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- Gluing_axiom abstract "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 functorF: 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 morphismU → 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 thatF(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 mapsF(U) → F(W)andF(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).".
- Gluing_axiom wikiPageID "1204294".
- Gluing_axiom wikiPageLength "10005".
- Gluing_axiom wikiPageOutDegree "57".
- Gluing_axiom wikiPageRevisionID "695094659".
- Gluing_axiom wikiPageWikiLink Abelian_group.
- Gluing_axiom wikiPageWikiLink Adjoint_functors.
- Gluing_axiom wikiPageWikiLink Algebraic_geometry.
- Gluing_axiom wikiPageWikiLink Algebraic_structure.
- Gluing_axiom wikiPageWikiLink Base_(topology).
- Gluing_axiom wikiPageWikiLink Browns_representability_theorem.
- Gluing_axiom wikiPageWikiLink Category:Differential_topology.
- Gluing_axiom wikiPageWikiLink Category:General_topology.
- Gluing_axiom wikiPageWikiLink Category:Homological_algebra.
- Gluing_axiom wikiPageWikiLink Category:Limits_(category_theory).
- Gluing_axiom wikiPageWikiLink Category:Mathematical_axioms.
- Gluing_axiom wikiPageWikiLink Category_of_abelian_groups.
- Gluing_axiom wikiPageWikiLink Category_of_local_rings.
- Gluing_axiom wikiPageWikiLink Category_of_sets.
- Gluing_axiom wikiPageWikiLink Complex_manifold.
- Gluing_axiom wikiPageWikiLink Cover_(topology).
- Gluing_axiom wikiPageWikiLink Differentiable_manifold.
- Gluing_axiom wikiPageWikiLink Existential_quantification.
- Gluing_axiom wikiPageWikiLink Functor.
- Gluing_axiom wikiPageWikiLink Grothendieck_topology.
- Gluing_axiom wikiPageWikiLink Group_object.
- Gluing_axiom wikiPageWikiLink Homotopy.
- Gluing_axiom wikiPageWikiLink Ieke_Moerdijk.
- Gluing_axiom wikiPageWikiLink Inclusion_map.
- Gluing_axiom wikiPageWikiLink Intersection_(set_theory).
- Gluing_axiom wikiPageWikiLink Inverse_element.
- Gluing_axiom wikiPageWikiLink Inverse_limit.
- Gluing_axiom wikiPageWikiLink Kripke_semantics.
- Gluing_axiom wikiPageWikiLink Lawvere-Tierney_topology.
- Gluing_axiom wikiPageWikiLink Limit_(category_theory).
- Gluing_axiom wikiPageWikiLink Local_ring.
- Gluing_axiom wikiPageWikiLink Mathematics.
- Gluing_axiom wikiPageWikiLink Morphism.
- Gluing_axiom wikiPageWikiLink Natural_transformation.
- Gluing_axiom wikiPageWikiLink Open_set.
- Gluing_axiom wikiPageWikiLink Partially_ordered_set.
- Gluing_axiom wikiPageWikiLink Product_(category_theory).
- Gluing_axiom wikiPageWikiLink Reflective_subcategory.
- Gluing_axiom wikiPageWikiLink Ring_(mathematics).
- Gluing_axiom wikiPageWikiLink Ringed_space.
- Gluing_axiom wikiPageWikiLink Section_(category_theory).
- Gluing_axiom wikiPageWikiLink Sheaf_(mathematics).
- Gluing_axiom wikiPageWikiLink Signature_(logic).
- Gluing_axiom wikiPageWikiLink Stalk_(sheaf).
- Gluing_axiom wikiPageWikiLink Subcategory.
- Gluing_axiom wikiPageWikiLink Subset.
- Gluing_axiom wikiPageWikiLink Tangent_bundle.
- Gluing_axiom wikiPageWikiLink Topological_space.
- Gluing_axiom wikiPageWikiLink Topos.
- Gluing_axiom wikiPageWikiLink Union_(set_theory).
- Gluing_axiom wikiPageWikiLink Vector_field.
- Gluing_axiom wikiPageWikiLinkText "Gluing axiom".
- Gluing_axiom wikiPageWikiLinkText "Gluing axiom#Sheaves on a basis of open sets".
- Gluing_axiom wikiPageWikiLinkText "gluing axiom".
- Gluing_axiom wikiPageWikiLinkText "gluing axiom#Sheaves on a basis of open sets".
- Gluing_axiom wikiPageUsesTemplate Template:EGA.
- Gluing_axiom wikiPageUsesTemplate Template:Reflist.
- Gluing_axiom subject Category:Differential_topology.
- Gluing_axiom subject Category:General_topology.
- Gluing_axiom subject Category:Homological_algebra.
- Gluing_axiom subject Category:Limits_(category_theory).
- Gluing_axiom subject Category:Mathematical_axioms.
- Gluing_axiom hypernym Presheaf.
- Gluing_axiom type Concept.
- Gluing_axiom type Physic.
- Gluing_axiom comment "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 functorF: O(X) → Cto a category C which initially one takes to be the category of sets.".
- Gluing_axiom label "Gluing axiom".
- Gluing_axiom sameAs Q5572389.
- Gluing_axiom sameAs m.04h29l.
- Gluing_axiom sameAs Q5572389.
- Gluing_axiom wasDerivedFrom Gluing_axiom?oldid=695094659.
- Gluing_axiom isPrimaryTopicOf Gluing_axiom.