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

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❯♥✐✈❡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 &gt; 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) &lt; 1. k(x) + 1

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

0 &lt; = k(y) &lt; 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] = &lt; [g, h] : g, h ∈ G &gt; ♦ 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 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,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

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♦✮✱ ❯❋❙❈✱ ✭✷✵✶✻✮✳