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 .

Attributes	Values
rdf:type	Thing yago:WikicatMathematicalAxioms yago:Abstraction100002137 yago:AuditoryCommunication107109019 yago:Communication100033020 yago:Maxim107152948 yago:Saying107151380 yago:Speech107109196
rdfs:label	Gluing axiom (en)
rdfs:comment	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)
rdfs:seeAlso	Categorification
dcterms:subject	Differential topology Homological algebra General topology Limits (category theory) Mathematical axioms
Wikipage page ID	1204294 (xsd:integer)
Wikipage revision ID	1124115152 (xsd:integer)
Link from a Wikipage to another Wikipage	Morphism of sheaves Topos theory Algebraic structure Vector field Inclusion map Mathematics Natural transformation Tangent bundle Full subcategory Morphism Equaliser (mathematics) Complex manifold Adjoint functors Category of sets Local ring Algebraic geometry Differential topology Gluing schemes Product (category theory) Ring (mathematics) Intersection (set theory) Invertible Homological algebra Abelian group General topology Lawvere–Tierney topology Limits (category theory) Homotopy theory Reflective subcategory Mathematical axioms Contravariant functor Grothendieck topology Group object Mayer–Vietoris axiom Ieke Moerdijk Open cover Open set Category of abelian groups Section (category theory) Sheaf (mathematics) Signature (logic) Smooth manifold Union (set theory) Existential quantification Partial order Topological space Subset Basis of a topological space Kripke–Joyal semantics Locally ringed space Left adjoint Projective limit Colimit Presheaf Sheaf space Stalk of a sheaf dbr:Category_of_local_rings
sameAs	Gluing axiom Gluing axiom Gluing axiom Gluing axiom
dbp:wikiPageUsesTemplate	dbt:Nobreak dbt:Reflist dbt:See_also dbt:Short_description dbt:EGA
has abstract	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)
gold:hypernym	Presheaf
prov:wasDerivedFrom	wikipedia-en:Gluing_axiom?oldid=1124115152&ns=0
page length (characters) of wiki page	10678 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Gluing_axiom
is Link from a Wikipage to another Wikipage of	List of axioms Pseudogroup Sheaf associated to the presheaf Glossary of arithmetic and diophantine geometry Partition of unity Stalk (sheaf) Base (topology) Germ (mathematics) Grothendieck topology Group functor Sheaf (mathematics)

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