feat(CategoryTheory): the monoidal category structure induced by a monoidal functor #12926
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…d by a monoidal functor
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eric-wieser
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May 16, 2024
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| noncomputable instance : MonoidalCategory (InducedCategory D F.obj) where | ||
| tensor_id X Y := by | ||
| dsimp [inducedCategoryMonoidal] | ||
| erw [tensor_id] | ||
| simp | ||
| rfl | ||
| tensor_comp := sorry | ||
| tensorHom_def := sorry | ||
| whiskerLeft_id X Y := by | ||
| dsimp [inducedCategoryMonoidal] | ||
| erw [whiskerLeft_id] | ||
| simp | ||
| rfl | ||
| id_whiskerRight X Y := by | ||
| dsimp [inducedCategoryMonoidal] | ||
| erw [id_whiskerRight] | ||
| simp | ||
| rfl | ||
| associator_naturality := sorry | ||
| leftUnitor_naturality := sorry | ||
| rightUnitor_naturality := sorry | ||
| pentagon X Y Z T := by | ||
| dsimp [inducedCategoryMonoidal, homMk] | ||
| simp only [LaxMonoidalFunctor.μ_natural_left, MonoidalFunctor.μ_inv_hom_id_assoc, | ||
| LaxMonoidalFunctor.μ_natural_right] | ||
| erw [← F.map_comp, ← F.map_comp, ← F.map_comp] | ||
| simp | ||
| triangle X Y := by | ||
| dsimp [inducedCategoryMonoidal, homMk] | ||
| simp only [LaxMonoidalFunctor.μ_natural_right, MonoidalFunctor.μ_inv_hom_id_assoc, | ||
| LaxMonoidalFunctor.μ_natural_left] | ||
| erw [← F.map_comp] | ||
| simp |
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Isn't this
noncomputable instance : MonoidalCategory (InducedCategory D F.obj) :=
CategoryTheory.Monoidal.induced (inducedFunctor F.obj) {
μIso := fun X Y => F.μIso X Y
εIso := F.εIso
}with some proof fields / extra simp lemmas / explicit casts? (and import Mathlib.CategoryTheory.Monoidal.Transport)
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Thanks very much indeed! I am sorry I had overlooked your remark: I was confused because of the two steps construction (first the MonoidalCategoryStruct and then the verification of the axioms).
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In this PR, given a monoidal functor
F : MonoidalFunctor C D, we define a monoidal category structure on the categoryInducedCategory D F.obj, which has the "same" objects asC, but the morphisms betweenXandYidentify toF.obj X ⟶ F.obj Y.