Theorems · Theorem · category theory

CategoryTheory.Presieve.isSheaf_iff_preservesFiniteProducts

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.FinitaryPreExtensive C]
  [CategoryTheory.FinitaryExtensive C] (F : CategoryTheory.Functor Cᵒᵖ (Type w)),
  CategoryTheory.Presieve.IsSheaf (CategoryTheory.extensiveTopology C) F ↔
    CategoryTheory.Limits.PreservesFiniteProducts F

A presheaf of sets on a category which is FinitaryExtensive is a sheaf iff it preserves finite products.

Defined in: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
Cited by: 1 results in Mathlib
Foundations: Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.FinitaryPreExtensive CategoryTheory.FinitaryExtensive

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Cites47

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.Homproof · cited by 32,603
CategoryTheory.Functor.objproof · cited by 19,642
CategoryTheory.CategoryStruct.compproof · cited by 17,999
CategoryTheory.Functorstatement and proof · cited by 16,252
Oppositestatement and proof · cited by 8,081
CategoryTheory.CategoryStruct.idproof · cited by 6,235
CategoryTheory.Isoproof · cited by 3,963
Finiteproof · cited by 3,029
CategoryTheory.Discreteproof · cited by 2,447
Opposite.unopproof · cited by 2,231
CategoryTheory.GrothendieckTopologyproof · cited by 1,415

Cited by1

Results whose statement or proof uses this declaration.

CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProductsproof · cited by 4