Theorems · Definition · category theory
CategoryTheory.Subobject.pullback
{C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{X Y : C} →
[CategoryTheory.Limits.HasPullbacks C] →
(X ⟶ Y) → CategoryTheory.Functor (CategoryTheory.Subobject Y) (CategoryTheory.Subobject X)When C has pullbacks, a morphism f : X ⟶ Y induces a functor Subobject Y ⥤ Subobject X,
by pulling back a monomorphism along f.
- Defined in
- Mathlib.CategoryTheory.Subobject.Basic
- Cited by
- 54 results in Mathlib
- Foundations
- Depth 35 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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.Functorstatement · cited by 16,252
- CategoryTheory.Limits.HasPullbacksstatement and proof · cited by 439
- CategoryTheory.Subobjectstatement · cited by 385
- CategoryTheory.MonoOver.pullbackproof · cited by 7
- CategoryTheory.Subobject.lowerproof · cited by 6
Cited by71
Results whose statement or proof uses this declaration.
- CategoryTheory.Dial.tensorObjImplproof · cited by 35
- Subobject.presheafproof · cited by 8
- CategoryTheory.Subobject.pullbackπstatement · cited by 8
- CategoryTheory.SubobjectRepresentableBy.isostatement · cited by 7
- CategoryTheory.Subobject.isPullbackstatement · cited by 5
- CategoryTheory.Regular.frobeniusStrongEpiMonoFactorisationstatement · cited by 4
- CategoryTheory.Dial.isoMkstatement and proof · cited by 4
- CategoryTheory.SubobjectRepresentableBy.homEquiv_eqstatement · cited by 3
- CategoryTheory.SubobjectRepresentableBy.iso_inv_hom_left_compstatement · cited by 3
- CategoryTheory.SubobjectRepresentableBy.iso_inv_left_πstatement and proof · cited by 3
- CategoryTheory.Subobject.inf_eq_map_pullback'statement · cited by 3
- CategoryTheory.Subobject.pullback_obj_mkstatement · cited by 3