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 for 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. Heuristic is the art and science of discovery and invention. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... A parameter is a measurement or value on which something else depends. ... Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... 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 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 Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...

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 comma category is a construction in category theory, a branch of mathematics. ...

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. In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ... In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...

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 mathematics, specifically in algebraic geometry, the Grothendieck-Riemann-Roch theorem is a far-reaching result on coherent cohomology. ... In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

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 a base topos. For discussion of topoi in literary theory, see literary topos. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...