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

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

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.

• This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f o p₁, g o 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 a terminal object, binary products and equalizers.

Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B 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.

Preimages

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

f : A → B

and

B₀ ⊆ B.

Let g be the inclusion map 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

f^-1 [ B₀ ] ↪ A

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

f^-1 [ B₀ ] → B₀.

Properties

• Whenever X ×_ZY exists, then so does Y ×_Z X and there is an isomorphism X ×_Z Y $\cong$ Y ×_ZX.
• Monomorphisms are stable under pullback: if the arrow f above is monic, then so is the arrow p₂. For example, in the category of sets, if X is a subset of Z, then, for any g : Y → Z, the pullback X ×_Z Y is the inverse image of X under g.
• Isomorphisms are also stable, and hence, for example, X ×_X Y $\cong$ Y for any map Y → X.
• Any category with pullbacks and products has equalizers.

See also

• The categorical dual of a pullback is a called a pushout.
• Product (Category Theory)
• Pullbacks in differential geometry
• Equijoin in relational algebra.

References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
• Cohn, Paul M.; Universal Algebra (1981), D.Reidel Publishing, Holland, ISBN 90-277-1213-1 (Originally published in 1965, by Harper & Row).

External links

• Interactive Web page which generates examples of pullbacks in the category of finite sets. Written by Jocelyn Paine.
• pullbacks on the N-Category Lab.