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. The pullback is written

P = X ×_{f, Z, g} Y.

Usually the morphisms Template:Mvar and Template:Mvar are omitted from the notation, and then the pullback is written

P = X ×_Z Y.

The pullback comes equipped with two natural morphisms P → X and P → Y. The pullback of two morphisms f and g need not exist, but if it does, it is essentially uniquely defined by the two morphisms. 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.

The dual concept of the pullback is the pushout.

Universal property

Explicitly, a 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

File:Categorical pullback.svg

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) 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

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

This situation is illustrated in the following commutative diagram.

File:Categorical pullback (expanded).svg

As with all universal constructions, a pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks (A, a₁, a₂) and (B, b₁, b₂) of the same cospan X → Z ← Y, there is a unique isomorphism between Template:Mvar and Template:Mvar respecting the pullback structure.

Pullback and product

The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms Template:Mvar and Template:Mvar exist, and forgetting that the object Template:Mvar exists. One is then left with a discrete category containing only the two objects Template:Mvar and Template:Mvar, 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" Template:Mvar, Template:Mvar, and Template:Mvar, one can also "trivialize" them by specializing Template:Mvar to be the terminal object (assuming it exists). Template:Mvar and Template:Mvar are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of Template:Mvar and Template:Mvar.

Examples

Commutative rings

File:Pullback commutative rings.svg

The category of commutative rings admits pullbacks.

In the category of commutative rings (with identity), the pullback is called the fibered product. Let Template:Mvar, Template:Mvar, and Template:Mvar 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

$A{\times }_{C}B=\{(a,b)\in A\times B|\alpha (a)=\beta (b)\}$

along with the morphisms

${\beta }^{\prime }:A{\times }_{C}B\to A,{\alpha }^{\prime }:A{\times }_{C}B\to B$

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

$\alpha \circ {\beta }^{\prime }=\beta \circ {\alpha }^{\prime }.$

Groups and modules

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.

Sets

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

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

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

Alternatively one may view the pullback in Set asymmetrically:

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

where $\coprod$ is the disjoint union of sets (the involved sets are not disjoint on their own unless Template:Mvar resp. Template:Mvar is injective). In the first case, the projection π₁ extracts the Template:Mvar index while π₂ forgets the index, leaving elements of Template:Mvar.

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 Template:Mvar and Template:Mvar 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 a binary product is equal to a pullback on the terminal object, and that an equalizer is a pullback involving a binary product).

Graphs of functions

A specific example of a pullback is given by the graph of a function. Suppose that $f:X\to Y$ is a function. The graph of Template:Mvar is the set $${\mathrm{\Gamma }}_f=\{(x,f(x)):x\in X\}\subseteq X\times Y.$$ The graph can be reformulated as the pullback of Template:Mvar and the identity function on Template:Mvar. By definition, this pullback is $$X{\times }_{f,Y,1_Y}Y=\{(x,y):f(x)=1_Y(y)\}=\{(x,y):f(x)=y\}\subseteq X\times Y,$$ and this equals ${\mathrm{\Gamma }}_f$.

Fiber bundles

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 Template:Mvar called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles. 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. All statements here hold true for differentiable manifolds as well. Differentiable maps f : M → N and g : P → N are transverse if and only if their product M × P → N × N is transverse to the diagonal of N. Thus, the pullback of two transverse differentiable 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 and intersections

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

Suppose f : A → B, B₀ ⊆ B. Let Template:Mvar be the inclusion map B₀ ↪ B. Then a pullback of Template:Mvar and Template:Mvar (in Set) is given by the preimage f⁻¹[B₀] together with the inclusion of the preimage in Template:Mvar

f⁻¹[B₀] ↪ A

and the restriction of Template:Mvar 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.

Least common multiple

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 $(\frac{\mathrm{lcm}(m,n)}{m},\frac{\mathrm{lcm}(m,n)}{n})$, where the numerators are both the least common multiple of m and n. The same pair is also the pushout.

Properties

• In any category with a terminal object Template:Mvar, the pullback X ×_T Y is just the ordinary product X × Y.
• Monomorphisms are stable under pullback: if the arrow Template:Mvar in the diagram is monic, then so is the arrow p₂. Similarly, if Template:Mvar is monic, then so is p₁.
• Isomorphisms are also stable, and hence, for example, X ×_X Y ≅ Y for any map Y → X (where the implied map X → X is the identity).
• In an abelian category all pullbacks exist, and they preserve kernels, in the following sense: if

File:Categorical pullback.svg

is a pullback diagram, then the induced morphism ker(p₂) → ker(f) is an isomorphism, and so is the induced morphism ker(p₁) → ker(g). Every pullback diagram thus gives rise to a commutative diagram of the following form, where all rows and columns are exact:

$\begin{array}{ccccccc}& & & & 0& & 0\\ & & & & \downarrow & & \downarrow \\ & & & & L& =& L\\ & & & & \downarrow & & \downarrow \\ 0& \to & K& \to & P& \to & Y\\ & & \parallel & & \downarrow & & \downarrow \\ 0& \to & K& \to & X& \to & Z\end{array}$

Furthermore, in an abelian category, if X → Z is an epimorphism, then so is its pullback P → Y, and symmetrically: if Y → Z is an epimorphism, then so is its pullback P → X. In these situations, the pullback square is also a pushout square.

• There is a natural isomorphism (A×_CB)×_B D ≅ A×_CD. Explicitly, this means:
  • if maps f : A → C, g : B → C and h : D → B are given and
  • the pullback of f and g is given by r : P → A and s : P → B, and
  • the pullback of s and h is given by t : Q → P and u : Q → D ,
  • then the pullback of f and gh is given by rt : Q → A and u : Q → D.

Graphically this means that two pullback squares, placed side by side and sharing one morphism, form a larger pullback square when ignoring the inner shared morphism.

$\begin{array}{ccccc}Q& \stackrel{t}{\to }& P& \stackrel{r}{\to }& A\\ {\downarrow }_u& & {\downarrow }_s& & {\downarrow }_f\\ D& \stackrel{h}{\to }& B& \stackrel{g}{\to }& C\end{array}$

• Any category with pullbacks and products has equalizers.

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 is not required to be unique.

See also

• Pullbacks in differential geometry
• Equijoin in relational algebra
• Fiber product of schemes

References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories (4.2MB PDF). Originally publ. John Wiley & Sons. Template:Isbn. (now free on-line edition).
• Cohn, Paul M.; Universal Algebra (1981), D. Reidel Publishing, Holland, Template:Isbn (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.
• Template:Nlab