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For a site $C$ there is an adjunction $a\colon PSh(C)\leftrightarrows Sh(C)\colon \iota$ with sheafification $a$ and inclusion $i$. $a$ preserves finite limits. The inclusion $\iota$ does not preserve finite colimits in general.

Let $\{F_i\}_{i\in I}$ be a family of sheaves. There coproduct $\coprod_{i\in I} F_i$ in $Sh(C)$ is given by $a\left( \coprod_{i\in I} \iota(F_i) \right)$, i.e. the sheafified presheaf-coproduct of the $F_i$s.

What is an example of such a presheaf-product $\coprod_{i\in I} \iota(F_i)$ that is not already a sheaf if all the $F_i$s were sheaves? Is this always true for a finite set $I$?

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2 Answers 2

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The empty coproduct is such an example if the site admits empty covers.

Concrete examples come from sheaves on topological spaces which must always send the empty set to the one point set.

In particular this means that only the "unary" coproduct of sheaves in $\text{PSh}$ is a sheaf.

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  • $\begingroup$ Do you know if $\iota$ preserves filtered colimits? I doubt so but I don't know an example. $\endgroup$
    – user8463524
    Commented May 7, 2015 at 7:04
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    $\begingroup$ @jeffrey: Yes if the site is quasi-compact, but not in general. Consider sheaves on $\mathbb{R}$ for instance. $\endgroup$
    – Martin Brandenburg
    Commented May 7, 2015 at 9:44
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For topological spaces (and probably something similar will hold for many sites) you can explicitly write down the coproduct of sheaves (which I assume to be $\mathsf{Set}$-valued here):

$$\left(\coprod_i F_i\right)(U) = \left\{\bigl((U_i)_i,(s_i)_i\bigr) : U = \coprod_i U_i,\,s_i \in F_i(U_i)\right\}$$

Here you can see that there is a huge difference between the coproduct of sheaves and the coproduct of the underlying presheaves! This is even more clear if you work with representable sheaves: $$\coprod_i \hom(-,X_i) = \hom(-,\coprod_i X_i)$$ in the category of sheaves, but of course this is far from being true "pointwise", i.e. for the underlying presheaves.

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  • $\begingroup$ Don't you have to quotient out by a suitable equivalence relation? $\endgroup$
    – Zhen Lin
    Commented May 7, 2015 at 12:53
  • $\begingroup$ @Zhen: By which one? $\endgroup$
    – Martin Brandenburg
    Commented May 7, 2015 at 12:55
  • $\begingroup$ The coproduct of stacks (which includes sheaves as an example) is described in Laumon-Moret-Bailly, Champs algébriques, (3.3). It coincides with my description. $\endgroup$
    – Martin Brandenburg
    Commented May 7, 2015 at 13:07
  • $\begingroup$ Well, consider something silly like decomposing the empty set into two copies of itself vs three copies of itself... $\endgroup$
    – Zhen Lin
    Commented May 7, 2015 at 13:23
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    $\begingroup$ I wonder about the claimed equality $\coprod_i \hom(-,X_i) = \hom(-,\coprod_i X_i)$. Is that really true? If one takes an indescrete site (covers given by isomorphisms only, which should lead to all presheaves being sheaves), this would claim $\coprod^{\mathrm{pre}}_i \hom(-,X_i) = \hom(-,\coprod_i X_i)$, which is not true (as "presheaves freely adjoin colimits"). Don't you need to assume something extra such as that the family of canonical inclusions $\{X_{i'} \rightarrow \coprod_i X_i\}_{i'}$ is a cover? $\endgroup$
    – Pavel Čoupek
    Commented Oct 24, 2020 at 0:02

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