mathlib documentation

topology.sheaves.stalks

Stalks #

For a presheaf F on a topological space X, valued in some category C, the stalk of F at the point x : X is defined as the colimit of the following functor

(nhds x)ᵒᵖ ⥤ (opens X)ᵒᵖ ⥤ C

where the functor on the left is the inclusion of categories and the functor on the right is F. For an open neighborhood U of x, we define the map F.germ x : F.obj (op U) ⟶ F.stalk x as the canonical morphism into this colimit.

Taking stalks is functorial: For every point x : X we define a functor stalk_functor C x, sending presheaves on X to objects of C. Furthermore, for a map f : X ⟶ Y between topological spaces, we define stalk_pushforward as the induced map on the stalks (f _* ℱ).stalk (f x) ⟶ ℱ.stalk x.

Some lemmas about stalks and germs only hold for certain classes of concrete categories. A basic property of forgetful functors of categories of algebraic structures (like Mon, CommRing,...) is that they preserve filtered colimits. Since stalks are filtered colimits, this ensures that the stalks of presheaves valued in these categories behave exactly as for Type-valued presheaves. For example, in germ_exist we prove that in such a category, every element of the stalk is the germ of a section.

Furthermore, if we require the forgetful functor to reflect isomorphisms and preserve limits (as is the case for most algebraic structures), we have access to the unique gluing API and can prove further properties. Most notably, in is_iso_iff_stalk_functor_map_iso, we prove that in such a category, a morphism of sheaves is an isomorphism if and only if all of its stalk maps are isomorphisms.

See also the definition of "algebraic structures" in the stacks project: https://stacks.math.columbia.edu/tag/007L

The stalk of a presheaf F at a point x is calculated as the colimit of the functor nbhds x ⥤ opens F.X ⥤ C

Equations

The germ of a section of a presheaf over an open at a point of that open.

Equations
@[simp]
theorem Top.presheaf.germ_res_apply {C : Type u} [category_theory.category C] [category_theory.limits.has_colimits C] {X : Top} (F : Top.presheaf C X) {U V : topological_space.opens X} (i : U V) (x : U) [I : category_theory.concrete_category C] (x_1 : (F.obj (opposite.op V))) :
(F.germ x) ((F.map i.op) x_1) = (F.germ (i x)) x_1
@[simp]
theorem Top.presheaf.germ_res {C : Type u} [category_theory.category C] [category_theory.limits.has_colimits C] {X : Top} (F : Top.presheaf C X) {U V : topological_space.opens X} (i : U V) (x : U) :
F.map i.op F.germ x = F.germ (i x)
theorem Top.presheaf.stalk_hom_ext {C : Type u} [category_theory.category C] [category_theory.limits.has_colimits C] {X : Top} (F : Top.presheaf C X) {x : X} {Y : C} {f₁ f₂ : F.stalk x Y} (ih : ∀ (U : topological_space.opens X) (hxU : x U), F.germ x, hxU⟩ f₁ = F.germ x, hxU⟩ f₂) :
f₁ = f₂

A morphism from the stalk of F at x to some object Y is completely determined by its composition with the germ morphisms.

@[ext]
theorem Top.presheaf.germ_ext {C : Type u} [category_theory.category C] [category_theory.limits.has_colimits C] {X : Top} [category_theory.concrete_category C] (F : Top.presheaf C X) {U V : topological_space.opens X} {x : X} {hxU : x U} {hxV : x V} (W : topological_space.opens X) (hxW : x W) (iWU : W U) (iWV : W V) {sU : (F.obj (opposite.op U))} {sV : (F.obj (opposite.op V))} (ih : (F.map iWU.op) sU = (F.map iWV.op) sV) :
(F.germ x, hxU⟩) sU = (F.germ x, hxV⟩) sV

For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits, every element of the stalk is the germ of a section.

theorem Top.presheaf.germ_eq {C : Type u} [category_theory.category C] [category_theory.limits.has_colimits C] {X : Top} [category_theory.concrete_category C] [category_theory.limits.preserves_filtered_colimits (category_theory.forget C)] (F : Top.presheaf C X) {U V : topological_space.opens X} (x : X) (mU : x U) (mV : x V) (s : (F.obj (opposite.op U))) (t : (F.obj (opposite.op V))) (h : (F.germ x, mU⟩) s = (F.germ x, mV⟩) t) :
∃ (W : topological_space.opens X) (m : x W) (iU : W U) (iV : W V), (F.map iU.op) s = (F.map iV.op) t

Let F be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms, preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.

For surjectivity, we are given an arbitrary section t and need to find a preimage for it. We claim that it suffices to find preimages locally. That is, for each x : U we construct a neighborhood V ≤ U and a section s : F.obj (op V)) such that f.app (op V) s and t agree on V.

Let F and G be sheaves valued in a concrete category, whose forgetful functor reflects isomorphisms, preserves limits and filtered colimits. Then if the stalk maps of a morphism f : F ⟶ G are all isomorphisms, f must be an isomorphism.

Let F and G be sheaves valued in a concrete category, whose forgetful functor reflects isomorphisms, preserves limits and filtered colimits. Then a morphism f : F ⟶ G is an isomorphism if and only if all of its stalk maps are isomorphisms.