In category theory, a branch of mathematics, a pullback is the limit of a diagramconsisting 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 universalwith 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 uniqueisomorphism between and respecting the pullback structure.
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 product, but with additional structure. Instead of "forgetting",, and, one can also "trivialize" them by specializing to be the terminal object. 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, the pullback is called the fibered product. Let,, and be commutative rings and and 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 and for all. We then have
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 is the disjoint union of sets. 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 a terminal object, binary products and equalizers.
s of sets under functions can be described as pullbacks as follows: Suppose,. Let be the inclusion map. Then a pullback of and 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.
Consider the multiplicativemonoid 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
In any category with a terminal object, the pullback is just the ordinary product.
Monomorphisms are stable under pullback: if the arrow in the diagram is monic, then so is the arrow. Similarly, if is monic, then so is.
Isomorphisms are also stable, and hence, for example, for any map .
In an abelian category all pullbacks exist, and they preserve kernels, in the following sense: if
* if maps f : A → C, g : B → C and h : D → B are given and
* the pullback of f and g is given by r : P → A and s : P → B, and
* the pullback of s and h is given by t : Q → P and u : Q → D,
* then the pullback of f and gh is given by rt : Q → A and u : Q → D.
Any category with pullbacks and products has equalizers.
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.
