Item 2b (01LP) — The Clowder Project

4.6.1 Direct Images

Let $f\colon X\to Y$ be a function.

The direct image function associated to $f$ is the function¹

\[ f_{!}\colon \mathcal{P}(X)\to \mathcal{P}(Y) \]

defined by²

\begin{align*} f_{!}(U) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ y\in Y\ \middle |\ \begin{aligned} & \text{there exists some $x\in U$}\\ & \text{such that $y=f(x)$} \end{aligned} \right\} \\ & = \left\{ y\in Y\ \middle |\ f^{-1}(y)\cap U\neq \text{Ø}\right\} \\ & = \left\{ f(x)\in Y\ \middle |\ x\in U\right\} \end{align*}

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

¹Further Notation: Also written simply $f\colon \mathcal{P}(X)\to \mathcal{P}(Y)$.
²Further Terminology: The set $f(U)$ is called the direct image of $U$ by $f$.

Notation 4.6.1.1.2 ▶ Further Notation for Direct Images

Sometimes one finds the notation

\[ \exists _{f}\colon \mathcal{P}(X)\to \mathcal{P}(Y) \]

for $f_{!}$. This notation comes from the fact that the following statements are equivalent, where $y\in Y$ and $U\in \mathcal{P}(X)$:

•
We have $y\in \exists _{f}(U)$.
•
There exists some $x\in U$ such that $f(x)=y$.

We will not make use of this notation elsewhere in Clowder.

ゴ

Warning 4.6.1.1.3 ▶ Notation for Direct Images Is Confusing

Notation for direct images between powersets is tricky:

1.
Direct images for powersets and presheaves are both adjoint to their corresponding inverse image functors. However, the direct image functor for powersets is a left adjoint, while the direct image functor for presheaves is a right adjoint:
1. (a)
  Powersets. Given a function $f\colon X\to Y$, we have an inverse image functor
  
  \[ f^{-1}\colon \mathcal{P}(Y)\to \mathcal{P}(X). \]
  
  The left adjoint of this functor is the usual direct image, defined above in Definition 4.6.1.1.1.
2. (b)
  Presheaves. Given a morphism of topological spaces $f\colon X\to Y$, we have an inverse image functor
  
  \[ f^{-1}\colon \mathsf{PSh}(Y)\to \mathsf{PSh}(X). \]
  
  The right adjoint of this functor is the direct image functor of presheaves, defined in .
2.
The presheaf direct image functor is denoted $f_{*}$, but the direct image functor for powersets is denoted $f_{!}$ (as it’s a left adjoint).
3.
Adding to the confusion, it’s somewhat common for $f_{!}\colon \mathcal{P}(X)\to \mathcal{P}(Y)$ to be denoted $f_{*}$.

We chose to write $f_{!}$ for the direct image to keep the notation aligned with the following similar adjoint situations:

Situation	Adjoint String
Functoriality of Powersets	$(f_{!}\dashv f^{-1}\dashv f_{*})\colon \mathcal{P}(X)\overset {\rightleftarrows }{\to }\mathcal{P}(Y)$
Functoriality of Presheaf Categories	$(f_{!}\dashv f^{-1}\dashv f_{*})\colon \mathsf{PSh}(X)\overset {\rightleftarrows }{\to }\mathsf{PSh}(Y)$
Base Change	$(f_{!}\dashv f^{}\dashv f_{})\colon \mathcal{C}_{/X}\overset {\rightleftarrows }{\to }\mathcal{C}_{/Y}$
Kan Extensions	$(F_{!}\dashv F^{}\dashv F_{})\colon \mathsf{Fun}(\mathcal{C},\mathcal{E})\overset {\rightleftarrows }{\to }\mathsf{Fun}(\mathcal{D},\mathcal{E})$

ゴ

Remark 4.6.1.1.4 ▶ Unwinding Definition 4.6.1.1.1

Identifying $\mathcal{P}(X)$ with $\mathsf{Sets}(X,\{ \mathsf{t},\mathsf{f}\} )$ via Item 2 of Proposition 4.5.1.1.4, we see that the direct image function associated to $f$ is equivalently the function

\[ f_{!}\colon \mathcal{P}(X)\to \mathcal{P}(Y) \]

defined by

\begin{align*} f_{!}(\chi _{U}) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Lan}}_{f}(\chi _{U})\\ & = \operatorname*{\operatorname {\mathrm{colim}}}((f\mathbin {\overset {\to }{\times }}\underline{(-_{1})})\overset {\operatorname {\mathrm{\mathrm{pr}}}}{\twoheadrightarrow }A\overset {\chi _{U}}{\to }\{ \mathsf{t},\mathsf{f}\} )\\ & = \operatorname*{\operatorname {\mathrm{colim}}}_{\substack {x\in X\\ \begin{bgroup} f(x)=-_{1} \end{bgroup}}}(\chi _{U}(x))\\ & = \bigvee _{\substack {x\in X\\ \begin{bgroup} f(x)=-_{1} \end{bgroup}}}(\chi _{U}(x)),\end{align*}

where we have used for the second equality. In other words, we have

\begin{align*} [f_{!}(\chi _{U})](y)& =\bigvee _{\substack {x\in X\\ \begin{bgroup} f(x)=y \end{bgroup}}}(\chi _{U}(x))\\ & =\begin{cases} \mathsf{true}& \text{if there exists some $x\in X$ such}\\ & \text{that $f(x)=y$ and $x\in U,$}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & =\begin{cases} \mathsf{true}& \text{if there exists some $x\in U$}\\ & \text{such that $f(x)=y$,}\\ \mathsf{false}& \text{otherwise} \end{cases}\end{align*}

for each $y\in Y$.

Proposition 4.6.1.1.5 ▶ Properties of Direct Images I

Let $f\colon X\to Y$ be a function.

1.
Functoriality. The assignment $U\mapsto f_{!}(U)$ defines a functor

\[ f_{!}\colon (\mathcal{P}(X),\subset )\to (\mathcal{P}(Y),\subset ). \]

In particular, for each $U,V\in \mathcal{P}(X)$, the following condition is satisfied:
- (★)
2.
Triple Adjointness. We have a triple adjunction
witnessed by:
1. (a)
  Units and counits of the form
  
  \[ \begin{aligned} \operatorname {\mathrm{id}}_{\mathcal{P}(X)} & \hookrightarrow f^{-1}\circ f_{!},\\ f_{!}\circ f^{-1} & \hookrightarrow \operatorname {\mathrm{id}}_{\mathcal{P}(Y)},\\ \end{aligned} \qquad \begin{aligned} \operatorname {\mathrm{id}}_{\mathcal{P}(Y)} & \hookrightarrow f_{*}\circ f^{-1},\\ f^{-1}\circ f_{*} & \hookrightarrow \operatorname {\mathrm{id}}_{\mathcal{P}(X)}. \end{aligned} \]
  
  In particular:
  - •
    For each $U\in \mathcal{P}(X)$, we have $U\subset f^{-1}(f_{!}(U))$.
  - •
    For each $U\in \mathcal{P}(X)$, we have $f^{-1}(f_{*}(U))\subset U$.
  - •
    For each $V\in \mathcal{P}(Y)$, we have $f_{!}(f^{-1}(V))\subset V$.
  - •
    For each $V\in \mathcal{P}(Y)$, we have $V\subset f_{*}(f^{-1}(V))$.

(b)
Bijections of sets

\begin{align*} \operatorname {\mathrm{Hom}}_{\mathcal{P}(Y)}(f_{!}(U),V) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(U,f^{-1}(V)),\\ \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(f^{-1}(U),V) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(U,f_{*}(V)), \end{align*}

natural in $U\in \mathcal{P}(X)$ and $V\in \mathcal{P}(Y)$ and (respectively) $V\in \mathcal{P}(X)$ and $U\in \mathcal{P}(Y)$. In particular:
1. (i)
  The following conditions are equivalent:
  1. (I)
    We have $f_{!}(U)\subset V$.
  2. (II)
    We have $U\subset f^{-1}(V)$.
2. (ii)
  The following conditions are equivalent:
  1. (I)
    We have $f^{-1}(U)\subset V$.
  2. (II)
    We have $U\subset f_{*}(V)$.

Interaction With Unions of Families of Subsets. The diagram

commutes, i.e. we have

\[ \bigcup _{U\in \mathcal{U}}f_{!}(U)=\bigcup _{V\in f_{!}(\mathcal{U})}V \]

for each $\mathcal{U}\in \mathcal{P}(X)$, where $f_{!}(\mathcal{U})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(f_{!})_{!}(\mathcal{U})$.

Interaction With Intersections of Families of Subsets. The diagram

commutes, i.e. we have

\[ \bigcap _{U\in \mathcal{U}}f_{!}(U)=\bigcap _{V\in f_{!}(\mathcal{U})}V \]

for each $\mathcal{U}\in \mathcal{P}(X)$, where $f_{!}(\mathcal{U})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(f_{!})_{!}(\mathcal{U})$.

Interaction With Binary Unions. The diagram

commutes, i.e. we have

\[ f_{!}(U\cup V)=f_{!}(U)\cup f_{!}(V) \]

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

Interaction With Binary Intersections. We have a natural transformation

with components

\[ f_{!}(U\cap V)\subset f_{!}(U)\cap f_{!}(V) \]

indexed by $U,V\in \mathcal{P}(X)$.

Interaction With Differences. We have a natural transformation

with components

\[ f_{!}(U)\setminus f_{!}(V)\subset f_{!}(U\setminus V) \]

indexed by $U,V\in \mathcal{P}(X)$.

Interaction With Complements. The diagram

commutes, i.e. we have

\[ f_{!}(U^{\textsf{c}})=f_{*}(U)^{\textsf{c}} \]

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

Interaction With Symmetric Differences. We have a natural transformation

with components

\[ f_{!}(U)\mathbin {\triangle }f_{!}(V)\subset f_{!}(U\mathbin {\triangle }V) \]

indexed by $U,V\in \mathcal{P}(X)$.

10.

Interaction With Internal Homs of Powersets. The diagram