Fiber product

The fiber product (also pullback, cartesian square or pullback square) is a term from the mathematical branch of category theory . The fiber product is of central importance in algebraic geometry .

The term fiber product is dual to the term pushout .

Fiber product of quantities

If and are two maps of sets, then the fiber product of and over the subset ${\displaystyle \xi \colon X\to S}$ ${\displaystyle \upsilon \colon Y\to S}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle S}$

${\displaystyle \{(x,y)\in X\times Y\mid \xi (x)=\upsilon (y)\}}$

of the Cartesian product of and . ${\displaystyle X}$ ${\displaystyle Y}$

Fiber products in any category

Definition via objects

If morphisms and are given in a category , then an object is called together with morphisms ${\displaystyle \xi :X\rightarrow S}$ ${\displaystyle v:Y\rightarrow S}$ ${\displaystyle X\times _{S}Y}$

${\displaystyle \mathrm {pr} _{1}\colon X\times _{S}Y\to X}$ and ${\displaystyle \mathrm {pr} _{2}\colon X\times _{S}Y\to Y,}$

the so-called canonical projections, a fiber product of X and Y over S, if and the following universal property is fulfilled: ${\displaystyle \xi \circ \mathrm {pr} _{1}=v\circ \mathrm {pr} _{2}}$

For each pair of morphisms from a test object T to X or Y , for the ${\displaystyle (f:T\rightarrow X,\,g:T\rightarrow Y)}$

${\displaystyle \xi f=\upsilon g}$(as morphisms ) ${\displaystyle T\rightarrow S}$

holds, there is exactly one morphism

${\displaystyle c\colon T\to X\times _{S}Y,}$

so that

${\displaystyle f=\mathrm {pr} _{1}\circ c}$ and ${\displaystyle g=\mathrm {pr} _{2}\circ c}$

applies.

In other words: the functors

${\displaystyle \mathrm {Hom} (T,X\times _{S}Y)}$ and ${\displaystyle \mathrm {Hom} (T,X)\times _{\mathrm {Hom} (T,S)}\mathrm {Hom} (T,Y)}$

are of course equivalent via pr ₁ and pr ₂ .

Definition via morphisms

In a more general approach, such pairs of morphisms and from an object to or from a fiber product, pullback, Cartesian, or pullback square are called, for which: ${\displaystyle f\operatorname {\colon } T\rightarrow X}$ ${\displaystyle g\operatorname {\colon } T\rightarrow Y}$ ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle Y}$

1. ${\displaystyle \xi f=\upsilon g}$(as morphisms ') ${\displaystyle T\to S}$
2. every further pair of morphisms and from an object to or for which applies is interchangeable with the first pair of morphisms via a uniquely determined morphism , i.e. H. and ${\displaystyle f'\operatorname {\colon } T'\rightarrow X}$ ${\displaystyle g'\operatorname {\colon } T'\rightarrow Y}$ ${\displaystyle T'}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \xi f'=\upsilon g'}$ ${\displaystyle e\operatorname {\colon } T'\rightarrow T}$ ${\displaystyle g'=ge}$ ${\displaystyle f'=fe.}$

The morphisms of pullbacks form a commutative diagram:

${\displaystyle {\begin{array}{rcl}T&{\xrightarrow[{}]{f}}&X\\g\!\downarrow &&\downarrow \!\xi \\Y&{\xrightarrow[{v}]{}}&S\\\end{array}}}$

This diagram represents a cone above the diagram with the “middle” arrow (the one between and ) omitted. The second condition expresses that the pullback is a limit of all such cones. It is said that it arises by pulling back from along and arises by withdrawing from along ${\displaystyle X{\xrightarrow {\xi }}S{\xleftarrow {\upsilon }}Y}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle f}$ ${\displaystyle \upsilon }$ ${\displaystyle \xi }$ ${\displaystyle g}$ ${\displaystyle \xi }$ ${\displaystyle \upsilon }$

Pullback cone

Occasionally, such pairs of morphisms ( ) from an object to or , for which only ${\displaystyle f\colon T\to X,g\colon T\to Y}$ ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle \xi f=\upsilon g}$(as morphisms ) ${\displaystyle T\to S}$

is known as a pullback cone ; Morphisms of pullback cones are defined using commutative diagrams. The fiber product is then an end object of the category of possible pullback cones above the diagram ${\displaystyle X{\xrightarrow {\xi }}S{\xleftarrow {\upsilon }}Y.}$

Uniqueness

The components and the fiber product from the definition via morphisms do not have to be clearly determined, but are unambiguous except for isomorphism. Ie., Is together with pictures and another such fiber product, so are and isomorphic and clearly through and determined. There can also be different possibilities for the morphisms and for one and the same object . The different variants are then in turn clearly determined by an isomorphism (of to itself) through one another. ${\displaystyle T,f}$ ${\displaystyle g}$ ${\displaystyle T'}$ ${\displaystyle f'}$ ${\displaystyle g'}$ ${\displaystyle T}$ ${\displaystyle T'}$ ${\displaystyle f'}$ ${\displaystyle g'}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle T}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle T}$

Also from the definition of objects, there is generally only one symbol for several possible objects that are isomorphic to one another. However, a standard representation is usually given for; z. B. in the category of quantities the quantity: ${\displaystyle X\times _{S}Y}$ ${\displaystyle X\times _{S}Y}$

${\displaystyle D=\{(x,y)|x\in X,y\in Y\ \mathrm {und} \ \xi (x)=\upsilon (y)\}\cong X\times _{S}Y}$

designation

The terms are not used consistently. More commonly, in mathematical texts, fiber product is used to describe the resulting object of product formation, while pullback is used to describe the resulting pair of images. In addition, there is the generalized designation of the fiber product as "Product about ...". With Cartesian or pullback square the overall construction or pullback diagram is then even more referred. Ultimately, however, the terms are interpreted synonymously and are only used differently in order to focus on a certain aspect of the fiber product.

properties

• If morphism is arbitrary, then is ${\displaystyle X\to Y}$

${\displaystyle X\times _{Y}Y\cong X}$.

• If and are injective set mappings (generally monomorphisms ), then the fiber product is the intersection (of the images) of and ${\displaystyle \xi }$ ${\displaystyle \upsilon }$ ${\displaystyle X}$ ${\displaystyle Y}$
• If the set is one element, the fiber product is isomorphic to the Cartesian product . The standard representation (see above) of the fiber product in the category of quantities is then identical to the Cartesian product. If an end object is general , the fiber product is isomorphic to the general categorical product . ${\displaystyle S}$ ${\displaystyle S}$
• The standard representation (see above) of a fiber product in the category of quantities is a subset of the Cartesian product. In general there is always a monomorphism from the fiber product to the general categorical product

${\displaystyle X\times _{S}Y\to X\times Y}$

(if both constructions exist).

• For an asymmetrical view of the fiber product, see change of base (fiber product) .

Examples

• The fiber product is a special Limes . Due to the continuity of the respective forget function , in the following categories - whose objects are always based on quantities - the underlying quantity of the fiber product (in this category) is the same as the fiber product (in the category of quantities) of the underlying quantities:

Groups, Abelian groups, rings, modules, vector spaces, topological spaces, Banach spaces.

• In the scheme category , the fiber product is given locally by tensor products . It's I. A. not the fiber product of the underlying topological spaces!
• The equality union in relational algebra .

Fiber products in algebraic geometry

The above categorical definition is used in particular in algebraic geometry to define the fiber product of two schemes with given morphisms . ${\displaystyle X\times _{S}Y}$ ${\displaystyle \xi \colon X\to S,\nu \colon Y\to S}$

If and are affine schemes , then there is also an affine scheme. From it follows namely ${\displaystyle X,Y}$ ${\displaystyle S}$ ${\displaystyle X\times _{S}Y}$ ${\displaystyle X=Spec(A),Y=Spec(B),S=Spec(R)}$

${\displaystyle X\times _{S}Y=Spec(A\otimes _{R}B)}$.

This gives an explicit description (and in particular proves the existence) of the fiber product of affine schemes.

An explicit description for fiber products of any scheme is obtained as follows. Be a covering by affine schemes, and all were ${\displaystyle S=\bigcup _{i}U_{i}}$ ${\displaystyle i}$

${\displaystyle \xi ^{-1}(U_{i})=\bigcup _{j}V_{ij},\nu ^{-1}(U_{i})=\bigcup _{k}W_{ik}}$

coverages by affine schemes, then is

${\displaystyle X\times _{S}Y=\bigcup _{i}\bigcup _{j,k}V_{ij}\times _{U_{i}}W_{ik}}$

a coverage by affine schemas, in particular, is thus defined as a schema. ${\displaystyle X\times _{S}Y}$

For each point in a scheme, designate the associated local ring . The points of the fiber product then correspond bijectively to the tuples with and a prime ideal . ${\displaystyle x}$ ${\displaystyle \kappa (x)}$ ${\displaystyle X\times _{S}Y}$ ${\displaystyle (x,y,s,{\mathfrak {p}})}$ ${\displaystyle \xi (x)=\nu (y)=s}$ ${\displaystyle {\mathfrak {p}}\subset \kappa (x)\otimes _{\kappa (s)}\kappa (y)}$

Individual evidence

1. ↑ ^{a b} R. Goldblatt u. a .: Topoi - The Categorial Analysis of Logic . Studies in Logic and the Foundations of Mathematics. Vol. 98. North-Holland Publishing Company, Amsterdam / New York / Oxford 1979, ISBN 0-444-85207-7 , chap. 3.13, p. 63 (English, description of pullbacks.).
2. ↑ R. Goldblatt u. a .: Topoi - The Categorial Analysis of Logic . Studies in Logic and the Foundations of Mathematics. Vol. 98. North-Holland Publishing Company, Amsterdam / New York / Oxford 1979, ISBN 0-444-85207-7 , chap. 3.11, p. 58 (English, description of limits and co-limits.).
3. ↑ Hartmut Ehrig, Michael Pfender and students of mathematics: Categories and automata . Walter de Gruyter, Berlin / New York 1972, ISBN 3-11-003902-8 , Def.334, p. 60 (Definition of pullbacks.).
4. ↑ R. Goldblatt u. a .: Topoi - The Categorial Analysis of Logic . Studies in Logic and the Foundations of Mathematics. Vol. 98. North-Holland Publishing Company, Amsterdam / New York / Oxford 1979, ISBN 0-444-85207-7 , chap. 3.6, p. 44 (English, definition of end objects.).
5. ↑ Hartmut Ehrig, Michael Pfender and students of mathematics: Categories and automata . Walter de Gruyter, Berlin / New York 1972, ISBN 3-11-003902-8 , Def.1.25, p. 19 (Definition of end objects.).
6. ↑ R. Goldblatt u. a .: Topoi - The Categorial Analysis of Logic . Studies in Logic and the Foundations of Mathematics. Vol. 98. North-Holland Publishing Company, Amsterdam / New York / Oxford 1979, ISBN 0-444-85207-7 (English).
7. ↑ Hartmut Ehrig, Michael Pfender and students of mathematics: Categories and automata . Walter de Gruyter, Berlin / New York 1972, ISBN 3-11-003902-8 .
8. ^ Saunders Mac Lane: Categories . Conceptual language and mathematical theory. Springer-Verlag, Berlin / Heidelberg / New York 1972, ISBN 3-540-05634-3 (American English: Categories. For the Working Mathematician. Translated by Klaus Schürger).
9. ↑ The stacks Project Lemma 25.17.3
10. ↑ The stacks Project Lemma 25.6.7
11. ↑ The stacks Project Lemma 25.17.4
12. ↑ The stacks Project Lemma 25.17.5