Currying of functors in three variables #
We study the equivalence of categories
currying₃ : (C₁ ⥤ C₂ ⥤ C₃ ⥤ E) ≌ C₁ × C₂ × C₃ ⥤ E
.
The equivalence of categories (C₁ ⥤ C₂ ⥤ C₃ ⥤ E) ≌ C₁ × C₂ × C₃ ⥤ E
given by the curryfication of functors in three variables.
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Uncurrying a functor in three variables.
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Currying a functor in three variables.
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Uncurrying functors in three variables gives a fully faithful functor.
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Given functors F₁ : C₁ ⥤ D₁
, F₂ : C₂ ⥤ D₂
, F₃ : C₃ ⥤ D₃
and G : D₁ × D₂ × D₃ ⥤ E
, this is the isomorphism between
curry₃.obj (F₁.prod (F₂.prod F₃) ⋙ G) : C₁ ⥤ C₂ ⥤ C₃ ⥤ E
and F₁ ⋙ curry₃.obj G ⋙ ((whiskeringLeft₂ E).obj F₂).obj F₃
.
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bifunctorComp₁₂
can be described in terms of the curryfication of functors.
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bifunctorComp₂₃
can be described in terms of the curryfication of functors.
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Flip the first and third arguments in a trifunctor.
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Flip the first and third arguments in a trifunctor, as a functor.
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Flip the second and third arguments in a trifunctor.
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Flip the second and third arguments in a trifunctor, as a functor.
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