Item 5a (02V1) — The Clowder Project

8.7.5 A Six-Functor Formalism for Sets, Part II

Explanation 8.7.5.1.1 ▶ A Six-Functor Formalism for Sets II

The assignment $X\mapsto \mathcal{P}(X)$ together with the functors $R_{!}$, $R_{-1}$, $R^{-1}$, and $R_{*}$ of Section 8.7.1, Section 8.7.2, Section 8.7.3, and Section 8.7.4 and the functors

\begin{align*} -_{1}\cap -_{2} & \colon \mathcal{P}(X)\times \mathcal{P}(X) \to \mathcal{P}(X),\\ [-_{1},-_{2}]_{X} & \colon \mathcal{P}(X)^{\mathsf{op}}\times \mathcal{P}(X) \to \mathcal{P}(X) \end{align*}

of Chapter 4: Constructions With Sets, Item 1 of Proposition 4.3.9.1.2 and Chapter 4: Constructions With Sets, Item 1 of Proposition 4.4.7.1.3 satisfy several properties reminiscent of a six functor formalism in the sense of .

We collect these properties in Proposition 8.7.5.1.2 below.

Proposition 8.7.5.1.2 ▶ A Six-Functor Formalism for Sets II

Let $X$ be a set.

1.
The Projection Formula I. The diagram
commutes, i.e. we have

\[ R_{!}(U\cap R_{-1}(V))=R_{!}(U)\cap V \]

for each $U\in \mathcal{P}(X)$ and each $V\in \mathcal{P}(Y)$.
2.
The Projection Formula II. We have a natural transformation
with components

\[ R_{*}(U)\cap V\subset R_{*}(U\cap R^{-1}(V)) \]

indexed by $U\in \mathcal{P}(X)$ and $V\in \mathcal{P}(Y)$.
3.
Interaction Between Co/Direct Images and Internal Homs of Powersets. The diagram
commutes, i.e. we have an equality of sets

\[ R_{!}([U,V]_{X})=[R_{*}(U),R_{!}(V)]_{Y}, \]

natural in $U,V\in \mathcal{P}(X)$.
4.
Interaction Between Co/Inverse Images and Internal Homs of Powersets. The diagram
commutes, i.e. we have an equality of sets

\[ R^{-1}([U,V]_{X})=[R_{-1}(U),R^{-1}(V)]_{Y}, \]

natural in $U,V\in \mathcal{P}(X)$.
5.
The External Tensor Product. We have an external tensor product

\[ -_{1}\boxtimes _{X\times Y}-_{2}\colon \mathcal{P}(X)\times \mathcal{P}(Y)\to \mathcal{P}(X\times Y) \]

given by

\begin{align*} U\boxtimes _{X\times Y}V & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{\mathrm{pr}}}^{-1}_{1}(U)\cap \operatorname {\mathrm{\mathrm{pr}}}^{-1}_{2}(V)\\ & = \left\{ (u,v)\in X\times Y\ \middle |\ \text{$u\in U$ and $v\in V$}\right\} . \end{align*}

This is the same map as the one in Chapter 4: Constructions With Sets, Item 5 of Proposition 4.4.1.1.4. Moreover, the following conditions are satisfied:

(a)
Interaction With Direct Images. Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X'$ and $S\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y'$ be relations. The diagram
commutes, i.e. we have

\[ [R_{!}\times S_{!}](U\boxtimes _{X\times Y}V)=R_{!}(U)\boxtimes _{X'\times Y'}S_{!}(V) \]

for each $(U,V)\in \mathcal{P}(X)\times \mathcal{P}(Y)$.

(b)

Interaction With Coinverse Images. Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X'$ and $S\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y'$ be relations. The diagram

commutes, i.e. we have

\[ [R_{-1}\times S_{-1}](U\boxtimes _{X'\times Y'}V)=R_{-1}(U)\boxtimes _{X\times Y}S_{-1}(V) \]

for each $(U,V)\in \mathcal{P}(X')\times \mathcal{P}(Y')$.

(c)

Interaction With Inverse Images. Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X'$ and $S\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y'$ be relations. The diagram

commutes, i.e. we have

\[ [R^{-1}\times S^{-1}](U\boxtimes _{X'\times Y'}V)=R^{-1}(U)\boxtimes _{X\times Y}S^{-1}(V) \]

for each $(U,V)\in \mathcal{P}(X')\times \mathcal{P}(Y')$.

(d)

Interaction With Codirect Images. Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X'$ and $S\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y'$ be relations. The diagram

commutes, i.e. we have

\[ [R_{*}\times S_{*}](U\boxtimes _{X\times Y}V)=R_{*}(U)\boxtimes _{X'\times Y'}S_{*}(V) \]

for each $(U,V)\in \mathcal{P}(X)\times \mathcal{P}(Y)$.

(e)

Interaction With Diagonals. The diagram

i.e. we have

\[ U\cap V=\Delta ^{-1}_{X}(U\boxtimes _{X\times X}V) \]

for each $U,V\in \mathcal{P}(X)$.

The Dualisation Functor. We have a functor

\[ D_{X}\colon \mathcal{P}(X)^{\mathsf{op}}\to \mathcal{P}(X) \]

given by

\begin{align*} D_{X}(U) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[U,\text{Ø}]_{X}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}} \end{align*}

for each $U\in \mathcal{P}(X)$, as in Chapter 4: Constructions With Sets, Item 5 of Proposition 4.4.7.1.3, satisfying the following conditions:

(a)
Duality. We have
(b)
Interaction With Internal Homs. The diagram
commutes, i.e. we have

\[ \underbrace{D_{X}(U\cap D_{X}(V))}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[U\cap [V,\text{Ø}]_{X},\text{Ø}]_{X}}=[U,V]_{X} \]

for each $U,V\in \mathcal{P}(X)$.
(c)
Interaction With Direct Images. The diagram
commutes, i.e. we have

\[ R_{!}(D_{X}(U))=D_{Y}(R_{*}(U)) \]

for each $U\in \mathcal{P}(X)$.
(d)
Interaction With Coinverse Images. The diagram
commutes, i.e. we have

\[ R_{-1}(D_{Y}(U))=D_{X}(R^{-1}(U)) \]

for each $U\in \mathcal{P}(X)$.
(e)
Interaction With Inverse Images. The diagram
commutes, i.e. we have

\[ R^{-1}(D_{Y}(U))=D_{X}(R_{-1}(U)) \]

for each $U\in \mathcal{P}(X)$.
(f)
Interaction With Codirect Images. The diagram
commutes, i.e. we have

\[ R_{*}(D_{X}(U))=D_{Y}(R_{!}(U)) \]

for each $U\in \mathcal{P}(X)$.

Proof of Proposition 8.7.5.1.2.

Item 1: The Projection Formula I

We have

\begin{align*} R_{!}(U\cap R_{-1}(V)) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ b\in Y\ \middle |\ R_{-1}(b)\cap (U\cap R_{-1}(V))\neq \text{Ø}\right\} \\ & = \left\{ b\in Y\ \middle |\ U\cap (R_{-1}(b)\cap R_{-1}(V))\neq \text{Ø}\right\} \\ & = \left\{ b\in Y\ \middle |\ U\cap R_{-1}(\left\{ b\right\} \cap V)\neq \text{Ø}\right\} \\ & = \left\{ b\in Y\ \middle |\ \text{$b\in V$ and $U\cap R_{-1}(b)\neq \text{Ø}$}\right\} \\ & = \left\{ b\in Y\ \middle |\ U\cap R_{-1}(b)\neq \text{Ø}\right\} \cap V\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{!}(U)\cap V. \end{align*}

where we have used Item 6 of Proposition 8.7.2.1.4 for the third equality.

Item 2: The Projection Formula II

We have

\begin{align*} R_{*}(U)\cap V & \subset R_{*}(U)\cap R_{*}(R^{-1}(V))\\ & = R_{*}(U\cap R^{-1}(V)), \end{align*}

where we have used: