Subsection 5.3.3 (01Q5): The Left Annihilator

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

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

as in the diagram

with components

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

given by $\zeta ^{\mathsf{Sets}}_{\ell |A}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{\mathrm{pr}}}_{1}$.

Invertibility

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

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

given by

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

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 $\zeta ^{\mathsf{Sets}}_{\ell |A} \circ \iota _{A} \colon \text{Ø}\to \text{Ø}$ 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 $\iota _{A}\circ \zeta ^{\mathsf{Sets}}_{\ell |A}$ is equal to the identity on every $(x,a)\in \text{Ø}\times A$, of which there are none.

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

Naturality

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

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

Being a Natural Isomorphism

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

The Clowder Project

5.3.3 The Left Annihilator