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 f : X → Z and g : Y → Z with a common codomain.
In many situations, X ×Z Y may intuitively be thought of as consisting of pairs of elements (x, y) with x in X, y in Y, and f(x) = g(y).
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.
[1] That is, for any other such triple (Q, q1, q2) where q1 : Q → X and q2 : Q → Y are morphisms with f q1 = g q2, there must exist a unique u : Q → P 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.
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" Z, f, and g, one can also "trivialize" them by specializing Z to be the terminal object (assuming it exists).
f and g are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of X and Y.
In the category of commutative rings (with identity), the pullback is called the fibered product.
Then the pullback of this diagram exists and is given by the subring of the product ring A × B defined by along with the morphisms given by
We then have 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.
This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f ∘ p1, g ∘ p2 : X × Y → Z where X × Y is the binary product of X and Y and p1 and p2 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 is equal to pullback on the terminal object, and that an equalizer is a pullback involving binary product).
The graph can be reformulated as the pullback of f and the identity function on Y.
A special case is the pullback of two fiber bundles E1, E2 → B.
In this case E1 × E2 is a fiber bundle over B × B, and pulling back along the diagonal map B → B × B gives a space homeomorphic (diffeomorphic) to E1 ×B E2, which is a fiber bundle over B.
Preimages of sets under functions can be described as pullbacks as follows: Suppose f : A → B, B0 ⊆ B.
Then a pullback of f and g (in Set) is given by the preimage f−1[B0] together with the inclusion of the preimage in A and the restriction of f to f−1[B0] Because of this example, in a general category the pullback of a morphism f and a monomorphism g can be thought of as the "preimage" under f of the subobject specified by g. Similarly, pullbacks of two monomorphisms can be thought of as the "intersection" of the two subobjects.
Consider the multiplicative monoid of positive integers Z+ as a category with one object.
In this category, the pullback of two positive integers m and n is just the pair
