Definition 5.3.4.1.1 (01Q8): The Right Annihilator of $\times $

The right annihilator of the product of sets is the natural isomorphism

\[ \zeta ^{\mathsf{Sets}}_{r} \colon \mathbb {0}^{\mathsf{Sets}}\circ \mathbf{\mu }^{\mathsf{Cats}_{\mathsf{2}}}_{4|\mathsf{Sets},\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}\circ (\mathbf{\epsilon }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}}\times \operatorname {\mathrm{id}}_{\mathsf{pt}}) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\times \circ (\operatorname {\mathrm{id}}_{\mathsf{Sets}}\times \mathbb {0}^{\mathsf{Sets}}) \]

as in the diagram

with components

\[ \zeta ^{\mathsf{Sets}}_{r|A}\colon A\times \text{Ø}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\text{Ø} \]

given by $\zeta ^{\mathsf{Sets}}_{r|A}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{\mathrm{pr}}}_{2}$.

Invertibility

The inverse of $\zeta ^{\mathsf{Sets}}_{r|A}$ is the map

\[ \zeta ^{\mathsf{Sets},-1}_{r|A}\colon \text{Ø}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A\times \text{Ø} \]

given by

\[ \zeta ^{\mathsf{Sets},-1}_{r|A}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\iota _{A}, \]

where $\iota _{A}$ is as defined in Chapter 4: Constructions With Sets, Construction 4.2.1.1.2:

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Invertibility I. The map $\zeta ^{\mathsf{Sets}}_{r|A}\circ \iota _{A}\colon \text{Ø}\to \text{Ø}$ is equal to $\operatorname {\mathrm{id}}_{\text{Ø}}$, as $\text{Ø}$ is the initial object of $\mathsf{Sets}$.
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Invertibility II. The map $\iota _{A}\circ \zeta ^{\mathsf{Sets}}_{r|A}$ is equal to the identity on every $(a,x)\in A\times \text{Ø}$, of which there are none.

Hence $\zeta ^{\mathsf{Sets}}_{r|A}$ is an isomorphism.

Naturality

We need to show that given a function $f\colon A\to B$, the diagram

commutes. But since $A\times \text{Ø}$ has no elements, this is trivially true.

Being a Natural Isomorphism

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

The Clowder Project

5.3.4 The Right Annihilator