Item 1 (00BH) — The Clowder Project

6.3.4 Pushouts

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

Definition 6.3.4.1.1 ▶ Pushouts of Pointed Sets

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

Construction 6.3.4.1.2 ▶ Construction of Pushouts of Pointed Sets

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

•
The Colimit. The pointed set $(X\coprod _{f,Z,g}Y,p_{0})$, where:
- •
  The set $X\coprod _{f,Z,g}Y$ is the pushout (of unpointed sets) of $X$ and $Y$ over $Z$ with respect to $f$ and $g$;
- •
  We have $p_{0}=[x_{0}]=[y_{0}]$.
•
The Cocone. The morphisms of pointed sets

\begin{align*} \mathrm{inj}_{1} & \colon (X,x_{0})\to (X\mathbin {\textstyle \coprod _{Z}}Y,p_{0}),\\ \mathrm{inj}_{2} & \colon (Y,y_{0})\to (X\mathbin {\textstyle \coprod _{Z}}Y,p_{0}) \end{align*}

given by

\begin{align*} \mathrm{inj}_{1}(x) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[(0,x)]\\ \mathrm{inj}_{2}(y) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[(1,y)]\end{align*}

for each $x\in X$ and each $y\in Y$.

Proof of .

Firstly, we note that indeed $[x_{0}]=[y_{0}]$, as we have

\begin{align*} x_{0} & = f(z_{0}),\\ y_{0} & = g(z_{0}) \end{align*}

since $f$ and $g$ are morphisms of pointed sets, with the relation $\mathord {\sim }$ on $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}_{Z}Y$ then identifying $x_{0}=f(z_{0})\sim g(z_{0})=y_{0}$.

We now claim that $(X\mathbin {\textstyle \coprod _{Z}}Y,p_{0})$ is the categorical pushout 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 pushout diagram commutes, i.e. that we have

Indeed, given $z\in Z$, we have

\begin{align*} [\mathrm{inj}_{1}\circ f](z) & = \mathrm{inj}_{1}(f(z))\\ & = [(0,f(z))]\\ & = [(1,g(z))]\\ & = \mathrm{inj}_{2}(g(z))\\ & = [\mathrm{inj}_{2}\circ g](z),\end{align*}

where $[(0,f(z))]=[(1,g(z))]$ by the definition of the relation $\mathord {\sim }$ on $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y$ (the coproduct of unpointed sets of $X$ and $Y$). Next, we prove that $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}_{Z}Y$ satisfies the universal property of the pushout. Suppose we have a diagram of the form

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

\[ \phi \colon (X\mathbin {\textstyle \coprod _{Z}}Y,p_{0})\to (P,*) \]

making the diagram

commute, being uniquely determined by the conditions

\begin{align*} \phi \circ \mathrm{inj}_{1} & = \iota _{1},\\ \phi \circ \mathrm{inj}_{2} & = \iota _{2}\end{align*}

via

\[ \phi (p)=\begin{cases} \iota _{1}(x) & \text{if $x=[(0,x)]$,}\\ \iota _{2}(y) & \text{if $x=[(1,y)]$} \end{cases} \]

for each $p\in X\mathbin {\textstyle \coprod _{Z}}Y$, where the well-definedness of $\phi $ is proven in the same way as in the proof of Chapter 4: Constructions With Sets, Definition 4.2.4.1.1. Finally, we show that $\phi $ is indeed a morphism of pointed sets, as we have

\begin{align*} \phi (p_{0}) & = \phi ([(0,x_{0})])\\ & = \iota _{1}(x_{0})\\ & = *, \end{align*}

or alternatively

\begin{align*} \phi (p_{0}) & = \phi ([(1,y_{0})])\\ & = \iota _{2}(y_{0})\\ & = *, \end{align*}

where we use that $\iota _{1}$ (resp. $\iota _{2}$) is a morphism of pointed sets.

Proposition 6.3.4.1.3 ▶ Properties of Pushouts of Pointed Sets

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

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

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

where $\mathcal{P}$ is the category that looks like this:
In particular, the action on morphisms of $-_{1}\mathbin {\textstyle \coprod _{-_{3}}}-_{1}$ is given by sending a morphism
in $\mathsf{Fun}(\mathcal{P},\mathsf{Sets}_{*})$ to the morphism of pointed sets

\[ \xi \colon (X\mathbin {\textstyle \coprod _{Z}}Y,p_{0})\overset {\exists !}{\to }(X'\mathbin {\textstyle \coprod _{Z'}}Y',p'_{0}) \]

given by

\[ \xi (p)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \phi (x) & \text{if $p=[(0,x)]$},\\ \psi (y) & \text{if $p=[(1,y)]$} \end{cases} \]

for each $p\in X\mathbin {\textstyle \coprod _{Z}}Y$, 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

\[ (X\mathbin {\textstyle \coprod _{W}}Y)\mathbin {\textstyle \coprod _{V}}Z\cong (X\mathbin {\textstyle \coprod _{W}}Y)\mathbin {\textstyle \coprod _{Y}}(Y\mathbin {\textstyle \coprod _{V}}Z) \cong X\mathbin {\textstyle \coprod _{W}}(Y\mathbin {\textstyle \coprod _{V}}Z), \]

where these pullbacks are built as in the diagrams