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- Grothendiecks_relative_point_of_view abstract "Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object. It is named after Alexander Grothendieck, who made extensive use of it in treating foundational aspects of algebraic geometry. Outside that field, it has been influential particularly on category theory and categorical logic.In the usual formulation, the language of category theory is applied, to describe the point of view as treating, not objects X of a given category C as such, but morphismsf: X → Swhere S is a fixed object. This idea is made formal in the idea of the slice category of objects of C 'above' S. To move from one slice to another requires a base change; from a technical point of view base change becomes a major issue for the whole approach (see for example Beck–Chevalley conditions).A base change 'along' a given morphismg: T → Sis typically given by the fiber product, producing an object over T from one over S. The 'fiber' terminology is significant: the underlying heuristic is that X over S is a family of fibers, one for each 'point' of S; the fiber product is then the family on T, which described by fibers is for each point of T the fiber at its image in S. This set-theoretic language is too naïve to fit the required context, certainly, from algebraic geometry. It combines, though, with the use of the Yoneda lemma to replace the 'point' idea with that of treating an object, such as S, as 'as good as' the representable functor it sets up.The Grothendieck–Riemann–Roch theorem from about 1956 is usually cited as the key moment for the introduction of this circle of ideas. The more classical types of Riemann–Roch theorem are recovered in the case where S is a single point (i.e. the final object in the working category C). Using other S is a way to have versions of theorems 'with parameters', i.e. allowing for continuous variation, for which the 'frozen' version reduces the parameters to constants.In other applications, this way of thinking has been used in topos theory, to clarify the role of set theory in foundational matters. Assuming that we don’t have a commitment to one 'set theory' (all toposes are in some sense equally set theories for some intuitionistic logic) it is possible to state everything relative to some given set theory that acts as a base topos.".
- Grothendiecks_relative_point_of_view wikiPageID "1816636".
- Grothendiecks_relative_point_of_view wikiPageLength "2972".
- Grothendiecks_relative_point_of_view wikiPageOutDegree "23".
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- Grothendiecks_relative_point_of_view wikiPageWikiLink Alexander_Grothendieck.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Algebraic_geometry.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Beck–Chevalley_conditions.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Categorical_logic.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Category:Category_theory.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Category:Scheme_theory.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Category_(mathematics).
- Grothendiecks_relative_point_of_view wikiPageWikiLink Category_theory.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Coefficient.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Comma_category.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Fiber_product.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Final_object.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Grothendieck–Riemann–Roch_theorem.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Heuristic.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Initial_and_terminal_objects.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Intuitionistic_logic.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Mathematical.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Mathematics.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Morphism.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Object_(category_theory).
- Grothendiecks_relative_point_of_view wikiPageWikiLink Parameter.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Pullback_(category_theory).
- Grothendiecks_relative_point_of_view wikiPageWikiLink Representable_functor.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Riemann–Roch_theorem.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Set_theory.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Slice_category.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Topos.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Topos_theory.
- Grothendiecks_relative_point_of_view wikiPageWikiLink Yoneda_lemma.
- Grothendiecks_relative_point_of_view wikiPageWikiLinkText ""relative" perspective".
- Grothendiecks_relative_point_of_view wikiPageWikiLinkText "Grothendieck approach".
- Grothendiecks_relative_point_of_view wikiPageWikiLinkText "Grothendieck relative point of view".
- Grothendiecks_relative_point_of_view wikiPageWikiLinkText "Grothendieck's relative point of view".
- Grothendiecks_relative_point_of_view wikiPageWikiLinkText "base change".
- Grothendiecks_relative_point_of_view wikiPageWikiLinkText "base scheme".
- Grothendiecks_relative_point_of_view wikiPageWikiLinkText "relative point of view".
- Grothendiecks_relative_point_of_view hasPhotoCollection Grothendiecks_relative_point_of_view.
- Grothendiecks_relative_point_of_view id "b/b015310".
- Grothendiecks_relative_point_of_view title "Base change".
- Grothendiecks_relative_point_of_view wikiPageUsesTemplate Template:Springer.
- Grothendiecks_relative_point_of_view subject Category:Category_theory.
- Grothendiecks_relative_point_of_view subject Category:Scheme_theory.
- Grothendiecks_relative_point_of_view comment "Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object. It is named after Alexander Grothendieck, who made extensive use of it in treating foundational aspects of algebraic geometry.".
- Grothendiecks_relative_point_of_view label "Grothendieck's relative point of view".
- Grothendiecks_relative_point_of_view sameAs Basiswechsel_(Faserprodukt).
- Grothendiecks_relative_point_of_view sameAs Schéma_produit.
- Grothendiecks_relative_point_of_view sameAs m.05z7ns.
- Grothendiecks_relative_point_of_view sameAs Q810248.
- Grothendiecks_relative_point_of_view sameAs Q810248.
- Grothendiecks_relative_point_of_view wasDerivedFrom Grothendiecks_relative_point_of_viewoldid=607159907.
- Grothendiecks_relative_point_of_view isPrimaryTopicOf Grothendiecks_relative_point_of_view.