About: Gluing axiom

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In mathematics, the gluing axiom is introduced to define what a sheaf on a topological space must satisfy, given that it is a presheaf, which is by definition a contravariant functor to a category which initially one takes to be the category of sets. Here is the partial order of open sets of ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism if is a subset of , and none otherwise. is the subset of With equal image in and .

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  • In mathematics, the gluing axiom is introduced to define what a sheaf on a topological space must satisfy, given that it is a presheaf, which is by definition a contravariant functor to a category which initially one takes to be the category of sets. Here is the partial order of open sets of ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism if is a subset of , and none otherwise. As phrased in the sheaf article, there is a certain axiom that must satisfy, for any open cover of an open set of . For example, given open sets and with union and intersection , the required condition is that is the subset of With equal image in In less formal language, a section of over is equally well given by a pair of sections : on and respectively, which 'agree' in the sense that and have a common image in under the respective restriction maps and . 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). (en)
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  • In mathematics, the gluing axiom is introduced to define what a sheaf on a topological space must satisfy, given that it is a presheaf, which is by definition a contravariant functor to a category which initially one takes to be the category of sets. Here is the partial order of open sets of ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism if is a subset of , and none otherwise. is the subset of With equal image in and . (en)
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  • Gluing axiom (en)
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