Pullback (category theory)

The category of commutative rings admits pullbacks.

In the category of commutative rings (with identity), the pullback is called the fibered product. Let A, B, and C be commutative rings (with identity) and α : A → C and β : B → C (identity preserving) ring homomorphisms. Then the pullback of this diagram exists and is given by the subring of the product ring A × B defined by

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

along with the morphisms

${\displaystyle \beta '\colon A\times _{C}B\to A,\qquad \alpha '\colon A\times _{C}B\to B}$

given by ${\displaystyle \beta '(a,b)=a}$ and ${\displaystyle \alpha '(a,b)=b}$ for all ${\displaystyle (a,b)\in A\times _{C}B}$. We then have

${\displaystyle \alpha \circ \beta '=\beta \circ \alpha '.}$

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.

In the category of sets, the pullback of functions f : X → Z and g : Y → Z always exists and is given by the set

${\displaystyle X\times _{Z}Y=\{(x,y)\in X\times Y|f(x)=g(y)\}=\bigcup _{z\in f(X)\cap g(Y)}f^{-1}[\{z\}]\times g^{-1}[\{z\}],}$

together with the restrictions of the projection maps π₁ and π₂ to X ×_Z Y.

Alternatively one may view the pullback in Set asymmetrically:

${\displaystyle X\times _{Z}Y\cong \coprod _{x\in X}g^{-1}[\{f(x)\}]\cong \coprod _{y\in Y}f^{-1}[\{g(y)\}]}$

where ${\displaystyle \coprod }$ is the disjoint union of sets (the involved sets are not disjoint on their own unless f resp. g is injective). In the first case, the projection π₁ extracts the x index while π₂ forgets the index, leaving elements of Y.

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 = pullback on the terminal object, and that an equalizer is a pullback involving binary product).

A specific example of a pullback is given by the graph of a function. Suppose that ${\displaystyle f\colon X\to Y}$ is a function. The graph of f is the set

$${\displaystyle \Gamma _{f}=\{(x,f(x))\colon x\in X\}\subseteq X\times Y.}$$

The graph can be reformulated as the pullback of f and the identity function on Y. By definition, this pullback is

$${\displaystyle X\times _{f,Y,1_{Y}}Y=\{(x,y)\colon f(x)=1_{Y}(y)\}=\{(x,y)\colon f(x)=y\}\subseteq X\times Y,}$$

and this equals ${\displaystyle \Gamma _{f}}$.

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 (formed in the category of topological spaces with continuous maps) X ×_B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles. This is also the case in the category of differentiable manifolds. 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. The pullback of two smooth transverse maps into the same differentiable manifold is also a differentiable manifold, and the tangent space of the pullback is the pullback of the tangent spaces along the differential maps.

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

Suppose f : A → B, 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.

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 (LCM(m, n)/m, LCM(m, n)/n), where the numerators are both the least common multiple of m and n. The same pair is also the pushout.