associated sheaf functor doesn't preserve arbitrary products

Question

I need example that associated sheaf functor doesn't preserve arbitrary products. I think that one can provide an example for sheaves over topological space. Thanks for your help.

Answer 1

Let $X$ be a space. An abelian group (or a set, if you prefer) $G$ determines a constant presheaf, call it $C_G$, and the associated sheaf is the sheaf of locally constant maps into $G$. Given a family $G_i$ with product $G$, the product of the presheaves $C_{G_i}$ is the presheaf $C_G$, but the product of the associated sheaves is not in general the sheaf of locally constant maps to $G$. (If $X$ is not locally connected then one can easily have a map to $G$ which projects to a locally constant map to $G_i$ for each $i$ but which is not locally constant itself. )

This is presumably a down-to-earth special case of what Todd Trimble is saying.

Answer 2

An example where sheafification does not preserve arbitrary products is where we take sheaves over a (sober) space $X$ that is not locally connected, for example the space of irrationals or Cantor space.

Recall that a Grothendieck topos $E$ is locally connected if the (essentially unique) geometric morphism $\Gamma = f_\ast: E \to Set$ has a left adjoint $f^\ast$ that in turn has a left adjoint. More generally, a geometric morphism $f_\ast: E \to F$ between toposes is an essential geometric morphism if its left adjoint $f^\ast$ has a left adjoint.

We have the following facts:

• A presheaf topos $Set^{C^{op}}$ is locally connected. Here the left adjoint to the global sections functor $\Gamma: Set^{C^{op}} \to Set$ is the diagonal functor $\Delta: Set \to Set^{C^{op}}$, which of course has a left adjoint.

• A geometric morphism $f_\ast: E \to F$ is essential if and only if $f^\ast$ preserves arbitrary products. (Of course $f^\ast$ is already left exact and so preserves equalizers, if it preserves small products, then it preserves small limits. Using the fact that Grothendieck toposes are cototal, this is enough to ensure that $f^\ast$ has a left adjoint.)

• Then in particular for a small site $(C, J)$, the inclusion functor $i: Sh(C, J) \to Set^{C^{op}}$ is an essential geometric morphism if and only if sheafification $a: Set^{C^{op}} \to Sh(C, J)$ preserves small products. Thus, putting the last two facts together, the composite geometric morphism $$Sh(C, J) \stackrel{i}{\to} Set^{C^{op}} \stackrel{\Gamma}{\to} Set$$ is essential, i.e., $Sh(C, J)$ is locally connected, if $a$ preserves products.

• In the case where the site is $(\text{Open}(X), J)$ where $J$ is the canonical Grothendieck topology given by covering families, $Sh(X) = Sh(C, J)$ is locally connected if and only if $X$ is a locally connected space. Thus $a: Set^{\text{Open}(X)^{op}} \to Sh(X)$ preserves small products only if $X$ is locally connected.