Pull back of sheaves has left adjoint if and only if the morphism between topological spaces is a local homeomorphism

Question

I am looking for a way to show that $f:X\rightarrow Y$ is a local homeomorphism of topological spaces if and only if $f^* :\operatorname{Sh}(Y)\rightarrow \operatorname{Sh}(X)$ has a left adjoint. Also can you please give an example?

Answer 1

Here is a counterexample.

Consider the Sierpiński space $S$: it has two points, say $0$ and $1$, and the open sets are precisely $\emptyset$, $\{ 1 \}$, and $S$. A sheaf $F$ on $S$ is completely determined by the data of $F (S)$, $F (\{ 1 \})$, and the restriction $F (S) \to F (\{ 1 \})$. Conversely, any data of that form determines a sheaf on $S$. Thus $\textbf{Sh} (S)$ is equivalent to the arrow category $[\mathbf{2}, \textbf{Set}]$.

Now, consider the inclusion $i : \{ 0 \} \hookrightarrow S$. The pullback functor $i^* : \textbf{Sh} (S) \to \textbf{Set}$ sends a sheaf $F$ on $S$ to the stalk $F_0$, which is naturally isomorphic to $F (S)$ (since the only open neighbourhood of $0$ is $S$). Under the identification of $\textbf{Sh} (S)$ with $[\mathbf{2}, \textbf{Set}]$, this corresponds to the functor $\operatorname{dom} : [\mathbf{2}, \textbf{Set}] \to \textbf{Set}$ that sends a map to its domain. This functor has a left adjoint, namely the functor that sends a set $X$ to the map $\textrm{id}_X$. But $\{ 0 \} \hookrightarrow S$ is not a local homeomorphism.