Pullbacks

Summary

The pullback (fibred product) of $f:Y\to Z$ and $g:X\to Z$ is the universal object $P$ with maps to $X$ and $Y$ such that the two composites to $Z$ agree. Pullbacks are limits of the diagram shape $\bullet \to \bullet \leftarrow \bullet$. Monomorphisms are characterised by pullback squares.

Overview

The pullback is the categorical version of the set-theoretic construction $\{(x,y):f(x)=g(y)\}$. It arises naturally in many contexts: intersection of subobjects, base change in algebraic geometry, cartesian squares in topology. Dually, the pushout is the colimit version.

Main Content

Definition 5.1.16: Pullback (Fibred Product)

A pullback of $f:Y\to Z$ and $g:X\to Z$ is an object $P$ with maps $p_1:P\to X$ and $p_2:P\to Y$ such that $f\circ p_1=g\circ p_2$ (the square commutes), and universal with this property: for any $Q$ with $q_1:Q\to X$ and $q_2:Q\to Y$ with $f{q}_1=g{q}_2$, there is a unique $u:Q\to P$ with $p_1\circ u=q_1$ and $p_2\circ u=q_2$.

$$\begin{array}{ccc}P& \stackrel{p_2}{\to }& Y\\ p_1\downarrow & & \downarrow f\\ X& \stackrel{g}{\to }& Z\end{array}$$

We write $P=X{\times }_{Z}Y$ (the fibred product of $X$ and $Y$ over $Z$).

A diagram of this shape is a pullback square or cartesian square.

Examples:

• In Set: $X{\times }_{Z}Y=\{(x,y)\in X\times Y:g(x)=f(y)\}$.
• In Top: same as Set, with subspace topology from $X\times Y$.
• In Grp: pullback of $f:H\to G\leftarrow K:g$ is $\{(h,k)\in H\times K:f(h)=g(k)\}$.
• In a preorder: pullback of $a\le c$ and $b\le c$ is $a\times b=\inf (a,b)$ (if exists).
• In Manifolds: pullback of bundles; in schemes: fibre product.

Example: Pullback in Set (BCT, Ch. 5.1)

For $f:\{a,b\}\to \{0,1\}$ with $f(a)=0,f(b)=1$ and $g:\{c,d\}\to \{0,1\}$ with $g(c)=0,g(d)=0$, the pullback is $\{(a,c),(a,d)\}$ — all pairs mapping to the same element.

Pullbacks and Monomorphisms

Lemma 5.1.32: Monos and Pullback Squares (BCT, Ch. 5.1)

A map $f:X\to Y$ is a monomorphism if and only if the following is a pullback square:

$$\begin{array}{ccc}X& \stackrel{1_X}{\to }& X\\ 1_X\downarrow & & \downarrow f\\ X& \stackrel{f}{\to }& Y\end{array}$$

i.e., if and only if the diagonal $X\to X{\times }_{Y}X$ is an isomorphism.

This characterises monomorphisms in purely categorical, limit-theoretic terms, without reference to elements.

Pullbacks as Limits

The pullback is the limit of the diagram $X\stackrel{g}{\to }Z\stackrel{f}{\leftarrow }Y$ (shape $\bullet \to \bullet \leftarrow \bullet$, called the cospan). All three objects $X,Y,Z$ and the two maps $f,g$ are part of the diagram; the pullback is the universal cone over it.

Pushouts (Dual Construction)

Definition: Pushout

Dually, the pushout of $f:Z\to Y$ and $g:Z\to X$ is an object $Q$ with maps $i_1:X\to Q$ and $i_2:Y\to Q$ such that $i_1\circ g=i_2\circ f$, universal with this property.

$$\begin{array}{ccc}Z& \stackrel{f}{\to }& Y\\ g\downarrow & & \downarrow i_2\\ X& \stackrel{i_1}{\to }& Q\end{array}$$

In Set: $Q=(X\bigsqcup Y)/\sim$ where $g(z)\sim f(z)$ for all $z\in Z$.

Examples:

• In Grp: pushout = amalgamated free product.
• In Top: pushout = adjunction space (gluing).
• In Ab: pushout = cofibre/mapping cone construction.

Pasting Lemma for Pullbacks

If you have a commutative grid of squares, the “pasting lemma” says: if the right square is a pullback, then the left square is a pullback if and only if the outer rectangle is a pullback.

Connections

• General limits (General Limits) subsume pullbacks under the uniform definition.
• Pushouts are Colimits — they are the dual notion.
• Monomorphisms (Products and Equalizers) are characterised by pullback squares.
• In sheaf theory and algebraic geometry, base change = pullback of geometric objects.

See Also

• Products and Equalizers — Other key special limits
• General Limits — The general definition
• Colimits — Pushouts and other colimits