Grothendieck's relative point of view
Encyclopedia
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 parameter
s, 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 morphism
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
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.
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
s, as the basic field of study, rather than a single such object. It is named after Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
, who made extensive use of it in treating foundational aspects of algebraic geometry
Algebraic geometry
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 occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. Outside that field, it has been influential particularly on category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
and categorical logic
Categorical logic
Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but more notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor...
.
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 morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s
- 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
Yoneda lemma
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
to replace the 'point' idea with that of treating an object, such as S, as 'as good as' the representable functor
Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
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
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
.
In other applications, this way of thinking has been used in topos theory, to clarify the role of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
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
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
) it is possible to state everything relative to some given set theory which acts as a base topos.