Gluing axiom

In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor

to a category C which initially one takes to be the category of sets. Here O(X) is the partial order of open sets of X ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism

if U is a subset of V, and none otherwise.

As phrased in the sheaf article, there is a certain axiom that F must satisfy, for any open cover of an open set of X. For example, given open sets U and V with union X and intersection W, the required condition is that

In less formal language, a section s of F over X is equally well given by a pair of sections (s′,s′′) on U and V respectively, which 'agree' in the sense that s′ and s′′ have a common image in F(W) 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.

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