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 composition of the inclusion of categories
(OpenNhds x)ᵒᵖ ⥤ (Opens X)ᵒᵖ
and the functor F : (Opens X)ᵒᵖ ⥤ C
.
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 stalkFunctor C x
,
sending presheaves on X
to objects of C
. Furthermore, for a map f : X ⟶ Y
between
topological spaces, we define stalkPushforward
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 MonCat
,
CommRingCat
,...) 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
Stalks are functorial with respect to morphisms of presheaves over a fixed X
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
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
- ℱ.stalk x = (TopCat.Presheaf.stalkFunctor C x).obj ℱ
Instances For
The germ of a section of a presheaf over an open at a point of that open.
Equations
- F.germ U x hx = CategoryTheory.Limits.colimit.ι ((TopologicalSpace.OpenNhds.inclusion x).op.comp F) (Opposite.op { obj := U, property := hx })
Instances For
The germ of a global section of a presheaf at a point.
Equations
- F.Γgerm x = F.germ ⊤ x True.intro
Instances For
A variant of germ_res
with op V ⟶ op U
so that the LHS is more general and simp fires more easier.
A morphism from the stalk of F
at x
to some object Y
is completely determined by its
composition with the germ
morphisms.
For a presheaf F
on a space X
, a continuous map f : X ⟶ Y
induces a morphisms between the
stalk of f _ * F
at f x
and the stalk of F
at x
.
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- One or more equations did not get rendered due to their size.
Instances For
Alias of TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_isOpenEmbedding
.
The morphism ℱ_{f x} ⟶ (f⁻¹ℱ)ₓ
that factors through (f_*f⁻¹ℱ)_{f x}
.
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- One or more equations did not get rendered due to their size.
Instances For
The morphism (f⁻¹ℱ)(U) ⟶ ℱ_{f(x)}
for some U ∋ x
.
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- One or more equations did not get rendered due to their size.
Instances For
The morphism (f⁻¹ℱ)ₓ ⟶ ℱ_{f(x)}
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The isomorphism ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ
.
Equations
- TopCat.Presheaf.stalkPullbackIso C f F x = { hom := TopCat.Presheaf.stalkPullbackHom C f F x, inv := TopCat.Presheaf.stalkPullbackInv C f F x, hom_inv_id := ⋯, inv_hom_id := ⋯ }
Instances For
If x
specializes to y
, then there is a natural map F.stalk y ⟶ F.stalk x
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Alias of TopCat.Presheaf.germ_stalkSpecializes
.
The stalks are isomorphic on inseparable points
Equations
- F.stalkCongr e = { hom := F.stalkSpecializes ⋯, inv := F.stalkSpecializes ⋯, hom_inv_id := ⋯, inv_hom_id := ⋯ }
Instances For
For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits, every element of the stalk is the germ of a section.
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.
Equations
- ⋯ = ⋯
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.