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Theorems · Theorem · category theory

CategoryTheory.IsIdempotentComplete.idempotents_split

∀ {C : Type u_1} {inst : CategoryTheory.Category.{v_1, u_1} C} [self : CategoryTheory.IsIdempotentComplete C] (X : C)
  (p : X ⟶ X),
  CategoryTheory.CategoryStruct.comp p p = p →
    ∃ Y i e,
      CategoryTheory.CategoryStruct.comp i e = CategoryTheory.CategoryStruct.id Y ∧
        CategoryTheory.CategoryStruct.comp e i = p

A category is idempotent complete iff all idempotent endomorphisms p split as a composition p = e ≫ i with i ≫ e = 𝟙 _

Defined in
Mathlib.CategoryTheory.Idempotents.Basic
Cited by
3 results in Mathlib
Foundations
Depth 3 from the axioms · uses no axioms
Assumes
CategoryTheory.IsIdempotentComplete
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

CategoryTheory.Idempotents.isIdempotentComplete_of_isIdempotentComplete_opposite · cited by 1Idempotents.isIdempotentC…CategoryTheory.Idempotents.Equivalence.isIdempotentComplete · cited by 1Equivalence.isIdempotentC…CategoryTheory.Idempotents.isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent · cited by 1Idempotents.isIdempotentC…CategoryTheory.Category · cited by 32673CategoryTheory.CategoryQuiver.Hom · cited by 32603Quiver.HomCategoryTheory.CategoryStruct.comp · cited by 17999CategoryStruct.compCategoryTheory.CategoryStruct.id · cited by 6235CategoryStruct.idCategoryTheory.IsIdempotentComplete · cited by 29CategoryTheory.IsIdempote…IsIdempotentComplete.idempote…CITED BYCITES

Cites5

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Cited by3

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