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feat(CategoryTheory/MorphismProperty): If P is stable under retracts, then P respects isomorphisms. #22274

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feat(CategoryTheory/MorphismProperty): If P is stable under retracts, then P respects isomorphisms. #22274

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5 changes: 5 additions & 0 deletions Mathlib/CategoryTheory/MorphismProperty/Retract.lean
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Expand Up @@ -87,6 +87,11 @@ lemma retracts_le_iff {P Q : MorphismProperty C} [Q.IsStableUnderRetracts] :
· intro h
exact le_trans (retracts_monotone h) Q.retracts_le

instance {P : MorphismProperty C} [P.IsStableUnderRetracts] :
P.RespectsIso :=
RespectsIso.of_respects_arrow_iso _
(fun _ _ e ↦ of_retract (Retract.ofIso e.symm))

end MorphismProperty

end CategoryTheory
5 changes: 5 additions & 0 deletions Mathlib/CategoryTheory/Retract.lean
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Expand Up @@ -66,6 +66,11 @@ def trans {Z : C} (h' : Retract Y Z) : Retract X Z where
i := h.i ≫ h'.i
r := h'.r ≫ h.r

/-- If `e : X ≅ Y`, then `X` is a retract of `Y`. -/
def ofIso (e : X ≅ Y) : Retract X Y where
i := e.hom
r := e.inv

end Retract

/--
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