Type valued presheaves
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,
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.
Show that the map induced on stalks by
to_sheafify is the inverse of
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.
The morphism from a presheaf to its sheafification, sending each section to its germs. (This forms the unit of the adjunction.)
The natural morphism from the stalk of the sheafification to the original stalk.
sheafify_stalk_iso we show this is an isomorphism.