Subsection 2.1.4 (0018): Pullbacks

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

2.1.4 Pullbacks

Let $A$, $B$, and $C$ be sets and let $f\colon A\to C$ and $g\colon B\to C$ be functions.

Definition 2.1.4.1.1 ▶ Pullbacks of Sets

The pullback of $A$ and $B$ over $C$ along $f$ and $g$^[1] is the pair ^[2] $\webleft (A\times _{C}B,\webleft\{ \text{pr}_{1},\text{pr}_{2}\webright\} \webright )$ consisting of:

The Limit. The set $A\times _{C}B$ defined by

\[ A\times _{C}B\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (a,b\webright )\in A\times B\ \middle |\ f\webleft (a\webright )=g\webleft (b\webright )\webright\} . \]
The Cone. The maps

\begin{align*} \text{pr}_{1} & \colon A\times _{C}B\to A,\\ \text{pr}_{2} & \colon A\times _{C}B\to B \end{align*}

defined by

\begin{align*} \text{pr}_{1}\webleft (a,b\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}a,\\ \text{pr}_{2}\webleft (a,b\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}b \end{align*}

for each $\webleft (a,b\webright )\in A\times _{C}B$.

Proof of Definition 2.1.4.1.1.

We claim that $A\times _{C}B$ is the categorical pullback of $A$ and $B$ over $C$ with respect to $\webleft (f,g\webright )$ in $\mathsf{Sets}$. First we need to check that the relevant pullback diagram commutes, i.e. that we have

Indeed, given $\webleft (a,b\webright )\in A\times _{C}B$, we have

\begin{align*} \webleft [f\circ \text{pr}_{1}\webright ]\webleft (a,b\webright ) & = f\webleft (\text{pr}_{1}\webleft (a,b\webright )\webright )\\ & = f\webleft (a\webright )\\ & = g\webleft (b\webright )\\ & = g\webleft (\text{pr}_{2}\webleft (a,b\webright )\webright )\\ & = \webleft [g\circ \text{pr}_{2}\webright ]\webleft (a,b\webright ),\end{align*}

where $f\webleft (a\webright )=g\webleft (b\webright )$ since $\webleft (a,b\webright )\in A\times _{C}B$. Next, we prove that $A\times _{C}B$ satisfies the universal property of the pullback. Suppose we have a diagram of the form

in $\mathsf{Sets}$. Then there exists a unique map $\phi \colon P\to A\times _{C}B$ making the diagram

commute, being uniquely determined by the conditions

\begin{align*} \text{pr}_{1}\circ \phi & = p_{1},\\ \text{pr}_{2}\circ \phi & = p_{2}\end{align*}

via

\[ \phi \webleft (x\webright )=\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright ) \]

for each $x\in P$, where we note that $\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright )\in A\times B$ indeed lies in $A\times _{C}B$ by the condition

\[ f\circ p_{1}=g\circ p_{2}, \]

which gives

\[ f\webleft (p_{1}\webleft (x\webright )\webright )=g\webleft (p_{2}\webleft (x\webright )\webright ) \]

for each $x\in P$, so that $\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright )\in A\times _{C}B$.

Example 2.1.4.1.2 ▶ Examples of Pullbacks of Sets

Here are some examples of pullbacks of sets.

Unions via Intersections. Let $A,B\subset X$. We have a bijection of sets

Proof of Example 2.1.4.1.2.

Item 1: Unions via Intersections

Indeed, we have

\begin{align*} A\times _{A\cup B}B & \cong \webleft\{ \webleft (x,y\webright )\in A\times B\ \middle |\ x=y\webright\} \\ & \cong A\cap B. \end{align*}

This finishes the proof.

Proposition 2.1.4.1.3 ▶ Properties of Pullbacks of Sets

Let $A$, $B$, $C$, and $X$ be sets.

Functoriality. The assignment $\webleft (A,B,C,f,g\webright )\mapsto A\times _{f,C,g}B$ defines a functor
\[ -_{1}\times _{-_{3}}-_{1}\colon \mathsf{Fun}\webleft (\mathcal{P},\mathsf{Sets}\webright )\to \mathsf{Sets}, \]

where $\mathcal{P}$ is the category that looks like this:

In particular, the action on morphisms of $-_{1}\times _{-_{3}}-_{1}$ is given by sending a morphism

in $\mathsf{Fun}\webleft (\mathcal{P},\mathsf{Sets}\webright )$ to the map $\xi \colon A\times _{C}B\overset {\exists !}{\to }A'\times _{C'}B'$ given by

\[ \xi \webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\phi \webleft (a\webright ),\psi \webleft (b\webright )\webright ) \]

for each $\webleft (a,b\webright )\in A\times _{C}B$, which is the unique map making the diagram

commute.
Associativity. Given a diagram

in $\mathsf{Sets}$, we have isomorphisms of sets

\[ \webleft (A\times _{X}B\webright )\times _{Y}C\cong \webleft (A\times _{X}B\webright )\times _{B}\webleft (B\times _{Y}C\webright ) \cong A\times _{X}\webleft (B\times _{Y}C\webright ), \]

where these pullbacks are built as in the diagrams
Unitality. We have isomorphisms of sets
Commutativity. We have an isomorphism of sets
Annihilation With the Empty Set. We have isomorphisms of sets
Interaction With Products. We have an isomorphism of sets
Symmetric Monoidality. The triple $\webleft (\mathsf{Sets},\times _{X},X\webright )$ is a symmetric monoidal category.

Proof of Proposition 2.1.4.1.3.

Item 1: Functoriality

This is a special case of functoriality of co/limits,

, with the explicit expression for $\xi $ following from the commutativity of the cube pullback diagram.

Item 2: Associativity

Indeed, we have

\begin{align*} \webleft (A\times _{X}B\webright )\times _{Y}C & \cong \webleft\{ \webleft (\webleft (a,b\webright ),c\webright )\in \webleft (A\times _{X}B\webright )\times C\ \middle |\ h\webleft (b\webright )=k\webleft (c\webright )\webright\} \\ & \cong \webleft\{ \webleft (\webleft (a,b\webright ),c\webright )\in \webleft (A\times B\webright )\times C\ \middle |\ \text{$f\webleft (a\webright )=g\webleft (b\webright )$ and $h\webleft (b\webright )=k\webleft (c\webright )$}\webright\} \\ & \cong \webleft\{ \webleft (a,\webleft (b,c\webright )\webright )\in A\times \webleft (B\times C\webright )\ \middle |\ \text{$f\webleft (a\webright )=g\webleft (b\webright )$ and $h\webleft (b\webright )=k\webleft (c\webright )$}\webright\} \\ & \cong \webleft\{ \webleft (a,\webleft (b,c\webright )\webright )\in A\times \webleft (B\times _{Y}C\webright )\ \middle |\ \text{$f\webleft (a\webright )=g\webleft (b\webright )$}\webright\} \\ & \cong A\times _{X}\webleft (B\times _{Y}C\webright ) \end{align*}

and

\begin{align*} \webleft (A\times _{X}B\webright )\times _{B}\webleft (B\times _{Y}C\webright ) & \cong \webleft\{ \webleft (\webleft (a,b\webright ),\webleft (b',c\webright )\webright )\in \webleft (A\times _{X}B\webright )\times \webleft (B\times _{Y}C\webright )\ \middle |\ b=b'\webright\} \\ & \cong \webleft\{ \webleft (\webleft (a,b\webright ),\webleft (b',c\webright )\webright )\in \webleft (A\times B\webright )\times \webleft (B\times C\webright )\ \middle |\ \begin{aligned} & \text{$f\webleft (a\webright )=g\webleft (b\webright )$, $b=b'$,}\\ & \text{and $h\webleft (b'\webright )=k\webleft (c\webright )$}\end{aligned}\webright\} \\ & \cong \webleft\{ \webleft (a,\webleft (b,\webleft (b',c\webright )\webright )\webright )\in A\times \webleft (B\times \webleft (B\times C\webright )\webright )\ \middle |\ \begin{aligned} & \text{$f\webleft (a\webright )=g\webleft (b\webright )$, $b=b'$,}\\ & \text{and $h\webleft (b'\webright )=k\webleft (c\webright )$}\end{aligned}\webright\} \\ & \cong \webleft\{ \webleft (a,\webleft (\webleft (b,b'\webright ),c\webright )\webright )\in A\times \webleft (\webleft (B\times B\webright )\times C\webright )\ \middle |\ \begin{aligned} & \text{$f\webleft (a\webright )=g\webleft (b\webright )$, $b=b'$,}\\ & \text{and $h\webleft (b'\webright )=k\webleft (c\webright )$}\end{aligned}\webright\} \\ & \cong \webleft\{ \webleft (a,\webleft (\webleft (b,b'\webright ),c\webright )\webright )\in A\times \webleft (\webleft (B\times _{B}B\webright )\times C\webright )\ \middle |\ \begin{aligned} & \text{$f\webleft (a\webright )=g\webleft (b\webright )$ and}\\ & \text{$h\webleft (b'\webright )=k\webleft (c\webright )$}\end{aligned}\webright\} \\ & \cong \webleft\{ \webleft (a,\webleft (b,c\webright )\webright )\in A\times \webleft (B\times C\webright )\ \middle |\ \text{$f\webleft (a\webright )=g\webleft (b\webright )$ and $h\webleft (b\webright )=k\webleft (c\webright )$}\webright\} \\ & \cong A\times _{X}\webleft (B\times _{Y}C\webright ), \end{align*}

where we have used Item 3 for the isomorphism $B\times _{B}B\cong B$.

Item 3: Unitality

Indeed, we have

\begin{align*} X\times _{X}A & \cong \webleft\{ \webleft (x,a\webright )\in X\times A\ \middle |\ f\webleft (a\webright )=x\webright\} ,\\ A\times _{X}X & \cong \webleft\{ \webleft (a,x\webright )\in X\times A\ \middle |\ f\webleft (a\webright )=x\webright\} , \end{align*}

which are isomorphic to $A$ via the maps $\webleft (x,a\webright )\mapsto a$ and $\webleft (a,x\webright )\mapsto a$.

Item 4: Commutativity

Clear.

Item 5: Annihilation With the Empty Set

Clear.

Item 6: Interaction With Products

Clear.

Item 7: Symmetric Monoidality

Omitted.