Theorems · Theorem · category theory
CategoryTheory.Limits.isIsoZero_iff_source_target_isZero
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[CategoryTheory.Limits.HasZeroObject C] (X Y : C),
CategoryTheory.IsIso 0 ↔ CategoryTheory.Limits.IsZero X ∧ CategoryTheory.Limits.IsZero YA zero morphism 0 : X ⟶ Y is an isomorphism if and only if
X and Y are zero objects.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CategoryTheory.Categorystatement and proof · cited by 32,673
- Quiver.Homstatement · cited by 32,603
- CategoryTheory.Isoproof · cited by 3,963
- Equiv.symmproof · cited by 3,681
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.Limits.HasZeroObjectstatement and proof · cited by 1,298
- CategoryTheory.IsIsostatement and proof · cited by 1,156
- CategoryTheory.Limits.IsZerostatement and proof · cited by 306
- CategoryTheory.Limits.IsZero.of_isoproof · cited by 35
- CategoryTheory.Limits.isZero_zeroproof · cited by 34
- CategoryTheory.Limits.IsZero.isoZeroproof · cited by 7
Cited by1
Results whose statement or proof uses this declaration.
- FDRep.simple_iff_end_is_rank_oneproof · cited by 1