Subsection 4.2.4 (00AH): Pullbacks

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4.2.4 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 4.2.4.1.1 ▶ 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 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*} \text{pr}_{1} & \colon \webleft (X\times _{Z}Y,\webleft (x_{0},y_{0}\webright )\webright )\to \webleft (X,x_{0}\webright ),\\ \text{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*} \text{pr}_{1}\webleft (x,y\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x,\\ \text{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 Definition 4.2.4.1.1.

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 \text{pr}_{1}\webright ]\webleft (x,y\webright ) & = f\webleft (\text{pr}_{1}\webleft (x,y\webright )\webright )\\ & = f\webleft (x\webright )\\ & = g\webleft (y\webright )\\ & = g\webleft (\text{pr}_{2}\webleft (x,y\webright )\webright )\\ & = \webleft [g\circ \text{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*} \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 $\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright )\in X\times Y$ 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 4.2.4.1.2 ▶ 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.

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.
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
Unitality. We have isomorphisms of pointed sets
Commutativity. We have an isomorphism of pointed sets
Interaction With Products. We have an isomorphism of pointed sets
Symmetric Monoidality. The triple $\webleft (\mathsf{Sets}_{*},\times _{X},X\webright )$ is a symmetric monoidal category.