English
The Day convolution associators satisfy the pentagon coherence identity; composing associators in the two alternate ways yields the same natural transformation.
Русский
Собрaжение ассоциаторов Day-конволюции удовлетворяет пентагональную когерентность; последовательности композиции дающих два пути сходятся.
LaTeX
$$$(\\text{associator} F G H) \\; \\text{and} \\; (\\text{associator} F (G ⊛ H) K)$ satisfy the pentagon relation.$$
Lean4
/-- The `CorepresentableBy` structure asserting that the Type-valued functor
`Y ↦ ((F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ Y)` is corepresented by
`(F ⊛ G) ⊛ H`. -/
@[simps]
def corepresentableBy₂' :
(whiskeringLeft _ _ _).obj (tensor C) ⋙
(whiskeringLeft _ _ _).obj ((tensor C).prod (𝟭 C)) ⋙ coyoneda.obj (.op <| (F ⊠ G) ⊠ H) |>.CorepresentableBy
((F ⊛ G) ⊛ H)
where
homEquiv :=
(corepresentableBy (F ⊛ G) H).homEquiv.trans <|
Functor.homEquivOfIsLeftKanExtension _ (extensionUnitLeft (F ⊛ G) (unit F G) H) _
homEquiv_comp := by aesop
