Gluing axiom
In mathematics, the gluing axiom is introduced to define what a sheaf F {\displaystyle {\mathcal {F}}} on a topological space X {\displaystyle X} must satisfy, given that it is a presheaf, which is by definition a contravariant functor F : O ( X ) → C {\displaystyle {\mathcal {F}}:{\mathcal {O}}(X)\rightarrow C} to a category C {\displaystyle C} which initially one takes to be the category of sets. Here O ( X ) {\displaystyle {\mathcal {O}}(X)} is the partial order of open sets of X {\displaystyle X} ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism U → V {\displaystyle U\rightarrow V} if U {\displaystyle U} is a subset of V {\displaystyle V} , and none otherwise.