Subsection 3.3.1 (01Q1): The Left Distributor

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

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

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

as in the diagram

whose component

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

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

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

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

The Clowder Project

3.3.1 The Left Distributor