nLab
pullback

Idea

In the category Set a ‘pullback’ is a subset of the cartesian product of two sets. Given a diagram of sets and functions like this:

$$\begin{array}{ccccccc}& & A& & & & B\\ & & & {}_f\searrow & & {\swarrow }_g& & \\ & & & & C& & & & \end{array}$$\array{ && A &&&& B \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& C &&&& }

the ‘pullback’ of this diagram is the subset $X\subseteq A\times B$ consisting of pairs $(a,b)$ such that the equation $f(a)=g(b)$ hold.

A pullback is therefore the categorical semantics of an equation.

This construction comes up, for example, when $A$ and $B$ are fiber bundles over $C$: then $X$ as defined above is the product of $A$ and $B$ in the category of fiber bundles over $C$. For this reason, a pullback is sometimes called a fibered product (or fiber product or fibre product).

In this case, the fiber of $A{\times }_{C}B$ over a (generalized) element $x$ of $C$ is the ordinary product of the fibers of $A$ and $B$ over $x$. In other words, the fiber product is the product taken fiber-wise. Of course, the fiber of $A$ at the generalized element $x:I\to C$ is itself a fibre product $I{\times }_{C}A$; the terminology depends on your point of view.

Note that there are maps $p_A:X\to A$, $p_B:X\to B$ sending any $(a,b)\in X$ to $a$ and $b$, respectively. These maps make this square commute:

$$\begin{array}{ccccc}& & & & X\\ & & & {}^{p_A}\swarrow & & {\searrow }^{p_B}\\ & & A& & & & B\\ & & & {}_f\searrow & & {\swarrow }_g& & \\ & & & & C& & & & \end{array}$$\array{ &&&& X \\& && {}^{p_A}\swarrow && \searrow^{p_B} \\ && A &&&& B \\ & && {}_f\searrow & & \swarrow_{g} && \\ &&&& C &&&& }

In fact, the pullback is the universal solution to finding a commutative square like this. In other words, given any commutative square

$$\begin{array}{ccccc}& & & & Y\\ & & & {}^{q_A}\swarrow & & {\searrow }^{q_B}\\ & & A& & & & B\\ & & & {}_f\searrow & & {\swarrow }_g& & \\ & & & & C& & & & \end{array}$$\array{ &&&& Y \\& && {}^{q_A}\swarrow && \searrow^{q_B} \\ && A &&&& B \\ & && {}_f\searrow & & \swarrow_{g} && \\ &&&& C &&&& }

there is a unique function $h:Y\to X$ such that

$$p_{A}h=q_A$$p_A h = q_A

and

$$p_{B}h=q_B.$$p_B h = q_B\,.

Since this universal property expresses the concept of pullback purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a limit.

Definition

In category theory

A pullback is a limit of a diagram like this:

$$\begin{array}{ccccccc}& & a& & & & b\\ & & & {}_f\searrow & & {\swarrow }_g& & \\ & & & & c& & & & \end{array}$$\array{ && a &&&& b \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& c &&&& }

Such a diagram is also called a pullback diagram* or a cospan. If the limit exists, we obtain a commutative square

$$\begin{array}{ccccc}& & & & x\\ & & & {}^{p_a}\swarrow & & {\searrow }^{p_b}\\ & & a& & & & b\\ & & & {}_f\searrow & & {\swarrow }_g& & \\ & & & & c& & & & \end{array}$$\array{ &&&& x \\& && {}^{p_a}\swarrow && \searrow^{p_b} \\ && a &&&& b \\ & && {}_f\searrow & & \swarrow_{g} && \\ &&&& c &&&& }

and the object $x$ is also called the pullback. It is well defined up to unique isomorphism. It has the universal property already described above in the special case of the category $\mathrm{Set}$.

The last commutative square above is called a pullback square.

The concept of pullback is dual to the concept of pushout: that is, a pullback in $C$ is the same as a pushout in the opposite category $C^{\mathrm{op}}$.

In type theory

In type theory a pullback $P$ in

$$\begin{array}{ccc}P& \to & A\\ \downarrow & & {\downarrow }^f\\ B& \stackrel{g}{\to }& C\end{array}$$\array{ P &\to& A \\ \downarrow && \downarrow^{\mathrlap{f}} \\ B &\stackrel{g}{\to}& C }

is given by the dependent sum over the dependent equality type

$$P=\sum _{a:A}\sum _{b:B}(f(a)=g(b)).$$P = \sum_{a : A} \sum_{b : B} (f(a) = g(b)) \,.

Properties

As an equalizer

If products exist in $C$, then the pullback

$$\begin{array}{ccc}a{\times }_{c}b& \to & a\\ \downarrow & & {\downarrow }^f\\ b& \stackrel{g}{\to }& c\end{array}$$\array{ a \times_c b &\to& a \\ \downarrow && \downarrow^{\mathrlap{f}} \\ b &\stackrel{g}{\to}& c }

is equivalently the equalizer

$$a{\times }_{c}b\to a\times b\stackrel{\stackrel{f{p}_1}{\to }}{\underset{g{p}_2}{\to }}c$$a \times_c b \to a \times b \stackrel{\overset{f p_1}{\to}}{\underset{g p_2}{\to}} c

of the two morphisms induced by $f$ and $g$ out of the product of $a$ with $b$.

Pasting of pullbacks

Consider a pasting diagram of the form

$$\begin{array}{ccccc}a& \to & b& \to & c\\ \downarrow & & \downarrow & & \downarrow \\ d& \to & e& \to & f\end{array}.$$\array{ a &\to& b &\to& c \\ \downarrow && \downarrow && \downarrow \\ d &\to& e &\to& f } \,.

There are three commuting squares: the two inner ones and the outer one.

Proposition

Suppose the right-hand inner square is a pullback: then the left-hand one is a pullback if and only if the outer square is.

Proof. Pasting a morphism $x\to a$ with the outer square gives rise to a commuting square over the (composite) bottom and right edges of the diagram. The square over the cospan in the left-hand inner square arising from $x\to a$ includes a morphism into $b$, which if $b$ is a pullback induces the same commuting square over $d\to e\to f$ and $c\to d$. So one square is universal iff the other is.

Proposition

The converse implication does not hold: it may happen that the outer and the left square are pullbacks, but not the right square.

For instance let $i:a\to b$ be a split monomorphism with retract $p:b\to a$ and consider

$$\begin{array}{ccccc}a& \stackrel{=}{\to }& a& \stackrel{=}{\to }& a\\ {\downarrow }^{=}& & {\downarrow }^i& & {\downarrow }^{=}\\ a& \stackrel{i}{\to }& b& \stackrel{p}{\to }& a\end{array}$$\array{ a & \stackrel{=}{\to} & a & \stackrel{=}{\to} & a \\ \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{=}} \\ a &\stackrel{i}{\to}& b &\stackrel{p}{\to}& a }

Then the left square and the outer rectangle are pullbacks but the right square cannot be a pullback unless $i$ was already an isomorphism.

Saturation

The saturation of the class of pullbacks is the class of limits over categories $C$ whose groupoid reflection ${\Pi }_1(C)$ is trivial and such that $C$ is L-finite.

Related concepts

• fiber product, base change

• wide pullback

• lax pullback, comma object

• (∞,1)-pullback, homotopy pullback

References

• Robert Paré, Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, CMS