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; it is the limit of the cospan X \rightarrow Z \leftarrow Y. The pullback is often written

 P = X \underset{Z}{\times} Y.\,

Contents

Universal property

Explicitly, the pullback of the morphisms f and g consists of an object P and two morphisms p1 : P → X and p2 : P → Y for which the diagram

commutes. Moreover, the pullback (P, p1, p2) must be universal with respect to this diagram. That is, for any other such triple (Q, q1, q2) for which the following diagram commutes, there must exist a unique u : Q → P (called mediating morphism) such that p_2 \circ u=q_2 and p_1\circ u=q_1

As with all universal constructions, the pullback, if it exists, is unique up to a unique isomorphism.

Weak pullbacks

A weak pullback of a cospan X → Z ← Y is a cone over the cospan that is only weakly universal, that is, the mediating morphism u : Q → P above need not be unique.

Examples

In the category of sets, a pullback of f and g is given by the set

X\underset{Z}{\times} Y = \{(x, y) \in X \times Y| f(x) = g(y)\},\,

together with the restrictions of the projection maps \pi_1 and \pi_2 to X × Z Y .

Alternatively one may view the pullback in Set asymmetrically:

X\underset{Z}{\times} Y

\cong
\coprod_{x\in X} g^{-1}[\{f(x)\}]

\cong
\coprod_{y\in Y} f^{-1}[\{g(y)\}]

where \coprod is the disjoint (tagged) union of sets (the involved sets are not disjoint on their own unless f resp. g is injective). In the first case, the projection \pi_1 extracts the x index while \pi_2 forgets the index, leaving elements of Y.

Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : EB and a continuous map f : X → B, the pullback X ×B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.

In any category with a terminal object Z, the pullback X ×Z Y is just the ordinary product X × Y.[1]

Preimages

Preimages of sets under functions can be described as pullbacks as follows: Suppose

f : AB

and

B0 ⊆ B.

Let g be the inclusion map B0B.

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

f-1 [ B0 ] ↪ A

and the restriction of f to f-1 [ B0 ]

f-1 [ B0 ] B0.

Properties

See also

Notes

  1. ^ Adámek, p. 197.

References

External links