English
Let F be a presheaf on a TopCat X and U a family of opens. If the fork constructed from the sheaf-condition equalizer on the pairwise intersections is an equalizer, then the induced cone F.mapCone (cocone U)ᵒᵖ is a limit cone.
Русский
Пусть F — канонический расслоитель на TopCat X, и элементарный набор открытых подмножеств U. Если ответвение, полученное из равноселивого условияSheaf на поперечных пересечениях, является равноселителем, тогда индуцированный конус F.mapCone (cocone U)ᵒᵖ является пределом.
LaTeX
$$$\text{IsLimit}\bigl(F.mapCone\bigl(\text{cocone }U\bigr)^{op}\bigr)\;\text{provided}\; \text{IsLimit}\bigl(\text{SheafConditionEqualizerProducts.fork }F U\bigr).$$$
Lean4
/-- If `SheafConditionEqualizerProducts.fork` is an equalizer,
then `F.mapCone (cone U)` is a limit cone.
-/
def isLimitMapConeOfIsLimitSheafConditionFork (P : IsLimit (SheafConditionEqualizerProducts.fork F U)) :
IsLimit (F.mapCone (cocone U).op) :=
IsLimit.ofIsoLimit ((IsLimit.ofConeEquiv (coneEquiv F U).symm).symm P)
{ hom :=
{ hom := 𝟙 _
w := by
intro x
induction x with
| op x => ?_
rcases x with ⟨⟩
· simp
rfl
· dsimp [coneEquivInverse, SheafConditionEqualizerProducts.res, SheafConditionEqualizerProducts.leftRes]
simp only [limit.lift_π, limit.lift_π_assoc, Category.id_comp, Fan.mk_π_app, Category.assoc]
rw [← F.map_comp]
rfl }
inv :=
{ hom := 𝟙 _
w := by
intro x
induction x with
| op x => ?_
rcases x with ⟨⟩
· simp
rfl
· dsimp [coneEquivInverse, SheafConditionEqualizerProducts.res, SheafConditionEqualizerProducts.leftRes]
simp only [limit.lift_π, limit.lift_π_assoc, Category.id_comp, Fan.mk_π_app, Category.assoc]
rw [← F.map_comp]
rfl } }
