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 → C {\displaystyle {\mathcal {F}}:{\mathcal {O}}\rightarrow C} to a category C {\displaystyle C} which initially one takes to be the category of sets. Wikipedia