Editing Gluing axiom (section)

==Sheafification==
{{See also|Categorification}}
{{See also|Sheaf (mathematics)#Turning a presheaf into a sheaf}}
To turn a given presheaf <math>\mathcal P</math> into a sheaf <math>\mathcal F</math>, there is a standard device called '''''sheafification''''' or '''''sheaving'''''. The rough intuition of what one should do, at least for a presheaf of sets, is to introduce an equivalence relation, which makes equivalent data given by different covers on the overlaps by refining the covers. One approach is therefore to go to the [[Stalk of a sheaf#Stalks of a sheaf|stalk]]s and recover the [[sheaf space]] of the ''best possible'' sheaf <math>\mathcal F</math> produced from <math>\mathcal P</math>.

This use of language strongly suggests that we are dealing here with [[adjoint functors]]. Therefore, it makes sense to observe that the sheaves on <math>X</math> form a [[full subcategory]] of the presheaves on <math>X</math>. Implicit in that is the statement that a [[morphism of sheaves]] is nothing more than a [[natural transformation]] of the sheaves, considered as functors. Therefore, we get an abstract characterisation of sheafification as [[left adjoint]] to the inclusion. In some applications, naturally, one does need a description.

In more abstract language, the sheaves on <math>X</math> form a [[reflective subcategory]] of the presheaves (Mac Lane–[[Ieke Moerdijk|Moerdijk]] ''Sheaves in Geometry and Logic'' p.&nbsp;86). In [[topos theory]], for a [[Lawvere–Tierney topology]] and its sheaves, there is an analogous result (ibid. p.&nbsp;227).