Definition 5.1.4.1.1 (01NS): The Associator of $\times $

The associator of the product of sets is the natural isomorphism

\[ \alpha ^{\mathsf{Sets}} \colon \mathord {\times }\circ {\webleft (\mathord {\times }\times \operatorname {\mathrm{id}}_{\mathsf{Sets}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathord {\times }\circ {\webleft (\operatorname {\mathrm{id}}_{\mathsf{Sets}}\times \mathord {\times }\webright )}\circ {\mathbf{\alpha }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}}, \]

as in the diagram

whose component

\[ \alpha ^{\mathsf{Sets}}_{X,Y,Z} \colon \webleft (X\times Y\webright )\times Z \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X\times \webleft (Y\times Z\webright ) \]

at $\webleft (X,Y,Z\webright )$ is given by

\[ \alpha ^{\mathsf{Sets}}_{X,Y,Z}\webleft (\webleft (x,y\webright ),z\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (x,\webleft (y,z\webright )\webright ) \]

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

Invertibility

The inverse of $\alpha ^{\mathsf{Sets}}_{X,Y,Z}$ is the morphism

\[ \alpha ^{\mathsf{Sets},-1}_{X,Y,Z} \colon X\times \webleft (Y\times Z\webright ) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\times Y\webright )\times Z \]

defined by

\[ \alpha ^{\mathsf{Sets},-1}_{X,Y,Z}\webleft (x,\webleft (y,z\webright )\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\webleft (x,y\webright ),z\webright ) \]

for each $\webleft (x,\webleft (y,z\webright )\webright )\in X\times \webleft (Y\times Z\webright )$. Indeed:

•
Invertibility I. We have

\begin{align*} \webleft [\alpha ^{\mathsf{Sets},-1}_{X,Y,Z}\circ \alpha ^{\mathsf{Sets}}_{X,Y,Z}\webright ]\webleft (\webleft (x,y\webright ),z\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\alpha ^{\mathsf{Sets},-1}_{X,Y,Z}\webleft (\alpha ^{\mathsf{Sets}}_{X,Y,Z}\webleft (\webleft (x,y\webright ),z\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\alpha ^{\mathsf{Sets},-1}_{X,Y,Z}\webleft (x,\webleft (y,z\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\webleft (x,y\webright ),z\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\operatorname {\mathrm{id}}_{\webleft (X\times Y\webright )\times Z}\webright ]\webleft (\webleft (x,y\webright ),z\webright ) \end{align*}

for each $\webleft (\webleft (x,y\webright ),z\webright )\in \webleft (X\times Y\webright )\times Z$, and therefore we have

\[ \alpha ^{\mathsf{Sets},-1}_{X,Y,Z}\circ \alpha ^{\mathsf{Sets}}_{X,Y,Z}=\operatorname {\mathrm{id}}_{\webleft (X\times Y\webright )\times Z}. \]
•
Invertibility II. We have

\begin{align*} \webleft [\alpha ^{\mathsf{Sets}}_{X,Y,Z}\circ \alpha ^{\mathsf{Sets},-1}_{X,Y,Z}\webright ]\webleft (x,\webleft (y,z\webright )\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\alpha ^{\mathsf{Sets}}_{X,Y,Z}\webleft (\alpha ^{\mathsf{Sets},-1}_{X,Y,Z}\webleft (x,\webleft (y,z\webright )\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\alpha ^{\mathsf{Sets}}_{X,Y,Z}\webleft (\webleft (x,y\webright ),z\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (x,\webleft (y,z\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\operatorname {\mathrm{id}}_{\webleft (X\times Y\webright )\times Z}\webright ]\webleft (x,\webleft (y,z\webright )\webright ) \end{align*}

for each $\webleft (x,\webleft (y,z\webright )\webright )\in X\times \webleft (Y\times Z\webright )$, and therefore we have

\[ \alpha ^{\mathsf{Sets},-1}_{X,Y,Z}\circ \alpha ^{\mathsf{Sets}}_{X,Y,Z}=\operatorname {\mathrm{id}}_{X\times \webleft (Y\times Z\webright )}. \]

Therefore $\alpha ^{\mathsf{Sets}}_{X,Y,Z}$ is indeed an isomorphism.

Naturality

We need to show that, given functions

\begin{align*} f & \colon X \to X',\\ g & \colon Y \to Y',\\ h & \colon Z \to Z’ \end{align*}

the diagram

commutes. Indeed, this diagram acts on elements as

and hence indeed commutes, showing $\alpha ^{\mathsf{Sets}}$ to be a natural transformation.

Being a Natural Isomorphism

Since $\alpha ^{\mathsf{Sets}}$ is natural and $\alpha ^{\mathsf{Sets},-1}$ is a componentwise inverse to $\alpha ^{\mathsf{Sets}}$, it follows from Chapter 11: Categories, Item 2 of Proposition 11.9.7.1.2 that $\alpha ^{\mathsf{Sets},-1}$ is also natural. Thus $\alpha ^{\mathsf{Sets}}$ is a natural isomorphism.

The Clowder Project

5.1.4 The Associator