Pullback (category theory)

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

The term of the fiber product is dual to the notion of pushout.

Fiber product of sets

Ξ is: X → S and υ: Y → S pictures of two sets, then the fiber product of X and Y over the subset S

The Cartesian product of X and Y.

Fiber products in any category

Definition about objects

Are morphisms ξ: X → S and υ: Y → S given in a category, it means an object X × S Y together with morphisms

Canonical projections, a fiber product of X and Y on S, when the following universal property is satisfied:

In other words, the functors

Pr1 and pr2 are via naturally equivalent.

Definition on morphisms

In a more general approach to such pairs of morphisms, and an object T by X and Y as a fiber product, pullback, Cartesian or pullback square are called, in which:

The morphisms of pullbacks form a commutative diagram:

This diagram shows a cone above the graph in which the "average " arrow ( between T and S ) has been omitted. The second condition is expressed that the pullback is a limit any such cone. It is said that f arises by pulling back ( engl. pull back ) of υ along ξ and g arises by retraction of ξ υ along.

Pullback cone

Occasionally, such pairs of morphisms ( f: T → X, g: T → Y) of an object T by X and Y, respectively, for the only

Applies, referred to as the pullback cone; Morphisms by pullback cones are defined by corresponding commutative diagrams. The fiber product is then a final object of the category of the possible pullback cone over the diagram

Unambiguity

The components T, f and g of the fiber product from the definition on morphisms need not be uniquely determined, but are unique up to isomorphism. That is, T ' together with pictures f' and g ' another such fiber product as T and T' are isomorphic and f ' and g' uniquely determined by f and g. For one and the same object T may also be different options for the morphisms f and g give. The different variants but are then in turn uniquely determined by each other by an isomorphism ( onto itself by T).

Also from the definition about objects is in general only a symbol of several possible, each mutually isomorphic objects. However, it is usually given a default representation for; For example, in the category of sets the amount:

Designation

The terms used are not fully consistent. Is commonly used in mathematical texts with fiber product rather referred to the resulting object in the product formation, whereas with pullback, the resulting pair is described by pictures. Added to this is the generalized name of the fiber product as " product about ... ". With Cartesian or pullback square the entire construction or the pullback diagram is then even more, respectively. Ultimately, the terms are, however, interpreted synonymously and are only used differently to each move to a particular aspect of the fiber product the center of attention.

Properties

• If X → Y is any morphism, then

• Are ξ and υ injective lot of pictures ( generally monomorphisms ), then the fiber product of the cut ( the images ) of X and Y.
• If S is a singleton, then the fiber product is isomorphic to the Cartesian product. The standard representation (see above) of the fiber product in the category of sets is then identical to the Cartesian product. Is generally a final object S, the fiber product is isomorphic to the general categorical product.
• The standard representation (see above) the fiber product in the category of sets is a subset of the Cartesian product. Generally, there is always a monomorphism from the fiber product in the general categorical product

• For an asymmetric view of the fiber product see the change of basis ( fiber product ).

Examples

• The fiber product is a special limit. Due to the continuity of the respective Vergissfunktors is in the following categories - whose objects always amounts are based on - the amount of the fiber product underlying ( in this category ) is equal to the fiber product ( in the category of sets ) of the underlying quantities:

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