Theorems · Theorem · category theory

CategoryTheory.mono_iff_isIso_fst

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X Y : C} {f : X ⟶ Y}
  {c : CategoryTheory.Limits.PullbackCone f f} (hc : CategoryTheory.Limits.IsLimit c),
  CategoryTheory.Mono f ↔ CategoryTheory.IsIso c.fst

Defined in: Mathlib.CategoryTheory.Limits.EpiMono
Cited by: 2 results in Mathlib
Foundations: Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category

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Cites23

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.compproof · cited by 17,999
CategoryTheory.Category.assocproof · cited by 6,433
CategoryTheory.CategoryStruct.idproof · cited by 6,235
CategoryTheory.Category.comp_idproof · cited by 2,119
CategoryTheory.Category.id_compproof · cited by 1,998
CategoryTheory.Limits.WalkingPairstatement · cited by 1,319
CategoryTheory.Limits.Cone.ptstatement and proof · cited by 1,298
CategoryTheory.IsIsostatement and proof · cited by 1,156
CategoryTheory.Monostatement · cited by 893
CategoryTheory.Limits.IsLimitstatement and proof · cited by 664

Cited by2

Results whose statement or proof uses this declaration.

CategoryTheory.JointlyReflectIsomorphisms.monoproof · cited by 1
CategoryTheory.mono_iff_isIso_sndproof · cited by 0