y

  ) (f ⊗ (g ⊙ h)σ

  ′

  ⊙Z

  ′

  ⊗(Y

  ′

  X

  ⊗ id Z )σ (X⊙Y )⊗Z

  X ⊗(Y ⊙Z) a X,Y,Z

  X ⊗Y

  ⊗Z ) a X,Y,Z (σ

  −1 Y

  ) (f ⊗ (g ⊙ h))(id X ⊗ σ

  ′

  ⊙Z

  ′

  ⊗(Y

  ′

  X

  = σ −1

  ⊗Z )a X,Y,Z (σ

  X ⊗ σ

  (id X ⊗ id Y )σ

  ′ f

  ′

  X ⊗X

  ′ )σ

  ′ f

  ′ (gf ⊗ g

  ⊗Z

  −1 Z

  ′ = σ

  ′ f

  ′ ) = gf ⊙ g

  ❡ ⊙(gf, g

  ⊗Z

  = id ⊙(X,Y )

  X ⊙Y

  = id

  = id X ⊗Y

  X ⊗Y

  σ

  −1 X ⊗Y

  X ⊗Y = σ

  X ⊗Y id X ⊗Y σ

  = σ −1

  X ⊗Y

  = σ −1 Z

  ′ (g ⊗ g

  (id

  ′ )σ

  −1 X ⊗(Y ⊙Z)

  X : X ⊙ 1 → X ♣♦r a X,Y,Z = σ

  X : 1 ⊙ X → X ❡ r

  ✱ l

  X,Y,Z : (X ⊙ Y ) ⊙ Z → X ⊙ (Y ⊙ Z)

  ▲♦❣♦✱ ⊙ é ✉♠ ❢✉♥t♦r✳ ❙❡❥❛♠ X, Y, Z ∈ Sk(C)✳ ❉❡✜♥✐♠♦s a

  )(f ⊙ f ′ ).

  = (g ⊙ g ′

  ′

  X ⊗X

  ′ (f ⊗ f

  ′ )(f ⊗ f

  ⊗Y

  −1 Y

  ′ σ

  Y ⊗Y

  ′ )σ

  ′ (g ⊗ g

  ⊗Z

  = σ −1 Z

  ′

  X ⊗X

  ′ )σ

  ⊗ id Z )σ (X⊙Y )⊗Z

  −1 −1 −1 = σ ′ ′ ′ (f ⊗ σ ′ ′ (g ⊗ h)σ Y σ )

  ⊗Z

  X Y ⊗Z Y ⊗Z ⊗(Y ⊙Z ) a X,Y,Z (σ ⊗ id Z )σ

  X ⊗Y (X⊙Y )⊗Z −1 −1

  = σ ′ ′ ′ (f ⊗ σ ′ ′ (g ⊗ h)) Y

  X ⊗Z ⊗(Y ⊙Z )

  ⊗ id a X,Y,Z (σ Z )σ X ⊗Y (X⊙Y )⊗Z

  −1 −1

  ′

  = σ ′ ′ ′ (id X ⊗ σ ′ ′ ) Y

  X ⊗Z ⊗(Y ⊙Z )

  (f ⊗ (g ⊗ h))a X,Y,Z (σ ⊗ id Z )σ

  X ⊗Y (X⊙Y )⊗Z

  (∗) −1 −1

  ′ ′ ′

  = σ ′ ′ ′ (id X ⊗ σ ) Y

  X ⊗(Y ⊙Z ) ⊗Z

  ′ ′ ′

  a X ,Y ,Z ((f ⊗ g) ⊗ h) (σ ⊗ id Z )σ

  X (X⊙Y )⊗Z ⊗Y

  (∗∗) −1 −1

  

′ ′ ′ ′

′ ′ ⊗ id

  = a X ,Y ,Z σ ′ ′ ′ (σ Z )

  X (X ⊙Y )⊗Z ⊗Y

  ((f ⊗ g) ⊗ h)(σ ⊗ id Z )σ X (X⊙Y )⊗Z

  ⊗Y −1

  ′ ′ ′

  = a X ,Y ,Z σ

  ′ ′ ′

  (X ⊙Y )⊗Z −1

  (σ ′ ′ (f ⊗ g)σ ⊗ h)σ X (X⊙Y )⊗Z

  X ⊗Y ⊗Y −1

  ′ ′ ′

  = a X ,Y ,Z σ ′ ′ ′ ((f ⊙ g) ⊗ h)σ (X⊙Y )⊗Z

  (X ⊙Y )⊗Z

  ′ ′ ′

  = a X ,Y ,Z (f ⊙ g) ⊙ h), ❡♠ (∗) ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a ❡ ❡♠ (∗∗) ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ a

  ✳ P♦rt❛♥t♦✱ a é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❆❣♦r❛✱ ♣r♦✈❡♠♦s q✉❡ l é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱ s❡❥❛ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ Sk(C)

  ✱ t❡♠♦s f l X = f l X (σ ⊗ id X )σ

  1 1⊗X

  = l Y (id ⊗ f )(σ ⊗ id X )σ

  1

  1 1⊗X

  = l Y (σ ⊗ f )σ

  1 1⊗X

  = l Y (σ id ⊗ id Y f )σ

  1 1 1⊗X = l Y (σ ⊗ id Y )(id ⊗ f )σ

  1 1 1⊗X (∗∗∗)

  = l Y (σ ⊗ id Y )σ (id ⊙ f )

  1 1⊗Y

  1 = l Y (id ⊙ f ).

  1 ⊙f

  ❡♠ (∗∗∗) ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ id ✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ❝♦♥❝❧✉í♠♦s

  1 q✉❡ r é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ▲♦❣♦✱ a✱ l ❡ r sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s✳

  ▼♦str❡♠♦s ♦s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦✳ P❛r❛ X, Y, Z, W ∈ Sk(C) ✱ t❡♠♦s a X,Y,Z a

  X ⊙W ⊙Y,Z,W

  −1 −1 = σ (id X ⊗ σ )a X,Y,Z

  ⊙W

  X Y ⊗(Y ⊙(Z⊙W )) ⊗(Z⊙W )

  ⊗ id (σ

  X Z ⊙W )σ ⊗Y (X⊙Y )⊗(Z⊙W )

  −1 −1 σ (id X ⊙Y ⊗ σ )

  Z (X⊙Y )⊗(Z⊙W ) ⊗W a X (σ ⊗ id W )σ

  ⊙Y,Z,W (X⊙Y )⊗Z ((X⊙Y )⊙Z)⊗W

  −1 −1 ⊗ σ

  = σ (id X )a X,Y,Z ⊙W

  X Y ⊗(Y ⊙(Z⊙W )) ⊗(Z⊙W )

  −1 (σ ⊗ σ )

  X ⊗Y Z

  ⊗W a X (σ ⊗ id W )σ

  ⊙Y,Z,W (X⊙Y )⊗Z ((X⊙Y )⊙Z)⊗W

  −1 −1 ⊗ σ

  = σ (id X )a X,Y,Z ⊙W

  X Y ⊗(Y ⊙(Z⊙W )) ⊗(Z⊙W )

  −1 (id X σ ⊗ σ id Z )

  ⊗Y X ⊗W ⊗Y Z

  ⊗W a X (σ ⊗ id W )σ

  ⊙Y,Z,W (X⊙Y )⊗Z ((X⊙Y )⊙Z)⊗W

  −1 −1 ⊗ σ

  = σ (id X )a X,Y,Z ⊙W

  X Y ⊗(Y ⊙(Z⊙W )) ⊗(Z⊙W )

  −1 (id X ⊗ σ )(σ ⊗ id Z )

  ⊗Y X ⊗W Z ⊗Y

  ⊗W a X (σ ⊗ id W )σ

  ⊙Y,Z,W (X⊙Y )⊗Z ((X⊙Y )⊙Z)⊗W

  −1 −1 ⊗ σ

  = σ (id X )a X,Y,Z ⊙W

  X Y ⊗(Y ⊙(Z⊙W )) ⊗(Z⊙W )

  −1 ((id X ⊗ id Y ) ⊗ σ )(σ ⊗ (id Z ⊗ id W ))

  X Z ⊗Y ⊗W a X (σ ⊗ id W )σ

  ⊙Y,Z,W (X⊙Y )⊗Z ((X⊙Y )⊙Z)⊗W

  (1) −1 −1 −1

  ⊗ σ ⊗ (id ⊗ σ = σ (id X )(id

  X Y ))

  X Y Z ⊗(Y ⊙(Z⊙W )) ⊗(Z⊙W ) ⊗W

  X,Y,Z

  X a ⊗W a ⊗Y,Z,W ((σ ⊗ id Z ) ⊗ id W )(σ ⊗ id W )σ

  X ⊗Y (X⊙Y )⊗Z ((X⊙Y )⊙Z)⊗W

  (2) −1 −1 −1

  ⊗ σ ⊗ σ = σ (id X (id Y ))

  Z X ⊗(Y ⊙(Z⊙W )) Y ⊗(Z⊙W ) ⊗W

  (id X ⊗ a Y,Z,W )a X,Y (a X,Y,Z ⊗ id W ) ⊗Z,W

  ((σ ⊗ id Z )σ ⊗ id W )σ

  X ⊗Y (X⊙Y )⊗Z ((X⊙Y )⊙Z)⊗W

  −1 −1 −1 = σ (id X ⊗ σ (id Y ⊗ σ )a Y,Z,W )

  X Y Z ⊗(Y ⊙(Z⊙W )) ⊗(Z⊙W ) ⊗W a X,Y

  ⊗Z,W (a X,Y,Z (σ ⊗ id Z )σ ⊗ id W )σ

  X ⊗Y (X⊙Y )⊗Z ((X⊙Y )⊙Z)⊗W

  (3) −1 −1

  = σ (id X ⊗ a Y,Z,W )(id X ⊗ σ )

  X ⊗(Y ⊙(Z⊙W )) (Y ⊙Z)⊗W

  −1 (id X ⊗ (σ ⊗ id W ))a X,Y ((id X ⊗ σ ) ⊗ id W )

  ⊗Z,W Y Y ⊗Z ⊗Z

  (σ ⊗ id W )(a X,Y,Z ⊗ id W )σ

  X ⊗(Y ⊙Z) ((X⊙Y )⊙Z)⊗W

  −1 −1 = σ (id X ⊗ a Y,Z,W )(id X ⊗ σ )

  X ⊗(Y ⊙(Z⊙W )) (Y ⊙Z)⊗W a X,Y

  ⊙Z,W (σ ⊗ id W )(a X,Y,Z ⊗ id W )σ

  X ⊗(Y ⊙Z) ((X⊙Y )⊙Z)⊗W

  (4) −1

  = (id X ⊙ a Y,Z,W )σ

  X ⊗((Y ⊙Z)⊙W )

  −1 (id X ⊗ σ )a X,Y (σ ⊗ id W )

  ⊙Z,W X ⊗(Y ⊙Z)

  (Y ⊙Z)⊗W σ (a X,Y,Z ⊙ id W )

  (X⊙(Y ⊙Z))⊗W (5)

  = (id X ⊙ a Y,Z,W )a X,Y (a X,Y,Z ⊙ id W ), ⊙Z,W

  ❡♠ (1) ✉s❛♠♦s ❞✉❛s ✈❡③❡s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a✱ ❡♠ (2) ✉s❛♠♦s ♦ ❛①✐♦♠❛ ❞♦ ♣❡♥tá❣♦♥♦✱ ❡♠ (3) ✉s❛♠♦s ❞✉❛s ✈❡③❡s ❛ ❞❡✜♥✐çã♦ ❞❡ a✱ ❡♠ (4) ✉s❛♠♦s

  X ⊙ a Y,Z,W X,Y,Z ⊙ id W ❛ ❞❡✜♥✐çã♦ ❞❡ id ❡ a ❡ ❡♠ (5) ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ a✳ ❆❣♦r❛✱ ✈❡r✐✜q✉❡♠♦s ♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦✳ ❚❡♠♦s r

  X ⊙ id Y −1

  = σ (r X ⊗ id Y )σ X ⊗Y

  (X⊙1)⊗Y −1

  = σ (r X (id X ⊗ σ )σ ⊗ id Y )σ X ⊗Y

  X

  1 ⊗1 (X⊙1)⊗Y

  −1 = σ (r X ⊗ id Y )((id X ⊗ σ ) ⊗ id Y )(σ ⊗ id Y )σ

  X ⊗Y

  1 X ⊗1 (X⊙1)⊗Y

  (1) −1

  ⊗ l ⊗ σ = σ (id

  X Y )a ((id X ) ⊗ id Y )

  X X,1,Y ⊗Y

  1 (σ ⊗ id Y )σ

  X ⊗1 (X⊙1)⊗Y

  (2) −1

  ⊗ l ⊗ (σ ⊗ id = σ (id

  X Y )(id

  X Y ))a

  X ⊗Y

  1 X,1,Y ⊗ id

  (σ Y )σ

  X ⊗1 (X⊙1)⊗Y

  (3) −1 −1 −1

  ⊗ l ⊗ σ ⊗ (σ ⊗ id = σ (id

  X Y )(id X )(id

  X Y ))

  X ⊗Y

  1 1⊗Y

  (id X ⊗ (σ ⊗ id Y ))a (σ ⊗ id Y )σ

  X

1 X,1,Y

  ⊗1 (X⊙1)⊗Y −1 −1

  ⊗ l ⊗ σ = σ (id

  X Y )(id X )a

  X ⊗Y X,1,Y

  1⊗Y (σ ⊗ id Y )σ

  X ⊗1 (X⊙1)⊗Y

  −1 −1 ⊙ l ⊗ σ

  = (id

  X Y )σ (id X )a

  X X,1,Y ⊗(1⊙Y ) 1⊗Y

  (σ ⊗ id Y )σ

  X ⊗1 (X⊙1)⊗Y

  (4) = (id X ⊙ l Y )a ,

X,1,Y

  ❡♠ ✭✶✮ ✉s❛♠♦s ♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦✱ ❡♠ ✭✷✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞♦ a✱ ❡♠ ✭✸✮ ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ l ❡ ❡♠ ✭✹✮ ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ a✳ P♦rt❛♥t♦✱ (Sk(C), ⊙, 1, a, l, r) é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❚❡♦r❡♠❛ ✸✳✶✳✷✷ ❚♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛✲ ❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡sq✉❡❧ét✐❝❛✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ (C, ⊗, 1, a, l, r)✱ s❡❣✉❡✱ ♣❡❧♦s ❧❡♠❛s ❛❝✐♠❛✱ q✉❡ (Sk(C), ⊙, 1, a, l, r) é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ C ❡ Sk(C) sã♦ ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✳ P❛r❛ t❛♥t♦✱ ♣r✐♠❡✐r❛♠❡♥t❡ ❞❡✜♥✐♠♦s ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s F ❡ G✳ ❙❡❥❛ F : C → Sk(C)

  ♣♦r −1

  F (X) = X f σ , ❡ F (f) = σ

  X Y ♣❛r❛ X ∈ C ❡ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ P❛r❛ X, Y ∈ C✱ ❞❡✜♥✐♠♦s

  F ζ : F (X) ⊙ F (Y ) → F (X ⊗ Y )

  X,Y ♣♦r F

  −1 −1 ζ = σ (σ ⊗ σ )σ = σ (σ ⊗ σ )σ . X,Y

  X X Y F

  X X Y ⊗Y (X)⊗F (Y ) ⊗Y

  X ⊗Y

  ❖❜s❡r✈❛♠♦s q✉❡✱ ❝♦♠♦ F (X) = X✱ ♣❛r❛ t♦❞♦ X ∈ C✱ ❡♥tã♦ F (X)⊙ F (Y ) = X ⊙ Y = X ⊗ Y

  ✭❞❡✜♥✐çã♦ ❞❛❞❛ ♥♦ ▲❡♠❛ ❉❛í✱

  −1

  σ σ σ

  X ⊗Y ⊗σ X ⊗Y

  X Y

  F X ⊗ Y X ⊗ Y = F (X ⊗ Y ).

  ζ : X ⊗ Y X ⊗ Y X,Y

  F ❡ ✐ss♦ ❡①♣❧✐❝❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ζ X,Y ✳

  F : ⊙ ◦ (F × F ) → F ◦ ⊗

  ▼♦str❡♠♦s q✉❡ ζ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ F

  ➱ ❝❧❛r♦ q✉❡ ζ X,Y é ✉♠ ✐s♦♠♦r✜s♠♦✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ ♣♦✐s é ′ ′

  ✉♠❛ ❝♦♠♣♦s✐çã♦ ❞❡ ✐s♦♠♦r✜s♠♦s✳ ❙❡❥❛♠ f : X → X ❡ g : Y → Y ♠♦r✜s♠♦s ❡♠ C✳ ❊♥tã♦

  F F (f ⊗ g)ζ

  X,Y F

  −1

  ′ ′

  = σ (f ⊗ g)σ ζ

  X X ⊗Y X,Y ⊗Y

  −1 = σ ′ ′ (f ⊗ g)(σ ⊗ σ )σ

  X Y X ⊗Y

  X ⊗Y −1

  ′ ′ ⊗ gσ

  = σ (f σ )σ

  X X Y ⊗Y

  X ⊗Y

  −1

  ′ ′

  = σ ′ ′ (σ F (f ) ⊗ σ F (g))σ

  X Y X ⊗Y

  X ⊗Y −1

  ′ ′ ′ ⊗ σ ′

  = σ (σ )(F (f ) ⊗ F (g))σ

  X X Y ⊗Y

  X ⊗Y

  −1

  ′ ′

  = σ ′ ′ (σ ⊗ σ )σ (F (f ) ⊙ F (g))

  X Y ′ ′ X ⊗Y

  X ⊗Y F

  X ,Y ❆❣♦r❛ ❞❡✜♥✐♠♦s

  ′ ′ = ζ (F (f ) ⊙ F (g)).

  F F φ = id .

  : 1 → F (1) ♣♦r φ

  1 F F , φ )

  ❱❡r✐✜q✉❡♠♦s q✉❡ (F, ζ é ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳ P❛r❛ X, Y ∈ C✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡ F F F F

  ⊙ ζ ⊙ id ζ (id )a = F (a X,Y,Z )ζ (ζ ),

  X,Y ⊗Z

  X Y,Z X,Y ,Z X ⊗Y,Z X,Y Z F F l = F (l X )ζ (φ ⊙ id )

  X X 1,X

  F F = F (r X )ζ (id ⊙ φ ). ❡ r

  X X,1

  X ❉❡ ❢❛t♦✱

  F F ⊙ ζ

  σ ζ (id )a

  X X,Y ⊗Z

  X Y,Z X,Y ,Z ⊗(Y ⊗Z)

  F = (σ ⊗ σ )σ (id ⊙ ζ )a

  X Y

  X Y,Z X,Y ,Z ⊗Z

  X ⊗Y ⊗Z

  F ⊗ σ ⊗ ζ

  = (σ )(id )σ a

  X Y ⊗Z

  X Y,Z X,Y ,Z

  X ⊗(Y ⊙Z)

  F = (σ ⊗ σ ζ )σ a

  X Y Y,Z ⊗Z X,Y ,Z

  X ⊗(Y ⊙Z) = (σ ⊗ (σ Y ⊗ σ Z )σ )σ a

  X X,Y ,Z Y ⊗Z X ⊗(Y ⊙Z)

  = (σ ⊗ (σ Y ⊗ σ Z ))(id ⊗ σ )σ a

  X X X,Y ,Z Y

  X ⊗Z ⊗(Y ⊙Z)

  (1) = (σ ⊗ (σ ⊗ σ ))a (σ ⊗ id )σ

  X Y Z X,Y ,Z Z

  X ⊗Y (X ⊙Y )⊗Z

  (2) = a X,Y,Z ((σ ⊗ σ ) ⊗ σ )(σ ⊗ id )σ

  X Y Z Z X ⊗Y (X ⊙Y )⊗Z

  F = a X,Y,Z (σ ⊗ σ )(ζ ⊗ id )σ

  X ⊗Y Z X,Y Z (X⊙Y )⊗Z

  F = a X,Y,Z (σ ⊗ σ )σ (ζ ⊙ id )

  X Z X,Y ⊗Y Z

  X ⊗Y ⊗Z F F

  = a X,Y,Z σ ζ (ζ ⊙ id )

  X X,Y Z (X⊗Y )⊗Z ⊗Y,Z

  (3) F F

  = σ X F (a X,Y,Z )ζ (ζ ⊙ id ).

  ⊗(Y ⊗Z)

  X X,Y ⊗Y,Z Z

  ❡♠ ✭✶✮ ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ a✱ ❡♠ ✭✷✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a ❡ −1

  X,Y,Z ) = σ a X,Y,Z σ ❡♠ ✭✸✮ ✉s❛♠♦s q✉❡ F (a ✱ ♦✉ s❡❥❛✱

  X ⊗(Y ⊗Z) (X⊗Y )⊗Z

  σ F (a X,Y,Z ) = a X,Y,Z σ ✳

  X ⊗(Y ⊗Z) (X⊗Y )⊗Z

  X ❈♦♠♦ σ ⊗(Y ⊗Z) é ✉♠ ✐s♦♠♦r✜s♠♦✱ s❡❣✉❡ ❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ ♣❛r❛

  F F , φ ) q✉❡ (F, ζ s❡❥❛ ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳ ❱❡❥❛♠♦s ❛ s❡❣✉♥❞❛ ✐❣✉❛❧❞❛❞❡

  σ l = σ l (σ ⊗ id )σ

  X X

  X X

  X

  1 1⊗X

  (1) = l X (id ⊗ σ )(σ ⊗ id )σ

  1 X

  X

  1 1⊗X

  = l X (σ ⊗ σ )σ

  X

  1 1⊗X

  F = l X σ ζ

  1⊗X 1,X (2)

  F = σ F (l X )ζ

  X 1,X

  F = σ F (l X )ζ (id ⊙ id )

  X X 1,X

  1 F F = σ F (l X )ζ (φ ⊙ id ),

  X X 1,X

  X ) = ❡♠ ✭✶✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ l ❡ ❡♠ ✭✷✮ ✉s❛♠♦s q✉❡ F (l

  −1 σ l X σ

  X ✳ 1⊗X

  F F ⊗ id

  = F (l X )ζ (φ ) ❈♦♠♦ σ X é ✉♠ ✐s♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ l

  X X ✳ 1,X

  F F , φ )

  ❆ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ é ♣r♦✈❛❞❛ ❛♥❛❧♦❣❛♠❡♥t❡✳ P♦rt❛♥t♦✱ (F, ζ é ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳

  ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ❢✉♥t♦r ✐♥❝❧✉sã♦ G : Sk(C) → C ❡ ♠♦str❡♠♦s q✉❡ ♦ ♠❡s♠♦ é ♠♦♥♦✐❞❛❧✱ ❡♠ q✉❡ G −1 G −1 ζ = σ , = σ . X,Y ♣❛r❛ X, Y ∈ Sk(C) ❡ φ

  X ⊗Y

  1 ❉❡ ❢❛t♦✱ ♣❛r❛ X, Y, Z ∈ Sk(C)✱ t❡♠♦s

  G G ⊗ ζ

  ζ (id G )a G X,Y ⊙Z (X) Y,Z (X),G(Y ),G(Z)

  −1 −1 = σ (id X ⊗ σ )a X,Y,Z

  X Y ⊗Z ⊗(Y ⊙Z)

  −1 −1 = a X,Y,Z σ (σ ⊗ id Z )

  X ⊗Y

  (X⊙Y )⊗Z G G

  = G(a X,Y,Z )ζ (ζ ⊗ id G )

  X X,Y (Z) ⊙Y,Z

  ❡ l = l

  X G (X) −1 −1

  = l X σ (σ ⊗ id X )

  1 1⊗X

  G −1

  = G(l X )ζ (σ ⊗ id G ) (X)

  1 1,X

  G G = G(l X )ζ (φ ⊗ id G ).

  (X) 1,X

  G G = G(r X )ζ (id ⊗ φ )

  ❆ ❝♦♥❞✐çã♦ ❡♥✈♦❧✈❡♥❞♦ r G G é ♣r♦✈❛❞❛ (X) (X)

  X,1 ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✳

  C Sk ❆❣♦r❛✱ ❞❡✜♥✐♠♦s α : G ◦ F → Id ❡ β : F ◦ G → Id (C) ♣♦r

  α X : X → X, α X = σ X , ♣❛r❛ t♦❞♦ X ∈ C,

  β X : X → X, β X = σ X , ♣❛r❛ t♦❞♦ X ∈ Sk(C).

  X = σ

  X ❏á t❡♠♦s q✉❡ α é ✉♠ ✐s♦♠♦r✜s♠♦✱ ♣❛r❛ t♦❞♦ X ∈ C ❡ ♣♦r✲ t❛♥t♦✱ r❡st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ α é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✳

  ❙❡❥❛ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C ❡ ♠♦str❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛ (G ◦ F )(X) Id C (X)

  P♦rt❛♥t♦✱ α é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❆❣♦r❛ ✈❡❥❛♠♦s q✉❡ α é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ ♦s ❞✐❛❣r❛♠❛s ❛❜❛✐①♦

  ζ

  IdC X,Y

  ζ

  Y

  ⊗α

  X

  α

  (G ◦ F )(X) ⊗ (G ◦ F )(Y ) Id C (X) ⊗ Id C (Y ) (G ◦ F )(X ⊗ Y ) Id C (X ⊗ Y )

  X = σ Y F (f ) = σ Y G(F (f )) = α Y (G ◦ F )(f ).

  (G ◦ F )(Y ) Id C (Y ) α

  −1 Y f σ

  X = σ Y σ

  X = id Y f σ

  é ❝♦♠✉t❛t✐✈♦✳ ❉❡ ❢❛t♦✱ t❡♠♦s f α X = f σ

  Y

  α

  f (G◦F )(f )

  X

  G ◦F

X,Y

  α

  F X,Y

  ,

  X ⊗Y

  σ −1

  X ⊗Y

  (σ X ⊗ σ Y )σ

  X ⊗Y

  −1

  (X),F (Y ) = σ

  )ζ G F

  X,Y = G(ζ

  (G ◦ F )(1) Id C

  G ◦F

  sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❆ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧ ❞❡ G ◦ F é ❞❛❞❛ ♣♦r ζ

  G ◦F

  φ

  IdC

  φ

  1

  1 α

  (1)

  X ⊗Y

  ♦✉ s❡❥❛✱ ζ

  ζ

  C .

  ❋✐♥❛❧♠❡♥t❡✱ ♣r♦✈❡♠♦s q✉❡ β é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✳ ❆ ♣r♦✈❛ ❞❡ q✉❡ β é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ é ❛♥á❧♦❣❛ à ♣r♦✈❛ ❢❡✐t❛ ♣❛r❛ α

  ✳ ❘❡st❛✲♥♦s ♠♦str❛r q✉❡ β é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✳ ▼♦str❡♠♦s q✉❡✱ ♣❛r❛ X, Y ∈ Sk(C)✱ ♦s ❞✐❛❣r❛♠❛s

  (F ◦ G)(X) ⊙ (F ◦ G)(Y ) Id Sk (C)

  (X) ⊙ Id Sk (C)

  (Y ) (F ◦ G)(X ⊙ Y )

  Id Sk

  (C) (X ⊙ Y )

  β

  X

  ⊙β

  Y

  ζ

  IdSk(C) X,Y

  F ◦G X,Y

  1 = φ

  β

  

X ⊙Y

  (F ◦ G)(1) Id Sk

  (C) (1)

  1 β

  

1

  φ

  IdSk(C)

  φ

  F ◦G

  sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❆ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧ ❞❡ F ◦ G é ❞❛❞❛ ♣♦r ζ

  F ◦G X,Y

  = F (ζ G

  X,Y )ζ

  Id

  1 = id

  G ◦F X,Y

  X ⊗Y

  = σ −1 X ⊗Y

  (σ X ⊗ σ Y ). ❆❧é♠ ❞✐ss♦✱

  φ G ◦F

  = G(φ F

  )φ G

  = id

  1 σ

  −1

  1 = σ

  −1

  1 .

  ▲♦❣♦✱ ζ

  Id C X,Y

  (α X ⊗ α Y ) = id

  (σ X ⊗ σ Y ) = σ

  −1

  X ⊗Y

  σ −1

  X ⊗Y

  (σ X ⊗ σ Y ) = σ X ⊗Y ζ

  G ◦F

  X,Y = α

  X ⊗Y

  ζ G ◦F

  X,Y ❡

  α

  1 φ

  G ◦F

  = σ

  1 σ

  F G (X),G(Y )

  = F (σ −1 X ⊗Y

  X ⊙Y

  X,Y = β

  ζ F

  −1 X ⊗Y

  X,Y = σ X ⊙Y σ

  ζ F

  X ⊙Y

  σ −1

  X,Y = σ

  ζ F

  ζ F

  X ⊙Y

  = id

  X ⊗Y

  Y )σ

  X ⊗ σ

  (σ

  X ⊙Y

  ◦G X,Y

  −1

  Sk (C) .

  C ❡ β : F ◦ G → Id Sk (C) ✳

  ❡ ✐s♦✲ ♠♦r✜s♠♦s ♥❛t✉r❛✐s ♠♦♥♦✐❞❛✐s α : G ◦ F → Id

  G )

  G , φ

  ) ❡ (G, ζ

  , φ F

  P♦rt❛♥t♦✱ ❡①✐st❡♠ ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s (F, ζ F

  Id

  ❡ β

  1 = φ

  1 = id

  −1

  1 σ

  = σ

  F ◦G

  1 φ

  X ⊗Y

  (σ X ⊙ σ Y ) = id X ⊙Y σ

  )ζ F

  X,Y ,

  X,Y .

  ζ F

  X ⊗Y

  −1

  X,Y = σ

  F ◦G

  ♦✉ s❡❥❛✱ ζ

  ζ F

  F ◦G

  X ⊗Y

  σ

  X ⊗Y

  σ −1

  X ⊗Y

  −1

  X,Y = σ

  ❆❧é♠ ❞✐ss♦✱ φ

  = F (φ G

  X ⊙Y

  ♦✉ s❡❥❛✱ φ

  X,Y (β X ⊙ β Y ) = id

  Sk (C)

  Id

  ▲♦❣♦✱ ζ

  1 .

  = σ −1

  F ◦G

  1 ,

  )φ F

  1 σ

  −1

  1 σ

  −1

  1 = σ

  1 )id

  = F (σ −1

  ❆ss✐♠✱ C ❡ Sk(C) sã♦ ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✳

  ❈❛♣ít✉❧♦ ✹ ▼❛❝ ▲❛♥❡✬s ❙tr✐❝t♥❡ss ❚❤❡♦r❡♠

  ◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ♠♦str❛♠♦s q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦✲ ♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡sq✉❡❧ét✐❝❛✳ ❆❣♦r❛ ♦ ♦❜❥❡t✐✈♦ é ♠♦str❛r q✉❡ t♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✲ ✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✳ ❊ss❡ r❡s✉❧t❛❞♦ é ♦ ❝♦♥❤❡❝✐❞♦ ✏▼❛❝ ▲❛♥❡✬s ❙tr✐❝t♥❡ss ❚❤❡♦r❡♠✑✳

  ❊st❡ t❡♦r❡♠❛ ❣❛r❛♥t❡ q✉❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝❛✲ t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ♣♦❞❡♠ s❡r r❡❞✉③✐❞♦s ❛♦ ❝❛s♦ ❡♠ q✉❡ t❛✐s ❝❛t❡❣♦r✐❛s sã♦ ❡str✐t❛s✱ ❝✉❥❛s ❡str✉t✉r❛s ♠♦♥♦✐❞❛✐s sã♦ ♠❛✐s s✐♠♣❧❡s✳ ❆❧é♠ ❞✐ss♦✱

  1 2 ♦❜✲ 1 , ..., X n ❥❡t♦s ♦❜t✐❞♦s ❝♦❧♦❝❛♥❞♦ ♣❛rê♥t❡s❡s ♥♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ X ✳ ❊♠ ♦ ▼❛❝ ▲❛♥❡✬s ❙tr✐❝t♥❡ss ❚❤❡♦r❡♠ é ✉s❛❞♦ ♣❛r❛ ♣r♦✈❛r q✉❡

  , ..., X n , P ∈ C s❡❥❛♠ X 1 ♦❜❥❡t♦s ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ C ❡ P

  1

  2 q✉❛✐sq✉❡r ✐s♦♠♦r✜s♠♦s ❡♥tr❡ P ❡ P ✱ ♦❜t✐❞♦s ♣❡❧♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ ✐❞❡♥t✐❞❛❞❡s ❡ ❞♦s ✐s♦♠♦r✜s♠♦s ❞❡ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ✉♥✐❞❛❞❡✱ sã♦ ✐❣✉❛✐s✳ ❊ss❡ r❡s✉❧t❛❞♦ é ❝❤❛♠❛❞♦ ▼❛❝ ▲❛♥❡ ❈♦❤❡r❡♥❝❡ ❚❤❡♦r❡♠✳

  ➱ s❛❜✐❞♦ ❞❛ á❧❣❡❜r❛ ♦r❞✐♥ár✐❛ q✉❡✱ ♣❛r❛ R ✉♠ ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡✱ R (R, R)

  ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ R ≃ Hom ✱ R (R, R)

  ❡♠ q✉❡ Hom sã♦ ♦s ❡♥❞♦♠♦r✜s♠♦s ❞❡ R ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ R✲ ♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❆ ♣r♦✈❛ ❞❡st❡ ❢❛t♦ é ❛♥á❧♦❣❛ à ❞❡♠♦♥str❛çã♦ q✉❡ ❛♣r❡s❡♥t❛♠♦s ♣❛r❛ ♦ ▼❛❝ ▲❛♥❡✬s ❙tr✐❝t♥❡ss ❚❤❡♦r❡♠✳

  

✹✳✶ ❈♦♥str✉çã♦ ❞❡ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧

❡str✐t❛

  ❙❡❥❛ (C, ⊗, 1, a, l, r) ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❈♦♥str✉í♠♦s ✉♠❛ ❝❛✲ t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛ ❛ss♦❝✐❛❞❛ ❛ C q✉❡ é ✉s❛❞❛ ♥♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ C (C)

  ❞❡ss❡ ❝❛♣ít✉❧♦✳ ❉❡♥♦t❛♠♦s ♣♦r End ❛ ❝❛t❡❣♦r✐❛ ❞❡✜♥✐❞❛ ♣❡❧♦ q✉❡ s❡❣✉❡✿ ✭✐✮ ❖❜❥❡t♦s✿ sã♦ ♣❛r❡s (F, c)✱ ❡♠ q✉❡ F : C → C é ✉♠ ❢✉♥t♦r ❡ c : ⊗ ◦ (F × Id C ) → F ◦ ⊗

  é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✱ ✐st♦ é✱ c X,Y : F (X) ⊗ Y → F (X ⊗ Y ), ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ t❛✐s q✉❡ ♦s ❞✐❛❣r❛♠❛s

  (F (X) ⊗ Y ) ⊗ Z a c ⊗id

  

X,Y Z F (X),Y,Z

  F (X ⊗ Y ) ⊗ Z F (X) ⊗ (Y ⊗ Z) c c

  ⊗Y,Z ⊗Z

  X X,Y

  F ((X ⊗ Y ) ⊗ Z) F (X ⊗ (Y ⊗ Z)) F (a )

X,Y,Z

  r

  F (X)

  F (X) F (X) ⊗ 1 c

  X,1

  F (r )

  X F (X ⊗ 1)

  sã♦ ❝♦♠✉t❛t✐✈♦s✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ C✱ ♦✉ s❡❥❛✱ c X,Y a F = F (a X,Y,Z )c X (c X,Y ⊗ id Z )

  ⊗Z (X),Y,Z ⊗Y,Z ✭✹✳✶✮ F = F (r X )c .

  ❡ r (X) X,1 ✭✹✳✷✮ F G F

  ), (G, c ) ∈ End C (C) ) ✭✐✐✮ ▼♦r✜s♠♦s✿ ♣❛r❛ (F, c ✱ ✉♠ ♠♦r✜s♠♦ ❞❡ (F, c

  G )

  ❡♠ (G, c é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ µ : F → G t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐s✲ q✉❡r X, Y ∈ C✱ ♦ ❞✐❛❣r❛♠❛ µ ⊗id

  

X Y

  F (X) ⊗ Y G(X) ⊗ Y

  

F G

  c c

X,Y X,Y

  F (X ⊗ Y ) G(X ⊗ Y ) µ

  

X ⊗Y

  é ❝♦♠✉t❛t✐✈♦✳ C (C)

  ✭✐✐✐✮ ▼♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡✿ ♣❛r❛ ❝❛❞❛ (F, c) ∈ End ✱ ♦ ♠♦r✜s♠♦ F : F → F

  ✐❞❡♥t✐❞❛❞❡ é ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ id ✳ ✭✐✈✮ ❈♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s✿ é ❛ ❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧ ❞❡ tr❛♥s❢♦r♠❛✲ çõ❡s ♥❛t✉r❛✐s✳

  ❆✜r♠❛çã♦ ✶✿ ❆ ❝♦♠♣♦s✐çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ F G H

  ), (G, c ), (H, c ) ∈ End C (C) ❙❡❥❛♠ (F, c ❡ µ : F → G, ν : G → H

  C (C) ♠♦r✜s♠♦s ❡♠ End ✳ Pr♦✈❡♠♦s q✉❡ ❛ ❝♦♠♣♦s✐çã♦ ✈❡rt✐❝❛❧ ν ◦ µ : F → H C (C)

  é ✉♠ ♠♦r✜s♠♦ ❡♠ End ✱ ✐st♦ é✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ ✈❡❥❛♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛

  (ν◦µ) ⊗id

  X Y

  F (X) ⊗ Y H(X) ⊗ Y

  

F H

  c c

X,Y X,Y

  F (X ⊗ Y ) H(X ⊗ Y ) (ν◦µ)

  X ⊗Y

  é ❝♦♠✉t❛t✐✈♦✳ ❉❡ ❢❛t♦✱ H H

  ⊗ id ⊗ id c X,Y ((ν ◦ µ)

  X Y ) = c X,Y (ν X µ

  X Y ) H

  = c (ν X ⊗ id Y )(µ X ⊗ id Y ) X,Y

  (1) G

  = ν X ⊗Y c (µ X ⊗ id Y ) X,Y

  (2) F

  = ν X µ X c ⊗Y ⊗Y X,Y

  F = (ν ◦ µ) X c ,

  ⊗Y X,Y C (C)

  ❡♠ ✭✶✮ ❡ ✭✷✮ ✉s❛♠♦s q✉❡ ν ❡ µ sã♦ ♠♦r✜s♠♦s ❡♠ End ✱ r❡s♣❡❝t✐✈❛✲ ♠❡♥t❡✳

  C (C) × End C (C) → End C (C) ❆❣♦r❛✱ ❞❡✜♥✐♠♦s ⊗ : End ♣♦r

  G F G F G ◦F

  ⊗((G, c ), (F, c )) = (G, c )⊗(F, c ) = (G◦F, c ) ❡ ⊗(ν, µ) = ν⊗µ = ν∗µ,

  G F )⊗(F, c ) ∈ End C (C) C (C)

  ♣❛r❛ q✉❛✐sq✉❡r (G, c ❡ ν, µ ♠♦r✜s♠♦s ❡♠ End ✳ G

  ◦F ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ c X,Y é ❞❡✜♥✐❞♦ ♣❡❧❛ ❝♦♠♣♦s✐çã♦

  ◦F G

  c

X,Y

  G(F (X)) ⊗ Y G(F (X ⊗ Y ))

  G F

  c G (c )

  F (X),Y X,Y

  G(F (X) ⊗ Y ) ♦✉ s❡❥❛✱

  G F G ◦F c = G(c )c .

  X,Y X,Y F (X),Y

  ❆✜r♠❛çã♦ ✷✿ ⊗ ❞❡✜♥✐❞♦ ❛❝✐♠❛ é ✉♠ ❢✉♥t♦r✳ ▼♦str❡♠♦s q✉❡ ⊗ ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ P❛r❛ ✐ss♦✱ ✈❡❥❛♠♦s q✉❡ (G ◦ G

  ◦F

  F, c ) ∈ End C (C) C (C) ❡ ν ∗µ é ✉♠ ♠♦r✜s♠♦ ❡♠ End ✱ ♣❛r❛ q✉❛✐sq✉❡r

  F G (F, c ), (G, c ) ∈ End C (C) C (C)

  ❡ µ, ν ♠♦r✜s♠♦s ❡♠ End ✳ ■♥✐❝✐❛♠♦s G

  ◦F ♠♦str❛♥❞♦ q✉❡ c s❛t✐s❢❛③ ❛s r❡❧❛çõ❡s

  G G G ◦F ◦F ◦F c a = (G ◦ F )(a X,Y,Z )c (c ⊗ id Z )

  X,Y (G◦F )(X),Y,Z

  X X,Y ⊗Z ⊗Y,Z

  G ◦F

  = (G ◦ F )(r X )c . ❡ r (G◦F )(X)

  X,1 ❉❡ ❢❛t♦✱

  G ◦F c a

  X,Y (G◦F )(X),Y,Z ⊗Z

  F G = G(c )c a G

  X,Y F (F (X)),Y,Z ⊗Z (X),Y ⊗Z

  (1) F G G

  = G(c )G(a )c (c ⊗ id Z ) X,Y F (X),Y,Z

  ⊗Z F (X)⊗Y,Z F (X),Y F G G

  ⊗ id = G(c a F )c (c Z )

  X,Y ⊗Z (X),Y,Z F F (X)⊗Y,Z (X),Y

  (2) F F G G

  = G(F (a X,Y,Z ))G(c )G(c ⊗ id Z )c (c ⊗ id Z )

  X X,Y F F ⊗Y,Z (X)⊗Y,Z (X),Y

  (3) F G F G

  = G(F (a X,Y,Z ))G(c )c (G(c ) ⊗ id Z )(c ⊗ id Z )

  X F X,Y F ⊗Y,Z (X⊗Y ),Z (X),Y

  F G F G = G(F (a X,Y,Z ))G(c )c (G(c )c ⊗ id Z )

  X F X,Y F ⊗Y,Z (X⊗Y ),Z (X),Y

  = (G ◦ F )(a X,Y,Z )c G ◦F

  (X),Y (ν F

  (3) = H(µ

  (X) ⊗ id Y )

  (X),Y (ν F

  (µ X ⊗ id Y ))c H F

  G X,Y

  Y ) = H(c

  (X) ⊗ id

  Y )c H F

  )H(c F X,Y

  X ⊗ id

  )H(µ

  G X,Y

  (2) = H(c

  ⊗ id Y )

  (H(µ X ) ⊗ id Y )(ν F (X)

  H G (X),Y

  X ⊗Y

  )c H F

  ((ν ∗ µ) X ⊗ id Y ) (1)

  J F (X),Y

  H ✱ ❡♠ ✭✸✮ ✉s❛♠♦s q✉❡ µ ∈ End

  F (X) ✱ ❡♠ ✭✷✮ ✉s❛✲ ♠♦s ❛ ♥❛t✉❛❧✐❞❛❞❡ ❞❡ c

  ◆❛ ✐❣✉❛❧❞❛❞❡ ✭✶✮ ✉s❛♠♦s q✉❡ (ν ∗ µ) X = H(µ X )ν

  X,Y .

  J ◦F

  X ⊗Y c

  = (ν ∗ µ)

  F X,Y )c

  (X),Y (ν F

  (X⊗Y ) J(c

  (5) = H(µ X ⊗Y )ν F

  J F (X),Y

  )ν F (X)⊗Y c

  )H(c F X,Y

  X ⊗Y

  (4) = H(µ

  (X) ⊗ id Y )

  = H(c G X,Y )c

  ◦G X,Y

  X ⊗Y,Z

  )c G F

  ) ∈ End C (C) ❡ q✉❡ G é ✉♠ ❢✉♥t♦r ❡ ♥❛ ✐❣✉❛❧❞❛❞❡

  ✭✺✮ ✉s❛♠♦s q✉❡ (F, c F

  ) ∈ End C (C) ✱ ❡♠ ✭✷✮ ❡

  ◆❛s ✐❣✉❛❧❞❛❞❡s ✭✶✮ ❡ ✭✹✮ ✉s❛♠♦s q✉❡ (G, c G

  X,1 .

  G ◦F

  (X),1 = (G ◦ F )(r X )c

  = G(F (r X ))G(c F X,1

  ✳ P♦rt❛♥t♦✱ (G ◦ F, c G

  (X),1 (5)

  )c G F

  = G(r F (X)

  (F (X)) (4)

  (G◦F )(X) = r G

  ⊗ id Z ) ❡ r

  (c G ◦F X,Y

  ✭✸✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ c G

  ◦F ) ∈ End C (C)

  é ❝♦♠✉t❛t✐✈♦✳ ❉❡ ❢❛t♦✱ c H

  ⊗id

  X ⊗Y

  (ν∗µ)

  J ◦F X,Y

  c

  H ◦G X,Y

  c

  Y

  

X

  ✳ ❆❣♦r❛✱ s❡❥❛♠ (F, c

  (ν∗µ)

  ✳ ❱❡r✐✜q✉❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛ J(F (X)) ⊗ Y H(G(X)) ⊗ Y J(F (X ⊗ Y )) H(G(X ⊗ Y ))

  ❡ ν : J → H ♠♦r✜s♠♦ ❡♠ End C (C)

  ✱ µ : F → G

  H ) ∈ End C (C)

  J ), (H, c

  G ), (J, c

  F ), (G, c

  C (C) ❡ q✉❡ H

  é ✉♠ ❢✉♥t♦r✱ ❡♠ ✭✹✮ ✉s❛♠♦s q✉❡ ν ∈ End C (C)

  C , c Id

  G ), (H, c

  F ), (G, c

  é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ (F, c

  Id C ))

  ▲❡♠❛ ✹✳✶✳✶ ❈♦♠ ❛ ♥♦t❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ (End C (C), ⊗, (Id C , c

  ) ∈ End C (C) ✳

  

C

  X, Y ∈ C ✳ ❉❛í✱ é ❝❧❛r♦ q✉❡ (Id

  ✳ Pr♦✈❡✲ ♠♦s q✉❡

  = id X ⊗Y ✱ ♣❛r❛ q✉❛✐sq✉❡r

  Id C X,Y

  ) ✱ ❝♦♥s✐❞❡r❛♠♦s c

  C , c Id C

  P♦rt❛♥t♦✱ ⊗ é ✉♠ ❢✉♥t♦r✳ ◆♦ ♣❛r (Id

  ) ◦ ⊗(µ, µ ′ ).

  = ⊗(ν, ν ′

  ′ )

  H ) ∈ End C (C)

  ((H, c H

  ) = (ν ∗ ν

  ) = (F, c F

  ) = (F, c F

  C ◦ F, c Id C ◦F

  ) ❡ (Id

  H ◦(G◦F )

  (H◦G)◦F ) = (H ◦ (G ◦ F ), c

  ❙❡❣✉♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ⊗✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ ((H ◦ G) ◦ F, c

  C ).

  ) ⊗ (Id C , c Id

  ) = (F, c F

  ) ⊗ (G, c G

  ) ⊗ (F, c F

  C

  C , c Id

  )) ❡ (Id

  ) ⊗ (F, c F

  ) ⊗ ((G, c G

  ) = (H, c H

  )) ⊗ (F, c F

  ′ ) ◦ (µ ∗ µ

  ◦ µ ′

  ❡ ❡♠ ✭✺✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ ν✳ P♦rt❛♥t♦✱ ν ∗ µ é ✉♠ ♠♦r✜s♠♦ ❡♠ End

  )) ) = ⊗(id

  = ⊗(id G , id F ) = id G ∗ id F = id G

  ) )

  F

  (F,c

  ) , id

  G

  (G,c

  F

  (G◦F,c

  ),(F,c

  G

  ⊗(id ((G,c

  ✳ ❊♥tã♦

  G ) ∈ End C (C)

  F ), (G, c

  ❙❡❣✉✐♠♦s ♠♦str❛♥❞♦ q✉❡ ⊗ é ✉♠ ❢✉♥t♦r✳ ❆ ♣r♦✈❛ ❞❡ss❡ ❢❛t♦ é s✐♠✐❧❛r ❛♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥♦ ❊①❡♠♣❧♦ ❙❡❥❛♠ (F, c

  C (C) ✳

  ◦F = id

  G ◦F

  = (ν ◦ µ) ∗ (ν ′

  )) .

  ′ )

  ′ ◦ µ

  ′ )) = ⊗(ν ◦ µ, ν

  ′ ) ◦ (µ, µ

  ✳ ❊♥tã♦ ⊗((ν, ν

  ♠♦r✜s♠♦s ❡♠ End C (C)

  , ν ′

  ❙❡❥❛♠ µ, ν, µ ′

  F

  ) = id

  ),(F,c

  G

  ⊗((G,c

  ) = id

  F

  )⊗(F,c

  G

  (G,c

  ) = (F ◦ Id C , c F ◦Id C ).

  C ◦ F = F = F ◦ Id C ❏á t❡♠♦s q✉❡ (H ◦ G) ◦ F = H ◦ (G ◦ F ) ❡ Id ✳

  H Id F F (H◦G)◦F ◦(G◦F ) C ◦F ◦Id C

  = c = c = c ❘❡st❛✲♥♦s ♠♦str❛r c ❡ c ✳ P❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ t❡♠♦s

  (H◦G)◦F F H ◦G c = (H ◦ G)(c )c

  X,Y X,Y F (X),Y

  F G H = H(G(c ))H(c )c

  X,Y F G (X),Y (F (X)),Y

  F G H = H(G(c )c )c

  X,Y F (X),Y (G◦F )(X),Y

  G H ◦F

  = H(c )c X,Y

  (G◦F )(X),Y H

  ◦(G◦F ) = c

  X,Y ❡

  Id C F Id C ◦F c = Id C (c )c

  X,Y X,Y F

  (X),Y F

  = c id F X,Y (X)⊗Y F

  = c X,Y

  F = id c

  F (X⊗Y ) X,Y F

  = F (id X )c ⊗Y X,Y

  Id

  C F

  = F (c )c Id

  X,Y C (X),Y F

  ◦Id C = c . X,Y

  ✹✳✷ ❚❡♦r❡♠❛ ❞❡ ▼❛❝ ▲❛♥❡

  C (C) ❚❡♥❞♦ ❞❡✜♥✐❞♦ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛ End ✱ ♣♦❞❡♠♦s

  ❛♣r❡s❡♥t❛r ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞♦ tr❛❜❛❧❤♦✳ ❚❡♦r❡♠❛ ✹✳✷✳✶ ❚❤❡♦r❡♠ ✷✳✽✳✺✱ ▼❛❝ ▲❛♥❡✬s ❙tr✐❝t♥❡ss ❚❤❡♦r❡♠✮✳ ❚♦❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✳

  C (C) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ (C, ⊗, 1, a, l, r) ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡ End ❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛ ❞❡✜♥✐❞❛ ♥♦ ▲❡♠❛ ❉❡✜♥✐♠♦s F : C → End C (C)

  ♣♦r F

  W

  F (W ) = (F W , c ) W → F V ,

  ❡ F(g) : F ♣❛r❛ q✉❛✐sq✉❡r W ∈ C ❡ g : W → V ♠♦r✜s♠♦ ❡♠ C✱ ❡♠ q✉❡ F

  W = W ⊗ − : C → C, W (X) = W ⊗ X W (g) = id W ⊗ g, ✐st♦ é, F ❡ F

  F

  W

  c = a W,X,Y X = g ⊗ id X , ❡ F(g)

  X,Y ♣❛r❛ q✉❛✐sq✉❡r X, Y, V ∈ C✳ ❖❜s❡r✈❛♠♦s q✉❡

  F F

  W W

  c : F W (X)⊗Y → F W (X⊗Y ), : (W ⊗X)⊗Y → W ⊗(X⊗Y ) X,Y ✐st♦ é , c X,Y

  F

  W

  = a W,X,Y ❡ ✐ss♦ ❡①♣❧✐❝❛ ♦ ♣♦rq✉ê ❞❡ c X,Y ❡ t❛♠❜é♠ g

  ⊗id

  X F −→ V ⊗ X.

  (g) X : F W (X) → F V (X), X : W ⊗ X ✐st♦ é F(g)

  ❆✜r♠❛çã♦ ✶✿ F é ✉♠ ❢✉♥t♦r✳ Pr✐♠❡✐r❛♠❡♥t❡ ♠♦str❡♠♦s q✉❡✱ ♣❛r❛ W ∈ C ❡ g ✉♠ ♠♦r✜s♠♦ ❡♠

  F

  W

  C W , c ) ∈ End C (C) C (C)

  ✱ F(W ) = (F ❡ F(g) é ✉♠ ♠♦r✜s♠♦ ❡♠ End ✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ C✱ ♦s ❞✐❛❣r❛♠❛s

  (F W (X) ⊗ Y ) ⊗ Z

  FW

  c ⊗id a

  Z X,Y

FW (X),Y,Z

  F F

  W (X ⊗ Y ) ⊗ Z W (X) ⊗ (Y ⊗ Z)

  FW FW

  c c

  ⊗Y,Z ⊗Z

  X X,Y

  F F W ((X ⊗ Y ) ⊗ Z) W (X ⊗ (Y ⊗ Z))

  F

  W (a X,Y,Z )

  r

FW (X)

  F F W W (X)

  (X) ⊗ 1

  FW F W (r X )

  c

  X,1

  F W (X ⊗ 1) sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ F✱ ♦s ❞✐❛❣r❛♠❛s t♦r♥❛♠✲s❡ ((W ⊗ X) ⊗ Y ) ⊗ Z a a

  ⊗id

  

W,X,Y Z W ⊗X,Y,Z

  (W ⊗ (X ⊗ Y )) ⊗ Z (W ⊗ X) ⊗ (Y ⊗ Z) a a

  W,X ⊗Y,Z W,X,Y ⊗Z

  W ⊗ ((X ⊗ Y ) ⊗ Z) W ⊗ (X ⊗ (Y ⊗ Z)) id

  ⊗a

  

W X,Y,Z

  r

  W ⊗X

  (W ⊗ X) ⊗ 1 W ⊗ X a id

  

W,X,1 ⊗r

W

  X W ⊗ (X ⊗ 1)

  ❡ sã♦ ❝♦♠✉t❛t✐✈♦s✱ ♣♦✐s sã♦ ♦s r❡s♣❡❝t✐✈♦s ❛①✐♦♠❛ ❞♦ ♣❡♥tá❣♦♥♦ ♣❛r❛ ♦s ♦❜❥❡t♦s W, X, Y, Z ❡ ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ ❞❛ Pr♦♣♦s✐çã♦ P♦rt❛♥t♦✱

  F

  W

  (F W , c ) ∈ End C (C) ✳

  ❆❣♦r❛✱ s❡❥❛ g : W → V ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❱❡r✐✜q✉❡♠♦s q✉❡ F(g) C (C)

  é ✉♠ ♠♦r✜s♠♦ ❡♠ End ✱ ♦✉ s❡❥❛✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ C

  F (g)

  

X ⊗id Y

  F F W (X) ⊗ Y V (X) ⊗ Y

FW FV

  c c

X,Y X,Y

  F F W (X ⊗ Y ) V (X ⊗ Y ).

  F (g)

  

X ⊗Y

  ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ F✱ ♦ ❞✐❛❣r❛♠❛ t♦r♥❛✲s❡ (g⊗id )⊗id

  X Y

  (W ⊗ X) ⊗ Y (V ⊗ X) ⊗ Y a a

W,X,Y V,X,Y

  W ⊗ (X ⊗ Y ) V ⊗ (X ⊗ Y ) g ⊗id

  X ⊗Y

  X ), id Y ) é ❝♦♠✉t❛t✐✈♦✱ ♣♦✐s é ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a ♣❛r❛ ((g, id ✳ ▲❡♠✲

  X = id X ⊗ id Y ❜r❡♠♦s q✉❡ id ⊗Y ✳ P♦rt❛♥t♦✱ F(g) é ✉♠ ♠♦r✜s♠♦ ❡♠ End C (C)

  ✳ ❈♦♥❝❧✉í♠♦s q✉❡ F ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ F : F → F

  ▲❡♠❜r❡♠♦s q✉❡ ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ✐❞❡♥t✐❞❛❞❡ id F ) X = id F

  ❡stá ❞❡✜♥✐❞❛ ♣♦r (id (X) ✱ ✈❡❥❛ ❊①❡♠♣❧♦ ❊ss❡ ❢❛t♦ é ♠✉✐t♦ ✉s❛❞♦ ❡ ♥ã♦ ❢❛r❡♠♦s ♥❡♥❤✉♠❛ ♠❡♥çã♦✳

  W ) : F W → ▼♦str❡♠♦s q✉❡ F é ✉♠ ❢✉♥t♦r✳ ❙❡❥❛ W ∈ C✳ ❊♥tã♦ F(id

  F W ❡✱ ♣❛r❛ ❝❛❞❛ X ∈ C✱ t❡♠♦s

  F ⊗ id

  (id W ) X = id W

  X W = id ⊗X = id

  F (X)

  W

  = (id F )

  X W = (id FW )

  X F ,c

  W

  = (id )

  X F (W ) W ) = id F

  ❡ ✐ss♦ ♥♦s ❞✐③ q✉❡ F(id (W ) ✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛✐sq✉❡r ♠♦r✲ ✜s♠♦s g : W → V ❡ h : V → U ❡♠ C ❡ X ∈ C✱ t❡♠♦s

  F (hg) X = hg ⊗ id

  X = (h ⊗ id X )(g ⊗ id X )

  F F = (h) X (g)

  X = (F(h) ◦ F(g))

  X ❡ ♣♦rt❛♥t♦ F(hg) = F(h) ◦ F(g) ❡ ✐ss♦ t❡r♠✐♥❛ ❛ ♣r♦✈❛ ❞❛ ❛✜r♠❛çã♦✳

  ●♦st❛rí❛♠♦s ❞❡ ❞❡✜♥✐r ✉♠❛ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧ ♣❛r❛ F✳ ❉❛ ❡str✉✲ C (C) t✉r❛ ♠♦♥♦✐❞❛❧ ❞❡ End t❡♠♦s✱ ♣❛r❛ W, V ∈ C✱ q✉❡

  F F

  W

  V F

  (W ) ⊗ F(V ) = (F W , c ) ⊗ (F V , c )

  F ◦F

  W

  V

  = (F W ◦ F V , c ), ❡✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱

  F F F ◦F

  W

  V V W

  F c = W (c )c X,Y X,Y F

  (X),Y

  V F W

  F = W (a V,X,Y )c

  V ⊗X,Y

  ⊗ a = (id W V,X,Y )a W,V ⊗X,Y . ❆❧é♠ ❞✐ss♦✱

  F

  W ⊗V

  F (W ⊗ V ) = (F W , c )

  ⊗V F

  W ⊗V

  = a W ⊗V,X,Y ❡✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✱ c X,Y ✳

  F F , φ )

  Pr❡❝✐s❛♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ t❡r♥❛ (F, ζ ✳ Pr✐♠❡✐r❛♠❡♥t❡ ✈❡✲ ♠♦s q✉❡✱ ♣❛r❛ W, V ∈ C✱

  F F ζ : F(W ) ⊗ F(V ) → F(W ⊗ V ), : F W ◦ F V → F W .

  W,V ♦✉ s❡❥❛, ζ W,V ⊗V ❉❡✜♥✐♠♦s✱ ♣❛r❛ t♦❞♦ X ∈ C✱

  F (ζ ) X : (F W ◦ F V )(X) = W ⊗ (V ⊗ X) → F W (X) = (W ⊗ V ) ⊗ X

  W,V ⊗V F −1

  ) X = a ♣♦r (ζ W,V ✳

  W,V,X F

  : ⊗ ◦ (F × F) → F ◦ ⊗ ❆✜r♠❛çã♦ ✷✿ ζ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳

  F F Pr❡❝✐s❛♠♦s ♣r♦✈❛r q✉❡ ζ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❡ q✉❡ ζ W,V

  C (C) é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ End ✱ ♣❛r❛ q✉❛✐sq✉❡r W, V ∈ C✳ ❙❡❥❛♠ g :

  ′ ′ W → W

  ❡ h : V → V ♠♦r✜s♠♦s ❡♠ C✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛

  F

  ζ

W,V

  F F W ◦ F

  V W ⊗V

  F F (g)⊗F(h)=F(g)∗F(h) (g⊗h)

  F ′ ′ F ′ ′ W ◦ F

  V W ⊗V

  F

  ζ

  W ′ ,V ′ ′ ′

  W ◦ F V → F W ◦ F

  V é ❝♦♠✉t❛t✐✈♦✳ ❘❡❝♦r❞❛♠♦s q✉❡ F(g) ∗ F(h) : F ❡ ❛ss✐♠✱ ♣❛r❛ X ∈ C✱ t❡♠♦s

  F ′ (F(g) ∗ F(h)) X = W (F(h) X )F(g)

  F (X)

  V

  = F W

  V

  End C (C) ✳ ❱❡❥❛♠♦s q✉❡ ζ

  F W,V

  : F W ◦ F V → F W ⊗V é ✉♠❛ tr❛♥s❢♦r✲

  ♠❛çã♦ ♥❛t✉r❛❧✱ ♦✉ s❡❥❛✱ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❝♦♠✉t❛✱ ♣❛r❛ f : X → Y ♠♦r✜s♠♦ ❡♠ C

  (F W ◦ F V )(X) F W

  ⊗V (X)

  (F W ◦ F V )(Y ) F W

  ⊗V (Y ). (ζ

  

F

W,V

  )

  X F W ⊗V (f ) (F W

  ◦F

  )(f ) (ζ

  F é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳

  

F

W,V

  )

  Y

  ❯s❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞❡ F ❡ ❞❡ ζ F W,V ✱ ♦ ❞✐❛❣r❛♠❛ t♦r♥❛✲s❡

  W ⊗ (V ⊗ X) (W ⊗ V ) ⊗ X W ⊗ (V ⊗ Y ) (W ⊗ V ) ⊗ Y a

  

−1

W,V,X

  id

  W ⊗V ⊗f

  id

  W

  ⊗(id

  V

  ⊗f ) a

  ❖ ♣ró①✐♠♦ ♣❛ss♦ é ♠♦str❛r♠♦s q✉❡ ζ F W,V é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠

  ✳ ▲♦❣♦✱ ζ

  ′

  V

  (h ⊗ id X )F(g)

  V ⊗X

  = (id W

  

  ⊗ (h ⊗ id X ))(g ⊗ id

  V ⊗X

  ) = g ⊗ (h ⊗ id X ). ❈♦♥s✐❞❡r❛♥❞♦ ♦ ♦❜❥❡t♦ X ∈ C ♥♦ ❞✐❛❣r❛♠❛ ❛♥t❡r✐♦r✱ t❡♠♦s

  (F W ◦ F

  V )(X) F

  W ⊗V (X) (F W

  ′

  ◦ F

  ′

  ♣❛r❛ (g, (h, id X ))

  )(X) F W

  ′

  ⊗V

  ′

  (X) a

  

−1

W,V,X

  (g⊗h)⊗id

  X

  g ⊗(h⊗id

  X )

  a

  

−1

W ′ ,V ′ ,X

  ❡ ♦ ♠❡s♠♦ é ❝♦♠✉t❛t✐✈♦ ❞❡✈✐❞♦ à ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a −1

  

−1

W,V,Y

  −1 W , (id V , f )) q✉❡ é ❝♦♠✉t❛t✐✈♦ ❞❡✈✐❞♦ à ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ a ♣❛r❛ (id ✳

  F F −1 ) X = a

  P♦rt❛♥t♦✱ ζ W,V é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❡ ❝♦♠♦ (ζ W,V W,V,X

  F é ✉♠ ✐s♦♠♦r✜s♠♦✱ ♣❛r❛ t♦❞♦ X ∈ C✱ s❡❣✉❡ q✉❡ ζ W,V é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳

  F C (C)

  P❛r❛ ❝♦♥❝❧✉✐r♠♦s q✉❡ ζ W,V é ✉♠ ♠♦r✜s♠♦ ❡♠ End ✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛

  F

  (ζ )

  X ⊗id Y W,V

  F ◦ F

  (F W V )(X) ⊗ Y W ⊗V (X) ⊗ Y

  FW ⊗V FW ◦FV

  c c

X,Y X,Y

  F ◦ F

  (F W V )(X ⊗ Y ) W ⊗V (X ⊗ Y )

  F

  (ζ )

  X ⊗Y

W,V

  F é ❝♦♠✉t❛t✐✈♦✱ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ζ ❡

  W,V F

  W ⊗V F W ◦F

  

V

  ❛s ✐❣✉❛❧❞❛❞❡s ♣❛r❛ c X,Y ❡ c X,Y ✱ ♦ ❞✐❛❣r❛♠❛ t♦r♥❛✲s❡

  −1

  a ⊗id

  Y W,V,X

  F ◦ F

  (F W V )(X) ⊗ Y W ⊗V (X) ⊗ Y a

  (id ⊗a )a W ⊗V,X,Y

  W V,X,Y W,V ⊗X,Y

  F (F W ◦ F V )(X ⊗ Y ) W (X ⊗ Y )

  ⊗V

  −1

  a

  W,V,X ⊗Y

  ❡ é ❝♦♠✉t❛t✐✈♦ ❞❡✈✐❞♦ ❛♦ ❛①✐♦♠❛ ❞♦ ♣❡♥tá❣♦♥♦ ♣❛r❛ ♦s ♦❜❥❡t♦s W, V, X, Y ✳ F

  C (C) P♦rt❛♥t♦✱ ζ W,V é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ End ❡ ✜♥❛❧✐③❛♠♦s ❛ ♣r♦✈❛ ❞❛ ❛✜r♠❛çã♦✳

  ❆❣♦r❛✱ ❞❡✜♥✐♠♦s F F −1

  φ : Id C → F = l ♣♦r φ X : X → 1 ⊗ X,

1 X

  ♣❛r❛ t♦❞♦ X ∈ C✳ F

  C (C) ❆✜r♠❛çã♦ ✸✿ φ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ End ✳

  F ➱ ❝❧❛r♦ q✉❡ φ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ❡✱ ♣❛r❛ t♦❞♦ X ∈ C✱

  F F

  φ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ C✳ P❛r❛ ♠♦str❛r♠♦s q✉❡ φ é ✉♠ ♠♦r✜s♠♦

  X

  ❡♠ End C (C)

  id

  −1

  X

  ⊗id

  Y

  a

  1,X,Y

  X ⊗Y

  Id C (X ⊗ Y ) F

  l

  

−1

X ⊗Y

  ❡ é ❝♦♠✉t❛t✐✈♦ ❞❡✈✐❞♦ ❛♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ❞❛ Pr♦♣♦s✐çã♦ ❆s✲ s✐♠✱ φ F

  é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ End C (C)

  ✳ ❆✜r♠❛çã♦ ✹✿ (F, ζ

  F , φ

  F )

  1 (X ⊗ Y ) l

  1 (X) ⊗ Y

  ✱ é ♥❡❝❡ssár✐♦ q✉❡ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❝♦♠✉t❡ Id C (X) ⊗ Y

  ⊗id

  F

  1 (X) ⊗ Y

  Id C (X ⊗ Y ) F

  1 (X ⊗ Y ),

  φ

  F

  X

  Y

  F

  c

  F1 X,Y

  c

  IdC X,Y

  φ

  

F

X ⊗Y

  ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C✳ ❖ ❞✐❛❣r❛♠❛ ❛❝✐♠❛ t♦r♥❛✲s❡ Id C (X) ⊗ Y

  é ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳

  ❙❡❣✉♥❞♦ ❛ ❉❡✜♥✐çã♦ ❞❡✈❡♠♦s ♠♦str❛r q✉❡✱ ♣❛r❛ W, V, U ∈ C✱ ♦s ❞✐❛❣r❛♠❛s

  F (W ),F(V ),F(U )

  F (a

  F V,U

  ⊗ζ

  F (W )

  id

  F W ⊗V,U

  ζ

  a

  F W,V ⊗U

  (U )

  ⊗id F

  F W,V

  (F(W ) ⊗ F(V )) ⊗ F(U ) F

  (W ) ⊗ F(V ⊗ U ) F

  ((W ⊗ V ) ⊗ U ) F

  (W ) ⊗ (F(V ) ⊗ F(U )) F

  (W ⊗ V ) ⊗ F(U ) F

  ) ζ

W,V,U

  (Id C , c Id C

  W (a −1 V,U,X

  W

  )id F

  = (id W ⊗ a −1 V,U,X

  ) V ⊗(U ⊗X)

  W

  )(id F

  = F

  −1 V,U,X

  V ◦F U )(X)

  ) (F

  W

  ) X )(id F

  F V,U

  F W ((ζ

  (V ⊗(U ⊗X)) = id W ⊗ a

  , ♣❛r❛ ❢❛❝✐❧✐t❛r✱ ❡st❛♠♦s ❛♣❧✐❝❛♥❞♦ ❛ ❞❡✜♥✐çã♦ (ν ∗ µ)

  )

  ⊗ id F

  X

  )

  U

  ∗ id F

  ) X = (ζ F W,V

  U

  F W,V

  X = H(µ X )ν F (X)

  F (U ) ) X = (ζ

  W,V ⊗ id

  (ζ F

  W ✳

  V ⊗U ❡ J = H = F

  ✱ G = F

  ♣❛r❛ F = F V ◦ F U

  X =

  ∗ ζ F V,U

  ) ⊗ F(W ) F

  F (W )

  F 1,W

  ζ

  (W )

  ⊗id F

  F

  φ

  (W ⊗ 1) l

  W

  (W ) ⊗ F(1) F

  (W ) F

  ) F

  (W ) ⊗ (Id C , c Id C

  (1 ⊗ W ) F

  (1) ⊗ F(W ) F

  (W ) F

  F (l

  ) r

  W

  ) ❝♦♠✉t❛♠✳ ❆♥t❡s ❞❡ ♣❛rt✐r♠♦s ♣❛r❛ ❛ ✈❡r✐✜❝❛çã♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❞✐❛❣r❛♠❛s✱ ❞❡t❡r♠✐♥❛♠♦s ❛s s❡❣✉✐♥t❡s ❝♦♠♣♦s✐çõ❡s ❤♦r✐③♦♥t❛✐s✳ P❛r❛ ❝❛❞❛ X ∈ C✱ t❡♠♦s q✉❡

  V,U ) = (id F

  ⊗ ζ F

  W

  V,U ) = (id F

  ⊗ ζ F

  (id F (W )

  W

  F (W )

  F (r

  (W ⊗ (V ⊗ U )) ζ

  ζ

  F

  ⊗φ

  (W )

  id F

  F W,1

  = F W

  ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ F = G = F W

  W

  ) X = (id F

  ⊗ φ F

  

W

  ) X = (id F

  ⊗ φ F

  (id F (W )

  X ✳

  ❡ H = F 1 ♥❛ ❞❡✜♥✐çã♦ ❞❡ (ν ∗ µ)

  ✱ J = Id C

  ⊗X ,

  )

  −1 W

  ⊗X = l

  −1 W

  = id 1⊗(W ⊗X) l

  F W ⊗X

  )(X) φ

  W

  ◦F

  

1

  = id (F

  ∗ φ F

  X =

  (F W (X)) φ

  X )id

  ❈♦♥s✐❞❡r❛♥❞♦ ♦ ♦❜❥❡t♦ X ∈ C ♥♦s ❞✐❛❣r❛♠❛s ❛❝✐♠❛✱ ♦ ❢❛t♦ ❞❡ q✉❡ End C (C)

  W ✳

  1 ❡ J = H = F

  ✱ G = F

  ♥❛ ❞❡✜♥✐çã♦ ❞❡ ν ∗ µ✱ F = Id C

  X ,

  ⊗ l −1

  (X) = id W

  W

  F

  −1

  F W (φ

  X = (id W ⊗ l

  )

  W

  X )(id F

  W (l −1

  = F

  C (X)

  ) Id

  W

  X )(id F

  F

  F W ⊗X

  

1

  ⊗V ((id F

  (X) =

  ⊗X = a

  −1 W,V,U

  (U ⊗X) a

  

W ⊗V

  F

  ⊗X = id

  W,V ) U

  )(ζ F

  (id U ⊗X

  F W ⊗V

  U

  ⊗X ,

  ) F

  F W,V

  (X) )(ζ

  U

  (id F

  F W ⊗V

  (X) =

  U

  W,V ) F

  ) X )(ζ F

  U

  −1 W,V,U

  ❛q✉✐ F = G = F U

  = id F

  X =

  F W ⊗X

  (X) )φ

  W

  1 (id F

  = F

  W (X)

  ) X )φ F F

  W

  1 ((id F

  F

  )

  ✱ J = F W ◦ F

  W

  ∗ id F

  ) X = (φ F

  W

  ⊗ id F

  ) X = (φ F

  ⊗ id F (W )

  (φ F

  X ✳

  W ⊗V ♥❛ ❞❡✜♥✐çã♦ ❞❡ (ν ∗µ)

  V ❡ H = F

  é ❡str✐t❛ ❡ ❛s ❞❡✜♥✐çõ❡s ❞❡ F✱ ❞♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡♠ F F End C (C)

  ✱ ❞❡ ζ ❡ ❞❡ φ ✱ ♦❜t❡♠♦s ♦s ❞✐❛❣r❛♠❛s ((F W ◦ F V ) ◦ F U )(X)

  −1

  a id

  W,V,U ⊗X (FW ◦FV ◦FU )(X)

  (F W ◦ F U )(X) (F W ◦ (F V ◦ F U ))(X) ⊗V

  −1 −1

  a id

  W ⊗a W ⊗V,U,X

V,U,X

  F (X) ◦ F

  (W ⊗V )⊗U (F W V ⊗U )(X)

  −1

  a ⊗id

  W,V,U

X a

W,V ⊗U,X

  F W ⊗(V ⊗U )(X) id id

  W ⊗X FW (X)

  F F (Id C ◦ F W )(X) W (X) (F W ◦ Id C )(X) W (X)

  −1 −1

  l l ⊗id id r ⊗id

  W

  X W ⊗l W

  X W ⊗X

  X F F

  (F ◦ F W )(X) (X) (F W ◦ F )(X) W (X) 1 1⊗W

  1 ⊗1

  

−1 −1

  a a

  1,W,X W,1,X

  ❡ ♦s ♠❡s♠♦s sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❆ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❤❡①á❣♦♥♦ s❡❣✉❡ ❞♦ ❛①✐♦♠❛ ❞♦ ♣❡♥tá❣♦♥♦ ♣❛r❛ ♦s ♦❜❥❡t♦s W, V, U, X ❡ ❛ ❞♦s q✉❛❞r❛❞♦s s❡❣✉❡✱ r❡s♣❡❝t✐✈❛♠♥t❡✱ ❞❛ Pr♦♣♦s✐çã♦ ❡ ❞♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦

  F F , φ )

  ♣❛r❛ ♦s ♦❜❥❡t♦s W, 1, X✳ P♦rt❛♥t♦✱ (F, ζ é ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳ ❆❣♦r❛✱ ❞❡✜♥✐♠♦s

  G : End C ,

  (C) → C, G(F, c) = F (1) ❡ G(µ) = µ

  1 C (C) C (C) ♣❛r❛ q✉❛✐sq✉❡r (F, c) ∈ End ❡ µ ✉♠ ♠♦r✜s♠♦ ❡♠ End ✳

  ❆✜r♠❛çã♦ ✺✿ G é ✉♠ ❢✉♥t♦r✳

  C (C) C (C) ❉❡ ❢❛t♦✱ s❡❥❛♠ (F, c) ∈ End ❡ µ, ν ♠♦r✜s♠♦s ❡♠ End ✳

  ❊♥tã♦ G G

  (id ) = (id F ) (F,c)

  = (id F )

  1 = id

  F (1)

  = id G (F,c)

  ❡ G

  (ν ◦ µ) = (ν ◦ µ)

  1 = ν µ

  1

  1 G = (ν)G(µ).

  ❆❣♦r❛✱ ❝♦♥s✐❞❡r❛♠♦s ✉♠❛ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧ ♣❛r❛ ♦ ❢✉♥t♦r G✳ P❛r❛ F

  ) ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✱ ❛❧❣✉♠❛s ✈❡③❡s ❡s❝r❡✈❡♠♦s (F, c s♦♠❡♥t❡ ❝♦♠♦ F ✳ ❉❡✜♥✐♠♦s

  G G ζ : G(G)⊗G(F ) → G(G⊗F ),

  G,F ♦✉ s❡❥❛✱ ζ G,F : G(1)⊗F (1) → G(F (1)), ♣♦r

  G G ζ = G(l F )c ,

  G,F (1)

  1,F (1) F G

  ), G = (G, c ) ∈ End C (C) ♣❛r❛ q✉❛✐sq✉❡r F = (F, c ✳

  G : ⊗ ◦ (G × G) → G ◦ ⊗

  ❆✜r♠❛çã♦ ✻✿ ζ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ F G G

  ), (G, c ) ∈ End C (C) ❏á t❡♠♦s q✉❡✱ ♣❛r❛ (F, c ✱ ζ é ✉♠ ✐s♦♠♦r✲

  G,F F G

  ), (G, c ), ✜s♠♦✱ ♣♦✐s é ❝♦♠♣♦s✐çã♦ ❞❡ ✐s♦♠♦r✜s♠♦s✳ ❆❣♦r❛✱ s❡❥❛♠ (F, c

  ′ ′ ′

  F G F F G ′ ′ ′

  (F , c ), (G , c ) ∈ End C (C) ) → (F , c ) ) → ❡ µ : (F, c ✱ ν : (G, c

  ′

  G ′

  (G , c ) C (C) ♠♦r✜s♠♦s ❡♠ End ✱ ♠♦str❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛

  

G

  ζ

G,F

  G F G F G G (G, c ) ⊗ G(F, c ) ((G, c ) ⊗ (F, c ))

  G G (ν)⊗G(µ)=G(ν∗µ) (ν⊗µ)=G(ν∗µ)

  ′ ′ ′ ′

  ′ G ′ F ′ G ′ F G G

  (G , c ) ⊗ G(F , c ) ((G , c ) ⊗ (F , c ))

  

G

  ζ

  

G′ ,F ′

  G é ❝♦♠✉t❛t✐✈♦ ❡ ✐st♦ ♥♦ ❞✐③ q✉❡ ζ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱

  G G G (ν ∗ µ)ζ = (ν ∗ µ) G(l )c

  F G,F 1 (1)

  1,F (1)

  = G ′

  G H,G

  ) é ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧✳

  ❉❡✈❡♠♦s ♠♦str❛r q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r (F, c F

  ), (G, c G

  ), (H, c H

  ) ∈ End C (C)

  ✱ ❛s ✐❣✉❛❧❞❛❞❡s ❛❜❛✐①♦ sã♦ ✈❡r❞❛❞❡✐r❛s ζ

  ◦F (id

  ❆✜r♠❛çã♦ ✼✿ (G, ζ G

  G (H) ⊗ζ

  G G,F

  )a G (H),G(G),G(F )

  = G(a H,G,F )ζ G H

  ◦G,F (ζ

  G H,G

  , φ G

  1 .

  ), l G (F )

  ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ l✱ ❡♠ ✭✸✮ ✉s❛♠♦s q✉❡ ν é ✉♠ ♠♦r✜s♠♦ ❡♠ End C (C)

  (G(ν) ⊗ G(µ)) = ζ

  G G

  ′

  ,F

′ (G(ν) ∗ G(µ)).

  ❊♠ ✭✶✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ ν ❝♦♠ ♦ ♠♦r✜s♠♦ l F

  (1) ✱ ❡♠ ✭✷✮

  ❡ ❡♠ ✭✹✮ ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ c G

  G = id

  ′

  ✳ ▲♦❣♦✱ ζ G

  é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❡ ✐ss♦ t❡r♠✐♥❛ ❛ ♣r♦✈❛ ❞❛ ❛✜r♠❛çã♦✳

  ❚❛♠❜é♠ ❞❡✜♥✐♠♦s φ

  G : 1 → G(Id

  C , c Id C

  ) = Id C (1) = 1 ♣♦r φ

  ⊗id G (F )

  = G

  ,F

  1 ⊗ ζ

  1 ) ⊗ ζ

  G G,F

  )a H

  (1),G(1),F (1) (1)

  = H(l G (F (1))

  )H(id

  G G,F

  )c H 1,(G◦F )(1)

  )c H 1,G(1)⊗F (1) a H

  (1),G(1),F (1) = H(l

  G (F (1))

  (id

  1 ⊗ ζ

  G G,F

  ))c H 1,G(1)⊗F (1) a

  (H(id

  = H(l (G◦F )(1)

  (l F )ζ G Id C ,F

  F,Id

  (φ G

  ⊗ id G (F )

  ), ❡ r

  G (F )

  = G

  (r F )ζ G

  C

  )a G (H),G(G),G(F )

  (id G (F )

  ⊗ φ G ).

  ❉❡ ❢❛t♦✱ ζ

  G H,G

  ◦F (id G

  (H) ⊗ ζ

  G G,F

  ′

  ′

  (µ

  (3) = G

  ′

  (1) (id

  1 ⊗ µ

  1 ))ν

  1⊗F (1) c

  G 1,F (1)

  ′ (l F

  (2) = G

  ′

  (1) )G

  ′ (id

  1 ⊗ µ

  1 )c

  G

  ′

  ′ (l F

  G 1,F (1)

  1 ⊗ id F

  1 )G

  1 )ν F

  (1) G(l F

  (1) )c

  G 1,F (1)

  (1) = G

  ′ (µ

  ′ (l

  1⊗F (1) c

  F (1)

  )ν 1⊗F (1) c

  G 1,F (1)

  = G ′

  (µ

  1 l F

  (1) )ν

  1,F (1) (ν

  (1) )

  = ζ G G

  ν

  1,F

  

  (1) (id G

  ′

  (id

  1 )

  1 ⊗ µ

  G

  1 )

  = ζ G G

  ′

  ,F

  ′ (ν

  1 ⊗ µ

  1 )

  

  (1) )c

  (4) = G

  

  ′ (l F

  ′

  (1) )c

  G

  

  1,F

  (1) (G

  ′

  ′ (id

  1 ) ⊗ µ

  1 )(ν

  1 ⊗ id F

  (1) )

  = G ′

  (l F

  H (1),G(1),F (1)

  (2) = H(ζ

  ,F =

  G (F )

  G ⊗ id

  ,F (φ

  

C

  (l F )ζ G Id

  = G

  ) (8)

  F (1)

  1 ⊗ id

  ,F (id

  C

  G Id

  G (id F )ζ

  C

  (F ) = r F

  Id

  = ζ G

  )c Id C 1,F (1)

  = Id C (l F (1)

  )id 1⊗F (1)

  = Id C (l F (1)

  = l F (1)

  ), l G (F )

  G (F )

  H,G ⊗ id

  (ζ G

  G H ◦G,F

  G (a H,G,F )ζ

  (7) =

  ) ❡ r G

  (1) (9)

  ⊗ id G

  F,Id

  ✱ ❡♠ ✭✷✮ ✉s❛♠♦s ❛ ♥❛t✉✲ r❛❧✐❞❛❞❡ ❞❡ l ❝♦♠ ♦ ♠♦r✜s♠♦ ζ G

  ❊♠ ✭✶✮ ❡ ✭✺✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ c H

  G ).

  (F ) ⊗ φ

  (id G

  C

  G F,Id

  G (r F )ζ

  (11) =

  1 )

  (1) ⊗ id

  (id F

  C

  (id F )ζ G

  = F (r

  = G

  C

  G F,Id

  (1) = ζ

  C

  F 1,Id

  (1) )c

  C

  = F (l Id

  F 1,1

  1 )c

  (10) = F (l

  F 1,1

  1 )c

  (F ) )

  (ζ G H,G

  G G,F l G

  (c H 1,G(1)

  H G (1),F (1)

  G,F )c

  = H(ζ G

  ) (5)

  ⊗ id F (1)

  (c H 1,G(1)

  H 1⊗G(1),F (1)

  (1) )c

  (1) ⊗ id F

  G,F )H(l G

  = H(ζ G

  ) (4)

  ⊗ id F (1)

  H 1⊗G(1),F (1)

  ) ⊗ id F (1)

  1,G(1),F (1) )c

  )H(l G (1)⊗F (1) a

  G G,F

  ) = H(ζ

  ⊗ id F (1)

  (c H 1,G(1)

  H 1⊗G(1),F (1)

  1,G(1),F (1) )c

  (1)⊗F (1) )H(a

  G,F )H(l G

  = H(ζ G

  (1),G(1),F (1) (3)

  H 1,G(1)⊗F (1) a H

  (1)⊗F (1) )c

  (H(l G (1)

  )(c H 1,G(1)

  G H ◦G,F

  (F ) )

  (H◦G)◦F )ζ

  G (id

  ) =

  ⊗ id G (F )

  (ζ G H,G

  G H ◦G,F

  1 ζ

  )

  = (id (H◦G)◦F

  (F ) )

  ⊗ id G

  G H,G

  ◦G,F (ζ

  = ζ G H

  H,G ⊗ id G

  ⊗ id F (1)

  (ζ G

  ◦G 1,F (1)

  )c H

  = (H ◦ G)(l F (1)

  (1) )

  ⊗ id F

  )c H 1,G(1)

  G (1)

  (1),F (1) (H(l

  )c H G

  G 1,F (1)

  (1) ))H(c

  = H(G(l F

  ) (6)

  G,F ❡ q✉❡ H é ✉♠ ❢✉♥t♦r✱ ❡♠ ✭✸✮ ✉s❛♠♦s H ) ∈ End C (C) q✉❡ (H, c ❡ ♣♦rt❛♥t♦ s❛t✐s❢❛③ ✭✹✳✶✮✱ ❡♠ ✭✹✮ ✉s❛♠♦s ♦ ♣r✐✲

  G ♠❡✐r♦ ❞✐❛❣r❛♠❛ ❞❛ Pr♦♣♦s✐çã♦ ❡♠ ✭✻✮ ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ζ

  G,F C (C)

  ❡ q✉❡ H é ✉♠ ❢✉♥t♦r✱ ❡♠ ✭✼✮✱ ✭✽✮ ❡ ✭✶✶✮ ✉s❛♠♦s q✉❡ End é ❡str✐t❛✱ F

  ) ∈ End C (C) ❡♠ ✭✾✮ ✉s❛♠♦s q✉❡ (F, c ❡ ♣♦rt❛♥t♦ s❛t✐s❢❛③ ✭✹✳✷✮ ❡ ❡♠ ✭✶✵✮ ✉s❛♠♦s ❛ Pr♦♣♦s✐çã♦ ✭✐✐✐✮✳

  C (C) ❋✐♥❛❧♠❡♥t❡✱ ❣♦st❛rí❛♠♦s ❞❡ ♣r♦✈❛r q✉❡ C ❡ End sã♦ ♠♦♥♦✐✲

  C ❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❡♠♦s α : G ◦ F → Id ❡ β : F ◦ G → Id End

  C (C) ❡ ❞❡✜♥✐♠♦s

  α W W = r W : W ⊗ 1 → W ♣♦r α

  F

  F F

  : F F → F ) X = F (l X )c , ❡ β (F,c ) ♣♦r (β (F,c )

  (1) 1,X F

  ) ∈ End C (C) ♣❛r❛ q✉❛✐sq✉❡r W, X ∈ C ❡ (F, c ✳ ◆♦t❛♠♦s q✉❡ (G ◦

  F

  W

  F )(W ) = G(F W , c ) = F W W

  (1) = W ⊗ 1 ❡ ❞❛í α : W ⊗ 1 → W ✱ ♦ W = r W q✉❡ ❥✉st✐✜❝❛ ❛ ❞❡✜♥✐çã♦ ❞❡ α ✳

  F F

  F (1) F

  F , c ) ) X : ❚❛♠❜é♠✱ (F ◦ G)(F, c ) = F(F (1)) = (F ❡ (β (F,c )

  (1) F F

  (X) = F (1) ⊗ X → F (X)✳ (1)

  C End ❏á s❛❜❡♠♦s q✉❡ F◦G✱ G◦F✱ Id ❡ Id C (C) sã♦ ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s✱

  ♦ q✉❡ ♣r♦✈❛ ♣❛rt❡ ❞❛ ❡q✉✐✈❛❧ê♥❝✐❛ q✉❡ q✉❡r❡♠♦s ❡st❛❜❡❧❡❝❡r ❡♥tr❡ C ❡ End C (C)

  ✳ ❆✜r♠❛çã♦ ✽✿ α é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✱ ✐st♦ é✱ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧ q✉❡ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ➱ ❝❧❛r♦ q✉❡ α é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ α é

  ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r W, V ∈ C✱ ♦s ❞✐❛❣r❛♠❛s

  α ⊗α

  W

  V

  (G ◦ F)(W ) ⊗ (G ◦ F)(V ) Id C (W ) ⊗ Id C (V )

  G ◦F

  ζ id

  W ⊗V W,V

  (G ◦ F)(W ⊗ V ) Id C (W ⊗ V ) α

  ⊗V W

1 G

  ◦F

  φ id

  1 Id C

  (G ◦ F)(1) (1) α

  1

  ❝♦♠✉t❛♠✳ ❉❡ ❢❛t♦✱ α W ⊗ α V = r W ⊗ r

W,1,F

  V (1)

  ζ G F

  −1

  ⊗V a

  (2) = r W

  V

  ,F

  W

  = (id W ⊗ r V )ζ G F

  V (1)

  1,F

  W

  )c F

  F

  V = (id W ⊗ r V )(r W ⊗ id

  (1) = (id W ⊗ r V )F W (l

  V

  (1) )a

  V

  F

  V )(id W ⊗ l

  = (id W ⊗ r

  )a W,1,V ⊗1

  V ⊗1

  V )(id W ⊗ l

  = (id W ⊗ r

  ) (1)

  V ⊗1

  ,F

W,V,1

  = r W ⊗V (ζ F

  ) :

  1 l

  −1

  1 = id

  1 .

  ❊♠ ✭✶✮ ✉s❛♠♦s ♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦✱ ❡♠ ✭✷✮ ✉s❛♠♦s ❛ Pr♦♣♦s✐çã♦ ✭✐✐✐✮✳

  ❆✜r♠❛çã♦ ✾✿ β é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✳ Pr✐♠❡✐r❛♠❡♥t❡ ♠♦str❡♠♦s q✉❡✱ ♣❛r❛ (F, c

  F ) ∈ End

  C (C)

  ✱ β (F,c

  F

  F F (1)

  1 id

  → F é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ End

  C (C) ✳ P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✱

  ❡s❝r❡✈❡♠♦s β F

  ✐♥✈és ❞❡ β (F,c

  F

  ) ✳ ❈♦♠❡ç❛♠♦s ♠♦str❛♥❞♦ q✉❡ β

  (F,c

  F

  ) é

  ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❙❡❥❛ f : X → Y ✉♠ ♠♦r✜s♠♦ ❡♠ C✳ ❉❡✈❡♠♦s

  W

  1 = l

  F )

  W,V )

  V

  1 ζ

  G F

  W

  ,F

  V

  = r W ⊗V

  G (ζ

  F W,V

  )ζ G F

  V

  ,F

  (3) = α W

  1 (φ

  ⊗V ζ

  G ◦F

  W,V ❡

  α

  1 φ

  G ◦F

  (4) = r

  1 G (φ

  F )φ

  G (5)

  = l

  W

  ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛ F F

  X

  é ✉♠ ✐s♦♠♦r✜s♠♦✱ ♣❛r❛ t♦❞♦ X ∈ C✳ P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ β

  F é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳

  ❱❡❥❛♠♦s q✉❡ β F

  é ✉♠ ♠♦r✜s♠♦ ❡♠ End C (C)

  ✳ ❙❡❥❛♠ X, Y ∈ C✳ ❊♥tã♦ ✈❡r✐✜q✉❡♠♦s q✉❡ ♦ ❞✐❛❣r❛♠❛

  F F (1)

  (X) ⊗ Y F (X) ⊗ Y

  F F (1)

  (X ⊗ Y ) F (X ⊗ Y )

  (β

  

F

  )

  ⊗id

  ✳ ➱ ❝❧❛r♦ q✉❡ (β

  Y

  c

  F X,Y

  c

  FF (1) X,Y

  (β

  

F

  )

  X ⊗Y

  é ❝♦♠✉t❛t✐✈♦✳ ❉❡ ❢❛t♦✱ c F X,Y

  ((β F ) X ⊗ id Y ) = c F X,Y

  (F (l X )c F 1,X

  F ) X = F (l X )c F 1,X

  ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ c F

  (1) (X)

  é ❝♦♠✉t❛t✐✈♦ ❡ ✐ss♦ ♥♦s ❞✐③ q✉❡ β F

  F (X) F F

  (1) (Y ) F (Y )

  (β

  F

  )

  X F (f )

  F

  F (1)

  (f ) (β

  F

  )

  Y

  é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱

  (f ), ❊♠ ✭✶✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ l ❡ q✉❡ F é ✉♠ ❢✉♥t♦r ❡ ❡♠ ✭✷✮

  F (f )(β F ) X = F (f )F (l X )c F 1,X

  = F (f l X )c F 1,X

  (1) = F (l Y )F (id

  1 ⊗ f )c

  F 1,X

  (2) = F (l Y )c

  F 1,Y

  (F (id

  1 ) ⊗ f )

  = (β F ) Y (id F (1)

  ⊗ f ) = (β F ) Y

  F F (1)

  ⊗ id Y )

  = c F X,Y

  1⊗X c

  C (C)

  (G, c G

  ) β

  F

  Id

  EndC(C)

  (µ) (F◦G)(µ) β

  G

  é ❝♦♠✉t❛t✐✈♦✱ ♦✉ s❡❥❛✱ β é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ ❉❡ ❢❛t♦✱ s❡❥❛ X ∈ C

  ✳ ❊♥tã♦ (µ ◦ β F ) X = µ X (β F )

  X = µ

  X F (l X )c F 1,X

  (1) = G(l X )µ

  F 1,X

  (F ◦ G)(G, c G

  (2) = G(l X )c

  G 1,X

  (µ

  1 ⊗ id

  X ) = (β G )

  X F (µ

  1 )

  X = (β

  G ◦ F(µ

  1 ))

  X = (β G ◦ F(G(µ)))

  X = (β G ◦ (F ◦ G)(µ)) X .

  ❊♠ ✭✶✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ µ ❡ ❡♠ ✭✷✮ ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ µ

  é ✉♠ ♠♦r✜s♠♦ ❡♠ End C (C)

  ) Id End

  F )

  (F (l X ) ⊗ id Y )(c F 1,X

  F

  ⊗ id Y ) (1)

  = F (l X ⊗ id Y )c F 1⊗X,Y

  (c F 1,X

  ⊗ id Y ) (2)

  = F (l

  X ⊗Y

  )F (a 1,X,Y

  )c F 1⊗X,Y

  (c F 1,X

  ⊗ id Y ) (3)

  = F (l

  X ⊗Y

  )c F 1,X⊗Y a F

  (1),X,Y = (β F ) X ⊗Y c

  F (1)

  End C (C) (F, c

  X,Y .

  ❊♠ ✭✶✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ c F

  ✱ ❡♠ ✭✷✮ ✉s❛♠♦s ❛ Pr♦♣♦s✐çã♦ ❡ q✉❡ F é ✉♠ ❢✉♥t♦r ❡ ❡♠ ✭✸✮ ✉s❛♠♦s q✉❡ (F, c

  F ) ∈ End C (C)

  ❡ ♣♦rt❛♥t♦ s❛t✐s❢❛③ ✭✹✳✶✮✳

  P♦rt❛♥t♦✱ β F

  é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ End C (C)

  ✳ ❋✐♥❛❧♠❡♥t❡✱ ♥❡❝❡ss✐✲ t❛♠♦s ✈❡r✐✜❝❛r q✉❡ β é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✳ ❙❡❥❛♠ (F, c

  F ), (G, c

  G ) ∈ End C (C)

  ❡ µ : F → G ✉♠ ♠♦r✜s♠♦ ❡♠ End C (C)

  ✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ ♦ ❞✐❛❣r❛♠❛

  (F ◦ G)(F, c F

  ) Id

  ✳

  ▲♦❣♦✱ µ ◦ β F = β G ◦ (F ◦ G)(µ)

  ) (G, c G

  β

  G ◦F

  ζ

  F ◦G

  ❡ ❝♦♠♦ β (F,c

  ) (F ◦ G)((G, c

  ) ⊗ (F, c F

  ) ⊗ (F ◦ G)(F, c F

  id

  (F ◦ G)(G, c G

  ) ∈ End C (C) ✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ ♦s ❞✐❛❣r❛♠❛s

  ), (G, c G

  ❙❡❥❛♠ (F, c F

  ✭✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✮✱ s❡❣✉❡ q✉❡ β é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳ ❘❡st❛✲♥♦s ✈❡r✐✜❝❛r q✉❡ β é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✳

  ) é ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ End C (C)

  F

  G ∗β F

G,F

  F )

  (ζ G G,F

  G G,F

  ◦F ) X (ζ

  X = (β G

  )

  F G (1),F (1)

  ⊗ id X )(ζ

  ◦F 1,X

  (1),F (1) )

  = (G ◦ F )(l X )c G

  (1),F (1),X (4)

  ⊗ id X )a −1 G

  G 1,F (1)

  (1) )c

  (1),X (G(l F

  )c G F

  ⊗ id X )(ζ F G

  X (5)

  (1),F (1),X = G(F (l X ))G(c

  ) ◦ ζ F G

  X

  )

  ◦ ζ F ◦G

  = (β G ◦F

  X (6)

  (G),G(F ) )

  G G,F

  = (β G ◦F

  ◦F ◦ F(ζ

  X = (β G

  )

  ) X (ζ F G (G),G(F )

  G G,F

  X F (ζ

  )

  F 1,X

  ⊗ id X )a −1 G

  G ) ⊗ (F, c

  φ

  = G((β F

  F (1) (X)

  X = G((β F ) X )(β G ) F

  ∗ β F )

  sã♦ ❝♦♠✉t❛t✐✈♦s✳ ❉❡ ❢❛t♦✱ s❡❥❛ X ∈ C✳ ❊♥tã♦ (β G

  F ◦G

  IdC

  X )(β

  id

  IdC

  β

  (F ◦ G)(Id C ) Id C Id C

  G ◦F

  β

  F )) (G, c

  )

  G ) F

  G 1,F (1)

  G 1⊗F (1),X

  (1) ) ⊗ id X )(c

  (1),X (G(l F

  = G((β F ) X )c G F

  (1),F (1),X (3)

  ⊗ id X )a −1 G

  (c G 1,F (1)

  (1) ⊗ id X )c

  (1)⊗X = G((β F ) X )G(l F

  G ) ⊗ (F, c

  )c G 1,F (1)⊗X

  −1 1,F (1),X

  (1) ⊗ id X )G(a

  (1) = G((β F ) X )G(l F

  G 1,F (1)⊗X

  (1)⊗X )c

  (2) = G((β F ) X )G(l F

  ❡ (β Id C ◦ φ

  ⊗ id

  ) ∈ End C (C) ❡ ♣♦rt❛♥t♦ s❛t✐s❢❛③ ✭✹✳✶✮✱ ❡♠

  ✭✸✮ ✉s❛♠♦s ❛ ♥❛t✉r❛❧✐❞❛❞❡ ❞❡ c G

  ✱ ❡♠ ✭✹✮ ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ c G

  ◦F ✱ ❡♠

  ✭✺✮ ✉s❛♠♦s F(ζ G

  G,F ) X = ζ

  G G,F

  X ❡ ❡♠ ✭✻✮ ❡ ✭✼✮ ✉s❛♠♦s ❛ ❉❡✜♥✐çã♦

  = (id Id C ) X . ❊♠ ✭✶✮ ✉s❛♠♦s ❛ Pr♦♣♦s✐çã♦ ❡ q✉❡ G é ✉♠ ❢✉♥t♦r✱ ❡♠ ✭✷✮

   P♦rt❛♥t♦✱ β é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ♠♦♥♦✐❞❛❧✳ ❆ss✐♠✱ ❡①✐st❡♠

  ❢✉♥t♦r❡s ♠♦♥♦✐❞❛✐s (F, ζ F

  , φ F

  ) ❡ (G, ζ

  G , φ

  G )

  ❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ♠♦✲ ♥♦✐❞❛✐s α : G ◦ F → Id

  C ❡ β : F ◦ G → Id End C (C) ✳ ▲♦❣♦✱ C ❡ End

  ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ (G, c G

  C (X)

  F ◦G

  F

  )

  X (7)

  = (β Id C ◦ F(φ G

  ) ◦ φ F

  )

  X = (β Id C )

  X F (id

  1 ) X φ

  X = Id C (l X )c

  X = id Id

  Id C 1,X

  (id

  1 ⊗ id X )l

  −1

  X = l X id

  1⊗X l

  −1

  X = id

  C (C) sã♦ ♠♦♥♦✐❞❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✳

  ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

  ❬✶❪ ❆❇❘❆▼❙❑❨✱ ❙✳❀ ❚❩❊❱❊▲❊❑❖❙✱ ◆✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈❛t❡✲ ❣♦r② ❚❤❡♦r② ❛♥❞ ❈❛t❡❣♦r✐❝❛❧ ▲♦❣✐❝✱ ◆❡✇ ❙tr✉❝t✉r❡s ❢♦r P❤②✲ s✐❝s ✽✶✸✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✷✵✶✶✱ ♣♣✳ ✸✲✾✺✳

  ❬✷❪ ❆❲❖❉❊❨✱ ❙✳ ❈❛t❡❣♦r② ❚❤❡♦r②✱ ❖①❢♦r❞✱ ✷✺✻♣✳ ✭✷✵✵✻✮✳ ❬✸❪ ❇❆❊❩✱ ❏✳❀ ❙❚❆❨✱ ▼✳ P❤②s✐❝s✱ ❚♦♣♦❧♦❣②✱ ▲♦❣✐❝ ❛♥❞ ❈♦♠♣✉✲ t❛t✐♦♥✿ ❆ ❘♦s❡tt❛ ❙t♦♥❡✱ ◆❡✇ ❙tr✉❝t✉r❡s ❢♦r P❤②s✐❝s✱ ▲❡❝t✉r❡

  ◆♦t❡s ✐♥ P❤②s✐❝s ✈♦❧✳ ✽✶✸✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✷✵✶✶✱ ♣♣✳ ✾✺✲✶✼✹✳ ❬✹❪ ❉˘A❙❈˘A▲❊❙❈❯✱ ❙✳❀ ◆˘A❙❚˘A❙❊❙❈❯✱ ❈✳❀ ❘❆■❆◆❯✱ ❙✳ ❍♦♣❢ ❆❧✲

  ❣❡❜r❛s✿ ❆♥ ■♥tr♦❞✉❝t✐♦♥✱ ◆❡✇ ❨♦r❦✿ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ✹✵✶♣✳ ✭✷✵✵✶✮✳

  ❬✺❪ ❊❚■◆●❖❋✱ P✳❀ ●❊▲❆❑■✱ ❙✳❀ ◆■❑❙❍❨❈❍✱ ❉✳❀ ❖❙❚❘■❑✱ ❱✳ ❚❡♥✲ s♦r ❈❛t❡❣♦r✐❡s✱ ▼❛t❤❡♠❛t✐❝❛❧ ❙✉r✈❡②s ❛♥❞ ▼♦♥♦❣r❛♣❤s✱ Pr♦✈✐✲ ❞❡♥❝❡✱ ❘❤♦❞❡ ■s❧❛♥❞✿ ❆▼❙✱ ✸✹✸♣✳ ✭✷✵✶✺✮✳

  ❬✻❪ ❋■❖❘❊✱ ▼✳❀ ▲❊■◆❙❚❊❘✱ ❚✳ ❆♥ ❛❜str❛❝t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❚❤♦♠♣s♦♥✬s ❣r♦✉♣ ❋✱ ❙❡♠✐❣r♦✉♣ ❋♦r✉♠ ✽✵✱ ✸✷✺✲✸✹✵ ✭✷✵✶✵✮✳

  ❬✼❪ ❍❯◆●❊❘❋❖❘❉✱ ❚✳ ❲✳ ❆❧❣❡❜r❛✱ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✱ ✺✵✷♣✳ ✭✷✵✵✵✮✳

  ❬✽❪ ▼❆❈ ▲❆◆❊✱ ❙✳ ❈❛t❡❣♦r✐❡s ❢♦r t❤❡ ❲♦r❦✐♥❣ ▼❛t❤❡♠❛t✐❝✐❛♥✱ ❙♣r✐♥❣❡r✱ ✭✶✾✼✶✮✳

  ❬✾❪ ▼❖▼❇❊▲▲■✱ ❏✳ ▼✳ ❯♥❛ ✐♥tr♦❞✉❝✐ó♥ ❛ ❧❛s ❝❛t❡❣♦rí❛s t❡♥s♦✲ r✐❛❧❡s ② s✉s r❡♣r❡s❡♥t❛❝✐♦♥❡s✱ ◆♦t❛s ❞❡ ❛✉❧❛✳ ❬✶✵❪ P■◆❚❊❘✱ ❙✳✱ ➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ tr❛♥ç❛❞❛s✱ ❉✐ss❡rt❛çã♦ ❞❡ ♠❡s✲ tr❛❞♦✱ ❯❋❙❈✱ ✭✷✵✶✸✮✳ ❬✶✶❪ P■◆❚❊❘✱ ❙✳✱ ❚❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ ✭❡♠ ❛♥❞❛♠❡♥t♦✮✱ ❯❋❙❈✱ ✭✷✵✶✻✮✳

Novo documento