Grothendieck's relative point of view

Grothendieck's relative point of view is a <a href="/facts/Heuristic/BcBEkv9c">heuristic</a> applied in certain abstract <a href="/facts/Mathematical/pxTouaz4">mathematical</a> situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on <a href="/facts/Parameter/zFvVFMv9">parameters</a>, as the basic field of study, rather than a single such object. It is named after <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a>, who made extensive use of it in treating foundational aspects of <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>. Outside that field, it has been influential particularly on <a href="/facts/Category_theory/6zS3DXPa">category theory</a> and <a href="/facts/Categorical_logic/u4CIDbss">categorical logic</a>.
In the usual formulation, the point of view treats not <a href="/facts/Object_(category_theory)/7WzxUThf">objects</a> X of a given category C, but <a href="/facts/Morphism/Kdpuyz3S">morphisms</a>

f: X → S
where S is a fixed object. This idea is made formal in the idea of the <a href="/facts/Slice_category/wTa3tURp">slice category</a> 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 <a href="/facts/Fiber_product/bC0QoX5U">fiber product</a>, 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 <a href="/facts/Yoneda_lemma/j04vXd6x">Yoneda lemma</a> to replace the 'point' idea with that of treating an object, such as S, as 'as good as' the <a href="/facts/Representable_functor/dILwN4oj">representable functor</a> it sets up.
The <a href="/facts/Grothendieck%25E2%2580%2593Riemann%25E2%2580%2593Roch_theorem/hh5nyYA9">Grothendieck–Riemann–Roch theorem</a> from about 1956 is usually cited as the key moment for the introduction of this circle of ideas. The more classical types of <a href="/facts/Riemann%25E2%2580%2593Roch_theorem/TRaNyUQA">Riemann–Roch theorem</a> are recovered in the case where S is a single point (i.e. the <a href="/facts/Final_object/oeckvXRj">final object</a> 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 <a href="/facts/Coefficient/5Hwq2Wuq">constants</a>.
In other applications, this way of thinking has been used in <a href="/facts/Topos_theory/zzttCIwC">topos theory</a>, to clarify the role of <a href="/facts/Set_theory/1ptfsvUP">set theory</a> in foundational matters. Assuming that we don't have a commitment to one 'set theory' (all topoi are in some sense equally set theories for some <a href="/facts/Intuitionistic_logic/zI7ulCnM">intuitionistic logic</a>) it is possible to state everything relative to some given set theory that acts as a base topos.

Grothendieck's relative point of view open-in-new

Grothendieck's relative point of view