Theorems · Theorem · category theory
CategoryTheory.Sieve.pullback_apply
∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (h : Y ⟶ X) (S : CategoryTheory.Sieve X) (x : C)
(sl : x ⟶ Y), (CategoryTheory.Sieve.pullback h S).arrows sl = S.arrows (CategoryTheory.CategoryStruct.comp sl h)- Defined in
- Mathlib.CategoryTheory.Sites.Sieves
- Cited by
- 20 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses propext, Quot.sound
- Assumes
- CategoryTheory.Category
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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 · cited by 17,999
- CategoryTheory.Sievestatement and proof · cited by 552
- CategoryTheory.Sieve.arrowsstatement and proof · cited by 446
- CategoryTheory.Sieve.pullbackstatement and proof · cited by 126
Cited by20
Results whose statement or proof uses this declaration.
- CategoryTheory.Sieve.pullback_compproof · cited by 14
- CategoryTheory.Sieve.pullback_idproof · cited by 9
- CategoryTheory.Sieve.mem_iff_pullback_eq_topproof · cited by 7
- CategoryTheory.Sieve.pullbackArrows_commproof · cited by 6
- CategoryTheory.regularTopology.mem_sieves_iff_hasEffectiveEpiproof · cited by 4
- CategoryTheory.Functor.coverPreserving_restrictedTopologyproof · cited by 3
- CategoryTheory.Sieve.pullback_interproof · cited by 2
- CategoryTheory.Precoverage.isSheaf_toGrothendieck_iffproof · cited by 2
- CategoryTheory.Sieve.functorPullback_pullbackproof · cited by 2
- CategoryTheory.PreOneHypercover.pullback_sieve₁proof · cited by 1
- CategoryTheory.Coverage.eq_top_pullbackproof · cited by 1