Sheaves #
We define sheaves on a topological space, with values in an arbitrary category with products.
The sheaf condition for a F : presheaf C X requires that the morphism
F.obj U ⟶ ∏ F.obj (U i) (where U is some open set which is the union of the U i)
is the equalizer of the two morphisms
∏ F.obj (U i) ⟶ ∏ F.obj (U i ⊓ U j).
We provide the instance category (sheaf C X) as the full subcategory of presheaves,
and the fully faithful functor sheaf.forget : sheaf C X ⥤ presheaf C X.
Equivalent conditions #
While the "official" definition is in terms of an equalizer diagram,
in src/topology/sheaves/sheaf_condition/pairwise_intersections.lean
and in src/topology/sheaves/sheaf_condition/open_le_cover.lean
we provide two equivalent conditions (and prove they are equivalent).
The first is that F.obj U is the limit point of the diagram consisting of all the F.obj (U i)
and F.obj (U i ⊓ U j).
(That is, we explode the equalizer of two products out into its component pieces.)
The second is that F.obj U is the limit point of the diagram constisting of all the F.obj V,
for those V : opens X such that V ≤ U i for some i.
(This condition is particularly easy to state, and perhaps should become the "official" definition.)
The sheaf condition for a F : presheaf C X requires that the morphism
F.obj U ⟶ ∏ F.obj (U i) (where U is some open set which is the union of the U i)
is the equalizer of the two morphisms
∏ F.obj (U i) ⟶ ∏ F.obj (U i) ⊓ (U j).
Equations
- F.is_sheaf = ∀ ⦃ι : Type v⦄ (U : ι → topological_space.opens ↥X), nonempty (category_theory.limits.is_limit (Top.presheaf.sheaf_condition_equalizer_products.fork F U))
The presheaf valued in punit over any topological space is a sheaf.
Transfer the sheaf condition across an isomorphism of presheaves.
A sheaf C X is a presheaf of objects from C over a (bundled) topological space X,
satisfying the sheaf condition.
Equations
The forgetful functor from sheaves to presheaves.