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gluin
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There are 5 letters in GLUIN ( G2I1L1N1U1 )
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GLUIN - 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 ...
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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) → Cto 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 → Vif 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). |