Theorems · Theorem · category theory

CategoryTheory.Idempotents.isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent

∀ (C : Type u_1) [inst : CategoryTheory.Category.{v_1, u_1} C],
  CategoryTheory.IsIdempotentComplete C ↔
    ∀ (X : C) (p : X ⟶ X),
      CategoryTheory.CategoryStruct.comp p p = p →
        CategoryTheory.Limits.HasEqualizer (CategoryTheory.CategoryStruct.id X) p

A category is idempotent complete iff for all idempotent endomorphisms, the equalizer of the identity and this idempotent exists.

Defined in: Mathlib.CategoryTheory.Idempotents.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites24

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.Category.assocproof · cited by 6,433
CategoryTheory.CategoryStruct.idstatement and proof · cited by 6,235
CategoryTheory.Category.comp_idproof · cited by 2,119
CategoryTheory.Category.id_compproof · cited by 1,998
CategoryTheory.Limits.Cone.ptproof · cited by 1,298
CategoryTheory.Limits.limit.lift_πproof · cited by 266
CategoryTheory.Limits.Fork.ιproof · cited by 162
CategoryTheory.Limits.Forkproof · cited by 85
CategoryTheory.Limits.Fork.ofιproof · cited by 66

Cited by1

Results whose statement or proof uses this declaration.

CategoryTheory.Idempotents.isIdempotentComplete_iff_idempotents_have_kernelsproof · cited by 0