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. The pullback is written

P = X \times f, Z, g Y

.

Usually the morphisms $f$ and $g$ are omitted from the notation, and then the pullback is written

P = X \times Z Y

.

The pullback comes equipped with two natural morphisms $P \to X$ and $P \to Y$ . 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, $X \times Z Y$ 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.

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

commutes. Moreover, the pullback $(P, p 1, p 2)$ must be universal with respect to this diagram.^[1] That is, for any other such triple $(Q, q 1, q 2)$ where $q 1 : Q \to X$ and $q 2 : Q \to Y$ are morphisms with $f q 1 = g q 2$ , there must exist a unique $u : Q \to P$ such that

p_{1}\circ u=q_{1},\qquad p_{2}\circ u=q_{2}.

This situation is illustrated in the following commutative diagram.

As with all universal constructions, a 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 $X \to Z \leftarrow Y$ , there is a unique isomorphism between $A$ and $B$ respecting the pullback structure.

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 $Y$ .

Commutative rings

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 $α : A \to C$ and $β : B \to C$ (identity preserving) ring homomorphisms. Then the pullback of this diagram exists and is given by the subring of the product ring $A \times B$ 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

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

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

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 $f : X \to Z$ and $g : Y \to Z$ 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 $π 1$ 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 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 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).

Graphs of functions

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

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 (formed in the category of topological spaces with continuous maps) $X \times B E$ 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 $E 1, E 2 \to B$ . In this case $E 1 \times E 2$ is a fiber bundle over $B \times B$ , and pulling back along the diagonal map $B \to B \times B$ gives a space homeomorphic (diffeomorphic) to $E 1 \times B E 2$ , 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 and intersections

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

.

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

In any category with a terminal object $T$ , the pullback $X \times T Y$ is just the ordinary product $X \times Y$ .^[2]
Monomorphisms are stable under pullback: if the arrow $f$ in the diagram is monic, then so is the arrow $p 2$ . Similarly, if $g$ is monic, then so is $p 1$ .^[3]
Isomorphisms are also stable, and hence, for example, $X \times X Y ≅ Y$ for any map $Y \to X$ (where the implied map $X \to X$ is the identity).
In an abelian category all pullbacks exist,^[4] and they preserve kernels, in the following sense: if

is a pullback diagram, then the induced morphism

ker(p 2) \to ker(f)

is an isomorphism,^[5] and so is the induced morphism

ker(p 1) \to 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&\rightarrow &K&\rightarrow &P&\rightarrow &Y\\&&\parallel &&\downarrow &&\downarrow \\0&\rightarrow &K&\rightarrow &X&\rightarrow &Z\end{array}}

Furthermore, in an abelian category, if

X \to Z

is an epimorphism, then so is its pullback

P \to Y

, and symmetrically: if

Y \to Z

is an epimorphism, then so is its pullback

P \to X

.^[6] In these situations, the pullback square is also a pushout square.^[7]

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&{\xrightarrow {t}}&P&{\xrightarrow {r}}&A\\\downarrow _{u}&&\downarrow _{s}&&\downarrow _{f}\\D&{\xrightarrow {h}}&B&{\xrightarrow {g}}&C\end{array}}

Any category with pullbacks and products has equalizers.

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.

[1]
Mitchell, p. 9
[2]
Adámek, p. 197.
[3]
Mitchell, p. 9
[4]
Mitchell, p. 32
[5]
Mitchell, p. 15
[6]
Mitchell, p. 34
[7]
Mitchell, p. 39

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).
Mitchell, Barry (1965). Theory of Categories. Academic Press.

Interactive web page which generates examples of pullbacks in the category of finite sets. Written by Jocelyn Paine.
pullback at the nLab

[1] [1]
Mitchell, p. 9

[2] [2]
Adámek, p. 197.

[3] [3]
Mitchell, p. 9

[4] [4]
Mitchell, p. 32

[5] [5]
Mitchell, p. 15

[6] [6]
Mitchell, p. 34

[7] [7]
Mitchell, p. 39

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