Pullback and Pushout

Pullbacks and pushouts are limits and colimits over diagrams of shape $\bullet \to \bullet \leftarrow \bullet$ and $\bullet \leftarrow \bullet \to \bullet$ , respectively. They generalize fiber products, intersections, and amalgamated sums, and play a central role in algebraic geometry (fiber products of schemes) and homological algebra (connecting exact sequences).

Pullback (Fiber Product)

Definition2.7Pullback

Given morphisms $f : A \to C$ and $g : B \to C$ in a category $\mathcal{C}$ , a pullback (or fiber product) is an object $A \times_C B$ together with morphisms $p_A : A \times_C B \to A$ and $p_B : A \times_C B \to B$ such that $f \circ p_A = g \circ p_B$ , and for every object $X$ with morphisms $\alpha : X \to A$ , $\beta : X \to B$ satisfying $f \circ \alpha = g \circ \beta$ , there exists a unique morphism $h : X \to A \times_C B$ with $p_A \circ h = \alpha$ and $p_B \circ h = \beta$ .

\begin{array}{ccc} A \times_C B & \xrightarrow{p_B} & B \\[6pt] \downarrow\scriptstyle{p_A} & & \downarrow\scriptstyle{g} \\[6pt] A & \xrightarrow{f} & C \end{array}

Pushout (Amalgamated Sum)

Definition2.8Pushout

Given morphisms $f : C \to A$ and $g : C \to B$ , a pushout (or amalgamated sum, or fibered coproduct) is an object $A \sqcup_C B$ with morphisms $\iota_A : A \to A \sqcup_C B$ and $\iota_B : B \to A \sqcup_C B$ such that $\iota_A \circ f = \iota_B \circ g$ , and satisfying the dual universal property.

\begin{array}{ccc} C & \xrightarrow{g} & B \\[6pt] \downarrow\scriptstyle{f} & & \downarrow\scriptstyle{\iota_B} \\[6pt] A & \xrightarrow{\iota_A} & A \sqcup_C B \end{array}

RemarkDuality

A pushout in $\mathcal{C}$ is a pullback in $\mathcal{C}^{\mathrm{op}}$ . Every result about pullbacks has a dual statement about pushouts.

Examples

ExamplePullback in Set

In $\mathbf{Set}$ , the pullback of $f : A \to C$ and $g : B \to C$ is

$A \times_C B = \{(a, b) \in A \times B : f(a) = g(b)\}$

with projections to $A$ and $B$ . When $f$ and $g$ are inclusion maps of subsets $A, B \subseteq C$ , the pullback is the intersection $A \cap B$ .

ExamplePullback as preimage

In $\mathbf{Set}$ , the pullback of $f : A \to B$ along the inclusion $\{b\} \hookrightarrow B$ is the fiber $f^{-1}(b)$ . More generally, pulling back along any map gives the "preimage" construction.

ExampleFiber product of schemes

In $\mathbf{Sch}$ , the pullback $X \times_S Y$ is the fiber product. For affine schemes, $\mathrm{Spec}\, A \times_{\mathrm{Spec}\, R} \mathrm{Spec}\, B = \mathrm{Spec}(A \otimes_R B)$ . This is the fundamental construction in algebraic geometry: base change, fibers, intersections, and self-products are all fiber products.

ExamplePullback in Grp

In $\mathbf{Grp}$ , the pullback of $f : G \to K$ and $g : H \to K$ is $G \times_K H = \{(g,h) \in G \times H : f(g) = g(h)\}$ with the induced group structure. When $f$ and $g$ are inclusion maps of subgroups, this recovers the intersection.

ExamplePullback in Top

In $\mathbf{Top}$ , the pullback of $f : X \to Z$ and $g : Y \to Z$ is the set $\{(x,y) : f(x) = g(y)\} \subseteq X \times Y$ with the subspace topology from the product topology on $X \times Y$ . The fiber $f^{-1}(z)$ is the pullback along the inclusion $\{z\} \hookrightarrow Z$ .

ExamplePushout in Set

In $\mathbf{Set}$ , the pushout of $f : C \to A$ and $g : C \to B$ is the quotient $(A \sqcup B) / {\sim}$ where $f(c) \sim g(c)$ for all $c \in C$ . This "glues" $A$ and $B$ along the common part specified by $C$ .

ExamplePushout in Ab

In $\mathbf{Ab}$ , the pushout of $f : C \to A$ and $g : C \to B$ is $(A \oplus B) / \{(f(c), -g(c)) : c \in C\}$ . This is the amalgamated sum or fiber sum.

ExamplePushout in Top: CW complexes

CW complexes are built by pushouts. Attaching an $n$ -cell to a space $X$ via a map $\varphi : S^{n-1} \to X$ is the pushout of $\varphi$ and the inclusion $S^{n-1} \hookrightarrow D^n$ :

$X \cup_\varphi D^n = X \sqcup_{S^{n-1}} D^n$

The entire CW structure is a sequence of pushouts.

ExamplePullback and kernel

In an abelian category, the kernel of $f : A \to B$ is the pullback of $f$ and the zero map $0 \to B$ :

$\ker f = A \times_B 0$

Dually, the cokernel is a pushout.

ExamplePullback of covering spaces

In $\mathbf{Top}$ , if $p : \tilde{X} \to X$ is a covering space and $f : Y \to X$ is any continuous map, the pullback $f^*\tilde{X} = Y \times_X \tilde{X}$ is a covering space of $Y$ . This is the "pullback" or "induced" covering space.

ExamplePushout in Ring

In $\mathbf{CRing}$ , the pushout of $R \to A$ and $R \to B$ is the tensor product $A \otimes_R B$ . This is because the pushout in $\mathbf{CRing}$ is the coproduct in the category of $R$ -algebras.

ExamplePullback preserves monomorphisms

If $g : B \to C$ is a monomorphism, then the pullback $p_A : A \times_C B \to A$ is also a monomorphism. This is because pullbacks "preserve" monomorphisms — a general property of limits.

Properties

Theorem2.3Pullback of monomorphisms

In any category, if the square

\begin{array}{ccc} P & \to & B \\ \downarrow & & \downarrow\scriptstyle{g} \\ A & \xrightarrow{f} & C \end{array}

is a pullback and $g$ is a monomorphism, then $P \to A$ is also a monomorphism.

Theorem2.4Pasting law for pullbacks

Given a commutative diagram

\begin{array}{ccccc} P & \to & Q & \to & R \\ \downarrow & & \downarrow & & \downarrow \\ A & \to & B & \to & C \end{array}

if the right square is a pullback, then the outer rectangle is a pullback if and only if the left square is a pullback.

RemarkLooking ahead

Pullbacks and pushouts are key to the construction of exact sequences in abelian categories and the Snake Lemma. In the theory of triangulated categories, the mapping cone triangle generalizes the pushout construction.