sheaf of sections
0.1 Presheaf Definition
Consider a rank vector bundle , whose typical fibre is defined with respect to a field . Let constitute a cover
for . Then, sections
of the bundle over some are defined
as continuous functions , which commute with the natural projection
map ; . Denote the space of sections of
the bundle over U to be . The space of sections is a vector
space
over the field by defining addition and scalar multiplication pointwise: for , and
Then, this forms a presheaf , a functor from
to the category of vector spaces, with restriction maps the natural
restriction of functions.
0.2 Sheaf Axioms
It is easy to see that it satisfies the sheaf axioms: for open and a cover of ,
-
1.
if and for all , then .
-
2.
if for all , such that for each with , , then there is an with for all .
The first follows from the fact that for any , there is always at least
one element of , the zero section, and that the transition
functions of the bundle are linear maps. The second follows by the construction of the bundle.
1 Sheafification
We may also see the vector bundle by applying associated sheaf construction
to the presheaf . First though, we show that the
stalk of the sheaf at a point is isomorphic to the fibre of the
bundle at the point. Let be a germ at , and define a map by
First, we show that the map is a vector space homomorphism. Consider two germs and in . These map to and respectively. We add the germs by finding an open set and adding the restrictions of the sections;
Of course, , so we have , since the
restriction maps are simply restriction of functions.
Now, it is easy to show that is injective.
Assume . Then
Now, we show that is surjective. For , let
open be isomorphic to some subset of . Then, is the
set of continuous maps , where is the typical fibre of ;
Then let be the constant function , and we
have constructed an isomorphism between and .
To construct the Étalé space, take the disjoint union of stalks,
, and endow it with
the following topology: the open sets shall be of the form
collection of germs of sections at points in .
Then, the associated sheaf to is the presheaf which assigns
continuous maps to each open . These are
maps where the preimage of is open. Clearly, this implies that
. To go the other way, note
that open sets of are the images of continuous maps . An open subset of may be written as a union of ;
. Then, by single-valuedness of maps,
a continuous map must map to for some , so we have .
Title | sheaf of sections |
---|---|
Canonical name | SheafOfSections |
Date of creation | 2013-03-22 15:46:36 |
Last modified on | 2013-03-22 15:46:36 |
Owner | guffin (12505) |
Last modified by | guffin (12505) |
Numerical id | 7 |
Author | guffin (12505) |
Entry type | Definition |
Classification | msc 55R25 |
Related topic | VectorBundle |
Defines | Sheaf of Sections |