Pullback (category theory)

In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often written
and comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square.
The dual concept of the pullback is the "pushout".
Universal property.
Explicitly, a pullback of the morphisms and consists of an object and two morphisms and for which the diagram
commutes. Moreover, the pullback must be universal with respect to this diagram. That is, for any other such triple where and are morphisms with , there must exist a unique such that 
This situation is illustrated in the following commutative diagram.
As with all universal constructions, a pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks and of the same cospan , there is a unique isomorphism between and respecting the pullback structure.
Pullback and product.
The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms and exist, and forgetting that the object exists. One is then left with a discrete category containing only the two objects and , and no arrows between them. This discrete category may be used as the index set to construct the ordinary binary product. Thus, the pullback can be thought of as the ordinary (Cartesian) product, but with additional structure. Instead of "forgetting" , , and , one can also "trivialize" them by specializing to be the terminal object (assuming it exists). and are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of and .
Examples.
Commutative rings.
In the category of commutative rings (with identity), the pullback is called the fibered product. Let , , and be commutative rings (with identity) and and (identity preserving) ring homomorphisms. Then the pullback of this diagram exists and given by the subring of the product ring defined by 
along with the morphisms 
given by formula_4 and formula_5 for all formula_6. We then have
Groups, Modules.
In complete analogy to the example of commutative rings above, one can show that all pullbacks exist in the category of groups and in the category of modules over some fixed ring.
Sets.
In the category of sets, the pullback of functions and always exists and is given by the set
together with the restrictions of the projection maps and to .
Alternatively one may view the pullback in asymmetrically:
where formula_10 is the disjoint union of sets (the involved sets are not disjoint on their own unless resp. is injective). In the first case, the projection extracts the index while forgets the index, leaving elements of .
This example motivates another way of characterizing the pullback: as the equalizer of the morphisms where is the binary product of and and and are the natural projections. This shows that pullbacks exist in any category with binary products and equalizers. In fact, by the existence theorem for limits, all finite limits exist in a category with binary products and equalizers; equivalently, all finite limits exist in a category with terminal object and pullbacks (by the fact that binary product = pullback on the terminal object, and that an equalizer is a pullback involving binary product).
Fiber bundles.
Another example of a pullback comes from the theory of fiber bundles: given a bundle map and a continuous map , the pullback (formed in the category of topological spaces with continuous maps) is a fiber bundle over called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
Preimages and intersections.
Preimages of sets under functions can be described as pullbacks as follows: 
Suppose , . Let be the inclusion map . Then a pullback of and (in ) is given by the preimage together with the inclusion of the preimage in 
and the restriction of to 
Because of this example, in a general category the pullback of a morphism and a monomorphism can be thought of as the "preimage" under of the subobject specified by . Similarly, pullbacks of two monomorphisms can be thought of as the "intersection" of the two subobjects.
Least common multiple.
Consider the multiplicative monoid of positive integers as a category with one object. In this category, the pullback of two positive integers and is just the pair , where the numerators are both the least common multiple of and . The same pair is also the pushout.
Properties.
formula_11
formula_12
Weak pullbacks.
A weak pullback of a cospan is a cone over the cospan that is only weakly universal, that is, the mediating morphism above is not required to be unique.