Item 4 (00AP) — The Clowder Project

6.2.4 Pullbacks

Let $(X,x_{0})$, $(Y,y_{0})$, and $(Z,z_{0})$ be pointed sets and let $f\colon (X,x_{0})\to (Z,z_{0})$ and $g\colon (Y,y_{0})\to (Z,z_{0})$ be morphisms of pointed sets.

Definition 6.2.4.1.1 ▶ Pullbacks of Pointed Sets

The pullback of $(X,x_{0})$ and $(Y,y_{0})$ over $(Z,z_{0})$ along $(f,g)$ is the pullback of $(X,x_{0})$ and $(Y,y_{0})$ over $(Z,z_{0})$ along $(f,g)$ in $\mathsf{Sets}_{*}$ as in , .

Construction 6.2.4.1.2 ▶ Construction of Pullbacks of Pointed Sets

Concretely, the pullback of $(X,x_{0})$ and $(Y,y_{0})$ over $(Z,z_{0})$ along $(f,g)$ is the pair consisting of:

•
The Limit. The pointed set $(X\times _{Z}Y,(x_{0},y_{0}))$.
•
The Cone. The morphisms of pointed sets

\begin{align*} \operatorname {\mathrm{\mathrm{pr}}}_{1} & \colon (X\times _{Z}Y,(x_{0},y_{0}))\to (X,x_{0}),\\ \operatorname {\mathrm{\mathrm{pr}}}_{2} & \colon (X\times _{Z}Y,(x_{0},y_{0}))\to (Y,y_{0}) \end{align*}

defined by

\begin{align*} \operatorname {\mathrm{\mathrm{pr}}}_{1}(x,y) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x,\\ \operatorname {\mathrm{\mathrm{pr}}}_{2}(x,y) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}y \end{align*}

for each $(x,y)\in X\times _{Z}Y$.

Proof of Construction 6.2.4.1.2.

We claim that $X\times _{Z}Y$ is the categorical pullback of $(X,x_{0})$ and $(Y,y_{0})$ over $(Z,z_{0})$ with respect to $(f,g)$ in $\mathsf{Sets}_{*}$. First we need to check that the relevant pullback diagram commutes, i.e. that we have

Indeed, given $(x,y)\in X\times _{Z}Y$, we have

\begin{align*} [f\circ \operatorname {\mathrm{\mathrm{pr}}}_{1}](x,y) & = f(\operatorname {\mathrm{\mathrm{pr}}}_{1}(x,y))\\ & = f(x)\\ & = g(y)\\ & = g(\operatorname {\mathrm{\mathrm{pr}}}_{2}(x,y))\\ & = [g\circ \operatorname {\mathrm{\mathrm{pr}}}_{2}](x,y),\end{align*}

where $f(x)=g(y)$ since $(x,y)\in X\times _{Z}Y$. Next, we prove that $X\times _{Z}Y$ satisfies the universal property of the pullback. Suppose we have a diagram of the form

in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets

\[ \phi \colon (P,*)\to (X\times _{Z}Y,(x_{0},y_{0})) \]

making the diagram

commute, being uniquely determined by the conditions

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

via

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

for each $x\in P$, where we note that $(p_{1}(x),p_{2}(x))\in X\times Y$ indeed lies in $X\times _{Z}Y$ by the condition

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

which gives

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

for each $x\in P$, so that $(p_{1}(x),p_{2}(x))\in X\times _{Z}Y$. Lastly, we note that $\phi $ is indeed a morphism of pointed sets, as we have

\begin{align*} \phi (*) & = (p_{1}(*),p_{2}(*))\\ & = (x_{0},y_{0}),\end{align*}

where we have used that $p_{1}$ and $p_{2}$ are morphisms of pointed sets.

Proposition 6.2.4.1.3 ▶ Properties of Pullbacks of Pointed Sets

Let $(X,x_{0})$, $(Y,y_{0})$, $(Z,z_{0})$, and $(A,a_{0})$ be pointed sets.

1.
Functoriality. The assignment $(X,Y,Z,f,g)\mapsto X\times _{f,Z,g}Y$ defines a functor

\[ -_{1}\times _{-_{3}}-_{1}\colon \mathsf{Fun}(\mathcal{P},\mathsf{Sets}_{*})\to \mathsf{Sets}_{*}, \]

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 $\mathsf{Fun}(\mathcal{P},\mathsf{Sets}_{*})$ to the morphism of pointed sets

\[ \xi \colon (X\times _{Z}Y,(x_{0},y_{0}))\overset {\exists !}{\to }(X'\times _{Z'}Y',(x'_{0},y'_{0})) \]

given by

\[ \xi (x,y)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(\phi (x),\psi (y)) \]

for each $(x,y)\in X\times _{Z}Y$, which is the unique morphism of pointed sets making the diagram
commute.
2.
Associativity. Given a diagram
in $\mathsf{Sets}_{*}$, we have isomorphisms of pointed sets

\[ (X\times _{W}Y)\times _{V}Z\cong (X\times _{W}Y)\times _{Y}(Y\times _{V}Z) \cong X\times _{W}(Y\times _{V}Z), \]

where these pullbacks are built as in the diagrams
3.
Unitality. We have isomorphisms of pointed sets