Theorems · Theorem · category theory
CategoryTheory.Idempotents.isIdempotentComplete_iff_idempotents_have_kernels
∀ (C : Type u_1) [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C],
CategoryTheory.IsIdempotentComplete C ↔
∀ (X : C) (p : X ⟶ X), CategoryTheory.CategoryStruct.comp p p = p → CategoryTheory.Limits.HasKernel pA preadditive category is pseudoabelian iff all idempotent endomorphisms have a kernel.
- Defined in
- Mathlib.CategoryTheory.Idempotents.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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.CategoryStruct.idproof · cited by 6,235
- CategoryTheory.Preadditivestatement and proof · cited by 3,309
- CategoryTheory.Limits.HasZeroMorphismsproof · cited by 3,275
- CategoryTheory.Limits.HasKernelstatement and proof · cited by 169
- sub_sub_cancelproof · cited by 105
- CategoryTheory.Limits.HasEqualizerproof · cited by 47
- CategoryTheory.IsIdempotentCompletestatement · cited by 29
- CategoryTheory.Preadditive.hasEqualizer_of_hasKernelproof · cited by 2
- CategoryTheory.Idempotents.idem_of_id_sub_idemproof · cited by 2
Cited by0
Results whose statement or proof uses this declaration.
Nothing cites this yet.