Item 2 (00AM) — The Clowder Project

6.2.4 Pullbacks

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

Definition 6.2.4.1.1 ▶ Pullbacks of Pointed Sets

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

Construction 6.2.4.1.2 ▶ Construction of Pullbacks of Pointed Sets

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

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

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

defined by

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

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

Proof of Construction 6.2.4.1.2.

We claim that $X\times _{Z}Y$ is the categorical pullback of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ over $\webleft (Z,z_{0}\webright )$ 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 $\webleft (x,y\webright )\in X\times _{Z}Y$, we have

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

where $f\webleft (x\webright )=g\webleft (y\webright )$ since $\webleft (x,y\webright )\in X\times _{Z}Y$. 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

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

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

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

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

\begin{align*} \phi \webleft (*\webright ) & = \webleft (p_{1}\webleft (*\webright ),p_{2}\webleft (*\webright )\webright )\\ & = \webleft (x_{0},y_{0}\webright ),\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 $\webleft (X,x_{0}\webright )$, $\webleft (Y,y_{0}\webright )$, $\webleft (Z,z_{0}\webright )$, and $\webleft (A,a_{0}\webright )$ be pointed sets.

1.
Functoriality. The assignment $\webleft (X,Y,Z,f,g\webright )\mapsto X\times _{f,Z,g}Y$ 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 morphism of pointed sets

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

given by

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

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

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

where these pullbacks are built as in the diagrams