Editing Gluing axiom (section)

{{Short description|Axiom specifying the requisites of a sheaf on a topological space}}
In [[mathematics]], the '''gluing axiom''' is introduced to define what a [[sheaf (mathematics)|sheaf]] <math>\mathcal F</math> on a [[topological space]] <math>X</math> must satisfy, given that it is a [[presheaf]], which is by definition a [[contravariant functor]]

:<math>{\mathcal F}:{\mathcal O}(X) \rightarrow C</math>

to a category <math>C</math> which initially one takes to be the [[category of sets]]. Here <math>{\mathcal O}(X)</math> is the [[partial order]] of [[open set]]s of <math>X</math> ordered by [[inclusion map]]s; and considered as a category in the standard way, with a unique [[morphism]]

:<math>U \rightarrow V</math>

if <math>U</math> is a [[subset]] of <math>V</math>, and none otherwise.

As phrased in the [[Sheaf (mathematics)|sheaf]] article, there is a certain axiom that <math>F</math> must satisfy, for any [[open cover]] of an open set of <math>X</math>. For example, given open sets <math>U</math> and <math>V</math> with [[union (set theory)|union]] <math>X</math> and [[intersection (set theory)|intersection]] <math>W</math>, the required condition is that

:<math>{\mathcal F}(X)</math> is the subset of <math>{\mathcal F}(U) \times {\mathcal F}(V)</math> With equal image in <math>{\mathcal F}(W)</math>

In less formal language, a [[Section (category theory)|section]] <math>s</math> of <math>F</math> over <math>X</math> is equally well given by a pair of sections :<math>(s', s'')</math> on <math>U</math> and <math>V</math> respectively, which 'agree' in the sense that <math>s'</math> and <math>s''</math> have a common image in <math>{\mathcal F}(W)</math> under the respective restriction maps

:<math>{\mathcal F}(U) \rightarrow {\mathcal F}(W)</math>

and

:<math>{\mathcal F}(V) \rightarrow {\mathcal F}(W)</math>.

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]]).