Theorems · Theorem · category theory
CategoryTheory.NatTrans.mono_of_mono_app
∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{F G : CategoryTheory.Functor C D} (α : G ⟶ F) [∀ (X : C), CategoryTheory.Mono (α.app X)], CategoryTheory.Mono αA natural transformation is a monomorphism if each component is.
- Defined in
- Mathlib.CategoryTheory.Functor.Category
- Cited by
- 6 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.
Cites10
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.Functor.objstatement and proof · cited by 19,642
- CategoryTheory.CategoryStruct.compproof · cited by 17,999
- CategoryTheory.Functorstatement and proof · cited by 16,252
- CategoryTheory.NatTrans.appstatement and proof · cited by 7,406
- CategoryTheory.Monostatement and proof · cited by 893
- CategoryTheory.cancel_monoproof · cited by 435
- CategoryTheory.NatTrans.ext'proof · cited by 340
- CategoryTheory.NatTrans.comp_appproof · cited by 27
Cited by6
Results whose statement or proof uses this declaration.
- CategoryTheory.NatTrans.mono_iff_mono_appproof · cited by 4
- CategoryTheory.IsGrothendieckAbelian.mono_of_isColimit_monoOverproof · cited by 2
- CategoryTheory.Sheaf.mono_of_injectiveproof · cited by 1
- CategoryTheory.IsGrothendieckAbelian.IsPresentable.surjectivityproof · cited by 1
- CategoryTheory.Limits.IsColimit.mono_ι_app_of_isFilteredproof · cited by 1
- CategoryTheory.NatTrans.mono_iff_mono_app'proof · cited by 0