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Concept
Gluing axiom
Summary
In mathematics, the gluing axiom is introduced to define what a sheaf on a topological space must satisfy, given that it is a presheaf, which is by definition a contravariant functor
to a category which initially one takes to be the . Here is the partial order of open sets of ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism
if is a subset of , and none otherwise.
As phrased in the sheaf article, there is a certain axiom that must satisfy, for any open cover of an open set of . For example, given open sets and with union and intersection , the required condition is that
is the subset of With equal image in
In less formal language, a of over is equally well given by a pair of sections : on and respectively, which 'agree' in the sense that and have a common image in under the respective restriction maps
and
The first major hurdle in sheaf theory is to see that this gluing or patching axiom is a correct abstraction from the usual idea in geometric situations. For example, a vector field is a section of a tangent bundle on a smooth manifold; this says that a vector field on the union of two open sets is (no more and no less than) vector fields on the two sets that agree where they overlap.
Given this basic understanding, there are further issues in the theory, and some will be addressed here. A different direction is that of the Grothendieck topology, and yet another is the logical status of 'local existence' (see Kripke–Joyal semantics).
To rephrase this definition in a way that will work in any category that has sufficient structure, we note that we can write the objects and morphisms involved in the definition above in a diagram which we will call (G), for "gluing":
Here the first map is the product of the restriction maps
and each pair of arrows represents the two restrictions
and
It is worthwhile to note that these maps exhaust all of the possible restriction maps among , the , and the .
The condition for to be a sheaf is that for any open set and any collection of open sets whose union is , the diagram (G) above is an equalizer.
Official source
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