Item 4 (01J7) — The Clowder Project

4.4.1 Foundations

Let $X$ be a set.

The powerset of $X$ is the set $\mathcal{P}(X)$ defined by

\[ \mathcal{P}(X) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ U\in P\ \middle |\ U\subset X\right\} , \]

where $P$ is the set in the axiom of powerset, of .

Remark 4.4.1.1.2 ▶ Powersets as Decategorifications of Co/Presheaf Categories

Under the analogy that $\{ \mathsf{t},\mathsf{f}\} $ should be the $(-1)$-categorical analogue of $\mathsf{Sets}$, we may view the powerset of a set as a decategorification of the category of presheaves of a category (or of the category of copresheaves):

•
The powerset of a set $X$ is equivalently (Item 2 of Proposition 4.5.1.1.4) the set

\[ \mathsf{Sets}(X,\{ \mathsf{t},\mathsf{f}\} ) \]

of functions from $X$ to the set $\{ \mathsf{t},\mathsf{f}\} $ of classical truth values.
•
The category of presheaves on a category $\mathcal{C}$ is the category

\[ \mathsf{Fun}(\mathcal{C}^{\mathsf{op}},\mathsf{Sets}) \]

of functors from $\mathcal{C}^{\mathsf{op}}$ to the category $\mathsf{Sets}$ of sets.

Let $X$ be a set.

1.
We write $\mathcal{P}_{0}(X)$ for the set of nonempty subsets of $X$.
2.
We write $\mathcal{P}_{\mathrm{fin}}(X)$ for the set of finite subsets of $X$.

Proposition 4.4.1.1.4 ▶ Elementary Properties of Powersets

Let $X$ be a set.

1.
Co/Completeness. The (posetal) category (associated to) $(\mathcal{P}(X),\subset )$ is complete and cocomplete:
1. (a)
  Products. The products in $\mathcal{P}(X)$ are given by intersection of subsets.
2. (b)
  Coproducts. The coproducts in $\mathcal{P}(X)$ are given by union of subsets.
3. (c)
  Co/Equalisers. Being a posetal category, $\mathcal{P}(X)$ only has at most one morphisms between any two objects, so co/equalisers are trivial.
2.
Cartesian Closedness. The category $\mathcal{P}(X)$ is Cartesian closed.
3.
Powersets as Sets of Relations. We have bijections

\begin{align*} \mathcal{P}(X) & \cong \mathrm{Rel}(\mathrm{pt},X),\\ \mathcal{P}(X) & \cong \mathrm{Rel}(X,\mathrm{pt}), \end{align*}

natural in $X\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$.

4.
Interaction With Products I. The map
is an isomorphism of sets, natural in $X,Y\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$ with respect to each of the functor structures $\mathcal{P}_{!}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{*}$ on $\mathcal{P}$ of Proposition 4.4.2.1.1. Moreover, this makes each of $\mathcal{P}_{!}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{*}$ into a symmetric monoidal functor.

Interaction With Products II. The map

where¹

\[ U\boxtimes _{X\times Y}V\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ (u,v)\in X\times Y\ \middle |\ \text{$u\in U$ and $v\in V$}\right\} \]

is an inclusion of sets, natural in $X,Y\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$ with respect to each of the functor structures $\mathcal{P}_{!}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{*}$ on $\mathcal{P}$ of Proposition 4.4.2.1.1. Moreover, this makes each of $\mathcal{P}_{!}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{*}$ into a symmetric monoidal functor.

Interaction With Products III. We have an isomorphism

\[ \mathcal{P}(X)\otimes \mathcal{P}(Y)\cong \mathcal{P}(X\times Y), \]

natural in $X,Y\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$ with respect to each of the functor structures $\mathcal{P}_{!}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{*}$ on $\mathcal{P}$ of Proposition 4.4.2.1.1, where $\otimes $ denotes the tensor product of suplattices of . Moreover, this makes each of $\mathcal{P}_{!}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{*}$ into a symmetric monoidal functor.

¹The set $U\boxtimes _{X\times Y}V$ is usually denoted simply $U\times V$. Here we denote it in this somewhat weird way to highlight the similarity to external tensor products in six-functor formalisms (see also Section 4.6.4).

Proof of Proposition 4.4.1.1.4.

Item 1: Co/Completeness

Omitted.

Item 2: Cartesian Closedness

See Section 4.4.7.

Item 3: Powersets as Sets of Relations

Indeed, we have

\begin{align*} \mathrm{Rel}(\mathrm{pt},X) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}(\mathrm{pt}\times X)\\ & \cong \mathcal{P}(X) \end{align*}

and

\begin{align*} \mathrm{Rel}(X,\mathrm{pt}) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathcal{P}(X\times \mathrm{pt})\\ & \cong \mathcal{P}(X), \end{align*}

where we have used Item 5 of Proposition 4.1.3.1.3.

Item 4: Interaction With Products I

The inverse of the map in the statement is the map

\[ \Phi \colon \mathcal{P}(X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y)\to \mathcal{P}(X)\times \mathcal{P}(Y) \]

defined by

\[ \Phi (S)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(S_{X},S_{Y}) \]

for each $S\in \mathcal{P}(X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y)$, where

\begin{align*} S_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ (0,x)\in S\right\} \\ S_{Y} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ y\in Y\ \middle |\ (1,y)\in S\right\} . \end{align*}

The rest of the proof is omitted.

Item 5: Interaction With Products II

Omitted.

Item 6: Interaction With Products III

Omitted.