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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 :''F'': ''O''(''X'') → ''C'' 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 :''U'' → ''V'' 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 :''F''(''X'') is the subset of ''F''(''U'')×''F''(''V'') with equal image in ''F''(''W''). 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 :''F''(''U'') → ''F''(''W'') and :''F''(''V'') → ''F''(''W''). 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). ==Removing restrictions on ''C''== To rephrase this definition in a way that will work in any category ''C'' that has sufficient structure, we note that we can write the objects and morphisms involved in the definition above in a diagram which we will call (G), for "gluing": : Here the first map is the product of the restriction maps :res''U'',''Ui'',:''F(U)''→''F(Ui)'' and each pair of arrows represents the two restrictions :res''Ui'',''Ui''∩''Uj'':''F(Ui)''→''F(Ui''∩''Uj)'' and :res''Uj'',''Ui''∩''Uj'':''F(Uj)''→''F(Ui''∩''Uj)''. It is worthwhile to note that these maps exhaust all of the possible restriction maps among ''U'', the ''Ui'', and the ''Ui''∩''Uj''. The condition for ''F'' to be a sheaf is exactly that ''F'' is the limit of the diagram. This suggests the correct form of the gluing axiom: :A presheaf ''F'' is a sheaf if for any open set ''U'' and any collection of open sets ''i''∈''I'' whose union is ''U'', ''F''(''U'') is the limit of the diagram (G) above. One way of understanding the gluing axiom is to notice that "un-applying" ''F'' to (G) yields the following diagram: : Here ''U'' is the colimit of this diagram. The gluing axiom says that ''F'' turns colimits of such diagrams into limits. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「gluing axiom」の詳細全文を読む スポンサード リンク
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