Subsection 5.3.1 (01Q1): The Left Distributor

The left distributor of the product of sets over the coproduct of sets is the natural isomorphism

\[ \delta ^{\mathsf{Sets}}_{\ell } \colon \mathord {\times }\circ (\operatorname {\mathrm{id}}_{\mathsf{Sets}}\times \mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}) \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ (\mathord {(\times )}\times \mathord {(\times )})\circ \mathbf{\mu }^{\mathsf{Cats}_{\mathsf{2}}}_{4|\mathsf{Sets},\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}\circ (\Delta _{\mathsf{Sets}}\times (\operatorname {\mathrm{id}}_{\mathsf{Sets}}\times \operatorname {\mathrm{id}}_{\mathsf{Sets}})) \]

as in the diagram

whose component

\[ \delta ^{\mathsf{Sets}}_{\ell |X,Y,Z}\colon X\times (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z)\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }(X\times Y)\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}(X\times Z) \]

at $(X,Y,Z)$ is defined by

\[ \delta ^{\mathsf{Sets}}_{\ell |X,Y,Z}(x,a)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} (0,(x,y)) & \text{if $a=(0,y)$,}\\ (1,(x,z)) & \text{if $a=(1,z)$} \end{cases} \]

for each $(x,a)\in X\times (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z)$.

Invertibility

The inverse of $\delta ^{\mathsf{Sets}}_{\ell |X,Y,Z}$ is the map

\[ \delta ^{\mathsf{Sets},-1}_{\ell |X,Y,Z}\colon (X\times Y)\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}(X\times Z) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X\times (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z) \]

given by

\[ \delta ^{\mathsf{Sets},-1}_{\ell |X,Y,Z}(a)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} (x,(0,y)) & \text{if $a=(0,(x,y))$,}\\ (x,(1,z)) & \text{if $a=(1,(x,z))$} \end{cases} \]

for $a\in (X\times Y)\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}(X\times Z)$. Indeed:

•
Invertibility I. The map $\delta ^{\mathsf{Sets},-1}_{\ell |X,Y,Z} \circ \delta ^{\mathsf{Sets}}_{\ell |X,Y,Z}$ acts on elements as

\begin{align*} & \mkern 40mu\mathclap {(x,(0,y))}\mkern 40mu \mapsto \mkern 40mu\mathclap {(0,(x,y))} \mkern 40mu \mapsto \mkern 40mu\mathclap {(x,(0,y))}\mkern 35mu\mathrlap {,}\\ & \mkern 40mu\mathclap {(x,(1,z))}\mkern 40mu \mapsto \mkern 40mu\mathclap {(1,(x,z))}\mkern 40mu \mapsto \mkern 40mu\mathclap {(x,(1,z))}\mkern 35mu\mathrlap {,} \end{align*}

but these are the two possible cases for elements of $X \times (Y \mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z)$. Hence the map is equal to the identity.
•
Invertibility II. The map $\delta ^{\mathsf{Sets}}_{\ell |X,Y,Z} \circ \delta ^{\mathsf{Sets},-1}_{\ell |X,Y,Z}$ acts on elements as

\begin{align*} & \mkern 40mu\mathclap {(0,(x,y))}\mkern 40mu \mapsto \mkern 40mu\mathclap {(x,(0,y))} \mkern 40mu \mapsto \mkern 40mu\mathclap {(0,(x,y))}\mkern 35mu\mathrlap {,}\\ & \mkern 40mu\mathclap {(1,(x,z))}\mkern 40mu \mapsto \mkern 40mu\mathclap {(x,(1,z))}\mkern 40mu \mapsto \mkern 40mu\mathclap {(1,(x,z))}\mkern 35mu\mathrlap {,} \end{align*}

but these are the two possible cases for elements of $(X \times Y) \mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}(X \times Z)$. Hence the map is equal to the identity.

Thus $\delta ^{\mathsf{Sets}}_{\ell |X,Y,Z}$ is an isomorphism for all $X,Y,Z$.

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

so it commutes, showing $\delta ^{\mathsf{Sets}}_{\ell }$ to be a natural transformation.

Being a Natural Isomorphism

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

The Clowder Project

5.3.1 The Left Distributor