Grothendieck's relative point of view

Grothendieck's relative point of view

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 morphisms

:"f": "X" → "S"

where "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 morphism

:"g": "T" → "S"

is 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 which acts as a base topos.

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Relative — can refer to: *Kinship, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be relatives Physics*Relativity as a concept in physics (for example Albert Einstein s… …   Wikipedia

  • Alexander Grothendieck — User:Geometry guy/InfoboxAlexander Grothendieck (born March 28, 1928 in Berlin, Germany) is considered to be one of the greatest mathematicians of the 20th century. He made major contributions to: algebraic topology, algebraic geometry, number… …   Wikipedia

  • Coherent duality — In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the local… …   Wikipedia

  • List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

  • List of algebraic geometry topics — This is a list of algebraic geometry topics, by Wikipedia page. Contents 1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 …   Wikipedia

  • Categorical logic — is a branch of category theory within mathematics, adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category,… …   Wikipedia

  • Glossary of scheme theory — This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of… …   Wikipedia

  • Scheme (mathematics) — In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider …   Wikipedia

  • mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

  • Algebraic geometry — This Togliatti surface is an algebraic surface of degree five. Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”