Sheafification of Type valued presheaves #
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We construct the sheafification of a Type valued presheaf,
as the subsheaf of dependent functions into the stalks
consisting of functions which are locally germs.
We show that the stalks of the sheafification are isomorphic to the original stalks,
via stalk_to_fiber which evaluates a germ of a dependent function at a point.
We construct a morphism to_sheafify from a presheaf to (the underlying presheaf of)
its sheafification, given by sending a section to its collection of germs.
Future work #
Show that the map induced on stalks by to_sheafify is the inverse of stalk_to_fiber.
Show sheafification is a functor from presheaves to sheaves, and that it is the left adjoint of the forgetful functor, following https://stacks.math.columbia.edu/tag/007X.
The prelocal predicate on functions into the stalks, asserting that the function is equal to a germ.
The local predicate on functions into the stalks, asserting that the function is locally equal to a germ.
Equations
The sheafification of a Type valued presheaf, defined as the functions into the stalks which
are locally equal to germs.
Equations
The morphism from a presheaf to its sheafification, sending each section to its germs. (This forms the unit of the adjunction.)
Equations
- F.to_sheafify = {app := λ (U : (topological_space.opens ↥X)ᵒᵖ) (f : F.obj U), ⟨λ (x : ↥(opposite.unop U)), F.germ x f, _⟩, naturality' := _}
The natural morphism from the stalk of the sheafification to the original stalk.
In sheafify_stalk_iso we show this is an isomorphism.
Equations
The isomorphism betweeen a stalk of the sheafification and the original stalk.
Equations
- F.sheafify_stalk_iso x = (equiv.of_bijective (F.stalk_to_fiber x) _).to_iso