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[Merged by Bors] - feat(category_theory/opposites): iso.unop #7400

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[Merged by Bors] - feat(category_theory/opposites): iso.unop #7400

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feat(category_theory/opposites): iso.unop
jcommelin Apr 28, 2021
1ef8168
remove duplicate namespace
jcommelin May 4, 2021
8c2086d
Merge branch 'master' into jcommelin-patch-2
jcommelin May 4, 2021
09f34b4
remove remove_op
jcommelin May 4, 2021
d8dae00
Update src/category_theory/opposites.lean
jcommelin May 4, 2021
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20 changes: 20 additions & 0 deletions src/category_theory/opposites.lean
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Expand Up @@ -271,6 +271,26 @@ protected def op (α : X ≅ Y) : op Y ≅ op X :=
hom_inv_id' := quiver.hom.unop_inj α.inv_hom_id,
inv_hom_id' := quiver.hom.unop_inj α.hom_inv_id }

/-- The isomorphism obtained from an isomorphism in the opposite category. -/
@[simps] def iso.unop {X Y : Cᵒᵖ} (f : X ≅ Y) : (Y.unop ≅ X.unop) :=
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{ hom := f.hom.unop,
inv := f.inv.unop,
hom_inv_id' := by simp only [← unop_comp, f.inv_hom_id, unop_id],
inv_hom_id' := by simp only [← unop_comp, f.hom_inv_id, unop_id] }

/-- The isomorphism obtained from an isomorphism in the opposite category. -/
@[simps] def iso.remove_op {X Y : C} (f : (op X) ≅ (op Y)) : Y ≅ X :=
{ hom := f.hom.unop,
inv := f.inv.unop,
hom_inv_id' := by simp only [← unop_comp, f.inv_hom_id, unop_id_op],
inv_hom_id' := by simp only [← unop_comp, f.hom_inv_id, unop_id_op] }
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@[simp] lemma iso.unop_op {X Y : Cᵒᵖ} (f : X ≅ Y) : f.unop.op = f :=
by ext; refl

@[simp] lemma iso.op_unop {X Y : C} (f : X ≅ Y) : f.op.unop = f :=
by ext; refl

end iso

namespace nat_iso
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