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\chapter{Locally Constant Objects}

Fix a topos $\mathscr{E}$.

\begin{definition}[Family version]
\label{definition: cover family version}
A collection of objects $\{f_{\alpha} : U_\alpha \to 1\}_{\alpha \in I}$ is called a \emph{cover} if the $f_\alpha$ constitute a jointly epimorphic family. That is to say, $\bigsqcup_{\alpha} f_{\alpha} : \bigsqcup_{\alpha} U_{\alpha} \twoheadrightarrow 1$ is an epimorphism.
\end{definition}

\begin{definition}[Single version]
\label{definition: cover single version}
An object $U \in \mathscr{E}$ is called a \emph{cover} if $U \twoheadrightarrow 1$ is an epimorphism.
\end{definition}

If $\mathscr{E}$ is cocomplete (has all colimits $D : I \to \mathscr{E}$, where $I$ is a small category), then \cref{definition: cover family version} and \cref{definition: cover single version} coincide.

\begin{definition}
Let $X,U \in \mathscr{E}$, $U \neq 0$. Then $X$ is said to be \emph{trivialized by $U$} if there exists a set $S \in \mathbf{Sets}$ such that $X \times U \cong \Delta S \times U$ in the slice topos $\mathscr{E}/U$. The set $S$ is called a \emph{local trivialization}, local to the object $U$.
\end{definition}

\begin{lemma}
If $X \in \mathscr{E}$ is trivialized by some $U$, say $X \times U \cong \Delta S \times U$ and $V \to U$ is any morphism, then $X \times V \cong \Delta S \times V$.
\end{lemma}
\begin{proof}
$X \times V$ is the pullback of $X \times U \to U \leftarrow V$ and $\Delta S \times V$ is the pullback of $\Delta S \times U \to U \leftarrow V$. But these are isomorphic, hence $X \times V \cong \Delta S \times V$ by uniqueness of the pullback.
\end{proof}

Note in particular that $V$ can be the initial object $0$, although the lemma does not give us any useful information in that case.

Recall that a well-pointed has a fully faithful global sections functor.

\begin{lemma}
Assume $\mathscr{E}$ is well-pointed. If $X$ is trivialized by some $U$ with two different local trivializations, say $\Delta T \times U \cong X \times U \cong \Delta S \times U$, then $S \cong T$ as $\mathbf{Sets}$.
\end{lemma}
\begin{proof}

\end{proof}

\begin{definition}
An object $X \in \mathscr{E}$ is \emph{locally constant} if there is a covering family $\{f_{\alpha} :U_{\alpha} \to 1\}_{\alpha \in I}$ such that for each $\alpha \in I$ the object $X$ is trivialized by $U_{\alpha}$.
\end{definition}

\begin{definition}
Let $U \in \mathscr{E}$ be a cover. An object $V \in \mathscr{E}$ is called a \emph{refinement} of $U$ if there is an epimorphism $V \twoheadrightarrow U$.
\end{definition}

\begin{example}
Let $X$ be a topological space. A cover of $X$ is a collection $\{U_\alpha\}_{\alpha \in I}$ of open subsets $U_{\alpha} \subseteq X$ for each $\alpha \in I$. Equivalently, it is a topological space $\bigsqcup_{\alpha} U_\alpha$ with a surjection to $X$. Hence $\bigsqcup_{\alpha} h_{U_{\alpha}} \twoheadrightarrow h_X = 1$ is an epimorphism, and therefore a cover in $\Sh(X)$. This example requires that $\mathbf{Top}$ be cocomplete.
\end{example}

\begin{lemma}
\label{lemma: common refinement of covers exists}
Let $U,V \in \mathscr{E}$ be covers. Then there exists a common refinement of both $U$ and $V$.
\end{lemma}
\begin{proof}
Take the pullback of $U \twoheadrightarrow 1 \twoheadleftarrow V$. Epimorphisms are stable under pullback by \cite[Theorem VII.7.3]{MacLaneMoerdijk91}
\end{proof}

\begin{definition}
An object $X \in \mathscr{E}$ is called \emph{constant} if there exists a set $S \in \mathbf{Sets}$ such that $X \cong \Delta S$.
\end{definition}

\begin{definition}
An object $X \in \mathscr{E}$ is called \emph{locally constant} if there exists a cover $U \twoheadrightarrow 1$ and a set $S \in \mathbf{Sets}$ such that $X \times U \cong \Delta S \times U$ in the slice topos $\mathscr{E}/U$.
\end{definition}

Note that a constant object $X \in \mathscr{E}$ is locally constant by taking $U = 1$.

\begin{lemma}
\label{lemma: refinement of locally constant object is locally constant}
If $X \in \mathscr{E}$ is locally constant, say $X \times U \cong \Delta S \times U$ in $\mathscr{E}/U$ and if $V \to U$ is any morphism to $U$, then $X \times V \cong \Delta S \times V$ in $\mathscr{E}/V$.
\end{lemma}
\begin{proof}
In the diagram below
\[ \begin{tikzcd}
X \times V \arrow[rrrr, dotted] \arrow[dr] \arrow[dddrr, bend right] &            &   &                   & \Delta S \times V \arrow[dl] \arrow[dddll, bend left] \\
           & X \times U \arrow[rr, "\sim"] \arrow[dr] &   & \Delta S \times U \arrow[dl] & \\
           &            & U &                   & \\
           &            & V \arrow[u] &
\end{tikzcd} \]
the bottom left area is a pullback, as is the bottom right area. Since there is a morphism from $X \times U$ to $\Delta S \times U$ we get an induced morphism from the pullback $X \times V \to \Delta S \times V$ as indicated by the dotted arrow. But the newly formed trapezoid is a pullback. Since both monomorphisms and epimorphisms are stable under pullback, the dotted arrow is also both mono and epi. Hence it is an isomorphism since $\mathscr{E}$ is balanced.
\end{proof}

\begin{lemma}
\label{lemma: different presentations of locally constant object}
If $X \in \mathscr{E}$ is locally constant with two different presentations, say $X \times U \cong \Delta S \times U$ and $X \times V \cong \Delta T \times V$ in $\mathscr{E}/U$ and $\mathscr{E}/V$, respectively, then $\Delta S \cong \Delta T$.
\end{lemma}
\begin{proof}
Let $W$ be a common refinement of the covers $U$ and $V$. By \cref{lemma: refinement of locally constant object is locally constant} we know that $X \times W \cong \Delta S \times W$ and $X \times W \cong \Delta T \times W$ in $\mathscr{E}/W$. Hence $\Delta S \times W \cong \Delta T \times W$ in $\mathscr{E}/W$, so $\Delta T \cong \Delta S$ in $\mathscr{E}$.
\end{proof}

\begin{lemma}
\label{lemma: terminal object is locally const}
The terminal object $1 \in \mathscr{E}$ is locally constant.
\end{lemma}
\begin{proof}
Take $U = 1$ and $S = *$.
\end{proof}

\begin{lemma}
\label{lemma: pullback of locally const is locally const}
Let $X,Y,Z$ be locally constant objects in $\mathscr{E}$. Suppose that there are morphisms $X \to Z \leftarrow Y$. Then the pullback $X \times_Z Y$ is locally constant.
\end{lemma}
\begin{proof}
Suppose that $X \times U_X \cong \Delta S_X \times U_X$, $Y \times U_Y \cong \Delta S_Y \times U_Y$ and $Z \times U_Z \cong \Delta S_Z \times U_Z$ in $\mathscr{E}/U_X$, $\mathscr{E}/U_Y$ and $\mathscr{E}/U_Z$, respectively. Take a common refinement $U$ of $U_X$, $U_Y$ and $U_Z$. Let $S = S_X \times_{S_Z} S_Y$. Then $\Delta S = \Delta S_X \times_{\Delta S_Z} \Delta S_Y$ because $\Delta$ is left exact. Since
\[ \begin{tikzcd}
 \left( X \times_Z Y \right) \times U \arrow[r] \arrow[d] & X \times U \cong \Delta S_X \times U \arrow[d] \\
 Y \times U \cong \Delta S_Y \times U \arrow[r] & Z \times U \cong \Delta S_Z \times U
\end{tikzcd} \]
and
\[ \begin{tikzcd}
 \Delta S \times U \arrow[r] \arrow[d] & X \times U \cong \Delta S_X \times U \arrow[d] \\
 Y \times U \cong \Delta S_Y \times U \arrow[r] & Z \times U \cong \Delta S_Z \times U
\end{tikzcd} \]
are both pullbacks of the same diagram, by uniqueness of pullbacks we must have an isomorphism $\left( X \times_Z Y \right) \times U \cong \Delta S \times U$.
\end{proof}

\begin{corollary}
All finite limits of locally constant objects are locally constant.
\end{corollary}

\begin{lemma}
\label{lemma: initial object is locally const}
The initial object $0 \in \mathscr{E}$ is locally constant.
\end{lemma}
\begin{proof}
Take $U = 1$ and $S = \emptyset$.
\end{proof}

\begin{lemma}
\label{lemma: locally constant coproducts}
If $X,Y$ are locally constant in $\mathscr{E}$, then so is $X + Y$.
\end{lemma}
\begin{proof}
Assume $X \times U_X \cong \Delta S_X \times U_X$ and $Y \times U_Y \cong \Delta S_Y \times U_Y$ in $\mathscr{E}/U_X$ and $\mathscr{E}/U_Y$, respectively. Take a common refinement $U$ of $U_X,U_Y$ and let $S = S_X \sqcup S_Y$. Then $\Delta S = \Delta S_X + \Delta S_Y$ because $\Delta$ preserves all colimits. Therefore
\begin{align*}
\Delta S \times U &\cong \left( \Delta S_X + \Delta S_Y \right) \times U \\
&\cong \left( \Delta S_X \times U \right) + \left( \Delta S_Y \times U \right) \\
&\cong \left(X \times U \right) + \left(Y \times U \right) \cong \left(X + Y\right) \times U
\end{align*}
as required.
\end{proof}

\begin{definition}
An object $X \in \mathscr{E}$ is called \emph{locally constant finite} if it is locally constant, say $X \times U \cong \Delta S \times U$ in $\mathscr{E}/U$, and moreover the set $S$ is finite.
\end{definition}

Let $\mathscr{E}_f$ be the full subcategory of $\mathscr{E}$ whose objects are locally constant finite.

\begin{proposition}
Let $p : \mathbf{Sets} \to \mathscr{E}$ be a point. Then $\mathscr{E}_f$ together with the inverse image part $p^*$ of $p$ is a Galois category.
\end{proposition}
\begin{proof}
We check the axioms of \cite[Definition 3.1]{lenstra08}. Axiom (G1) is true by \cref{lemma: terminal object is locally const} and \cref{lemma: pullback of locally const is locally const}.
\end{proof}