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 $f : X \to Z$ and $g : Y \to Z$ with a common codomain; it is the limit of the cospan $X \to Z \leftarrow Y$ . The pullback is often written

P = X \times Z Y

The categorical dual of a pullback is a called a pushout. Remarks opposite to the above apply: the pushout is a coproduct with additional structure.

Universal property

Explicitly, the pullback of the morphisms $f$ and $g$ consists of an object $P$ and two morphisms $p 1 : P \to X$ and $p 2 : P \to Y$ for which the diagram

commutes. Moreover, the pullback $(P, p 1, p 2)$ must be universal with respect to this diagram. That is, for any other such triple $(Q, q 1, q 2)$ for which the following diagram commutes, there must exist a unique $u : Q \to P$ (called a mediating morphism) such that

p_2 \circ u=q_2, \qquad p_1\circ u=q_1.

As with all universal constructions, the pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks $(A, a 1, a 2)$ and $(B, b 1, b 2)$ of the same cospan, there is a unique isomorphism between $A$ and $B$ respecting the pullback structure.

Weak pullbacks

A weak pullback of a cospan $X \to Z \leftarrow Y$ is a cone over the cospan that is only weakly universal, that is, the mediating morphism $u : Q \to P$ above is not required to be unique.

Pullback and product

The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms $f$ and $g$ exist, and forgetting that the object $Z$ exists. One is then left with a discrete category containing only the two objects $X$ and $Y$ , 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" $Z$ , $f$ , 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 $Y$ .

Examples

Commutative rings

Pullback of commutative rings admits this diagram commutes.

In the category of commutative rings (with identity), denoted $CRing$ , the pullback is called the fibered product. Let

A, B, C \in Ob(CRing)

α : A \to C \in Hom(CRing)

β : B \to C \in Hom(CRing)

So $A$ , $B$ , and $C$ are commutative rings with identity and $α$ and $β$ are ring homomorphisms. Then the pullback of these objects and morphisms is defined to be the subset of the Cartesian product $A \times B$ defined by

A \times_{C} B = \left\{(a,b) \in A \times B \; \big| \; \alpha(a) = \beta(b) \right\}

along with the morphisms

\beta' \colon A \times_{C} B \to A, \qquad \alpha'\colon A \times_{C} B \to B

such that

\alpha \circ \beta' = \beta \circ \alpha'.

Sets

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

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

together with the restrictions of the projection maps $π 1$ and $π 2$ to $X \times Z Y$ .

Alternatively one may view the pullback in $Set$ asymmetrically:

X\times_Z 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 $π 1$ extracts the $x$ index while $π 2$ forgets the index, leaving elements of $Y$ .

This example motivates another way of characterizing the pullback: as the equalizer of the morphisms $f \circ p 1, g \circ p 2 : X \times Y \to Z$ where $X \times Y$ is the binary product of $X$ and $Y$ and $p 1$ and $p 2$ 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.

Fiber bundles

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

Categories with a terminal object

In any category with a terminal object $T$ , the pullback $X \times T Y$ is just the ordinary product $X \times Y$ .^[1]

Preimages

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

Suppose $f : A \to B$ , $B 0 \subseteq B$ . Let $g$ be the inclusion map $B 0 ↪ B$ . Then a pullback of $f$ and $g$ (in $Set$ ) is given by the preimage $f -1 [B 0]$ together with the inclusion of the preimage in $A$

f -1 [B 0] ↪ A

and the restriction of $f$ to $f -1 [B 0]$

f -1 [B 0] \to B 0

Properties

Whenever $X \times Z Y$ exists, then so does $Y \times Z X$ and there is an isomorphism $X \times Z Y ≅ Y \times Z X$ .
Monomorphisms are stable under pullback: if the arrow $f$ above is monic, then so is the arrow $p 2$ . For example, in the category of sets, if $X$ is a subset of $Z$ , then, for any $g : Y \to Z$ , the pullback $X \times Z Y$ is the inverse image of $X$ under $g\circ p_2$ .
Isomorphisms are also stable, and hence, for example, $X \times X Y ≅ Y$ for any map $Y \to X$ .
Any category with pullbacks and products has equalizers.

Notes

↑ Adámek, p. 197.

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.