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Hey there, I was reading 'An introduction to homological algebra' by Rotman, and on page 279 in the section about sheaves, example 5.64, Rotman gives an example of a constant presheaf $\mathcal{P}$ that's not sheaf, the presheaf of constant real-valued functions on $\mathbb{R}^{2}$. Let the topological space $X = \mathbb{R}^{2}$ and for each $U\subseteq\mathbb{R}^{2}$ define

$\mathcal{P}(U) = $ {$f:U\rightarrow\mathbb{R}\mid$ $f$ is constant}.

If $U\subseteq V$, $\rho_{U}^{V}:\mathcal{P}(V)\rightarrow\mathcal{P}(U)$ is the restriction map $\sigma\mapsto\sigma\mid U$. Now for example let $U=U_{1}\bigcup U_{2}$, where $U_{1}$ and $U_{2}$ are disjoint nonempty sets, define $\sigma_{1}\in P(U_{1})$ by $\sigma_{1}(u_{1})=0$ for all $u_{1}\in U_{1}$, and $\sigma_{2}\in P(U_{2})$ by $\sigma_{2}(u_{2})=5$ for all $u_{2}\in U_{2}$. The overlap condition is vacuous since $U_{1}\bigcap U_{2}=\textrm{Ø}$, but there is no constant function $\sigma\in P(U)$ such that $\sigma\mid U_{i}=\sigma_{i}$, for $i=1,2$ (aka the gluing condition is not satisfied), therefore $\mathcal{P}$ is not a sheaf.

My confusion was with the overlap. How can he apply the gluing condition ff the sets $U_{1}$ and $U_{2}$ don't even overlap? In such a case wouldn't $\mathcal{P}$ satisfy the gluing condition and be a sheaf?

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By your definition $\mathcal P(\emptyset)$ consists of a unique element so all restriction maps to the empty set map all element to the same. –  Torsten Ekedahl Aug 26 '11 at 15:45
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btw, the sheafification of this presheaf is the sheaf of locally constant functions. –  Pietro Majer Aug 26 '11 at 16:33
    
Okay, thanks guys, sort of as a follow-up, is the sheafification of the presheaf of constant functions on a site (category with a grothendieck topology) also the sheaf of locally constant functions? would it hold too in this case? –  Mario Carrasco Aug 26 '11 at 20:57
    
Reading the title I thought this would be a trivial question. But after reading the very nice answers by RR and NS, I see there were some subtleties I didn't consider. +1 –  Qfwfq Nov 10 '11 at 20:04
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2 Answers 2

up vote 7 down vote accepted

As you say yourself, the overlap condition is vacuous and thus automatically true. However, the sheaf condition for presheaves has two parts:

  • The overlap condition: that $\sigma_i | (U_i \cap U_j) = \sigma_j | (U_i \cap U_j)$. This is true, since the intersection is empty.
  • The gluing condition: that there is some $\sigma$ such that $\sigma | U_i = \sigma_i$. This is false, since $\sigma_1 = 0$ and $\sigma_2 = 5$, but $\sigma$ must be constant.

The fact that the overlap condition is vacuous means that you can proceed directly to checking the gluing condition, but the gluing condition itself is rarely vacuous.

Basically, you want to avoid thinking that "the sets don't even overlap". Statements in mathematical logic are always either true or false, but not void: if the intersection is empty, then it is still meaningful to discuss it; the sentence doesn't just go away.

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Awesome, thanks Ryan, this had gone completely over my head –  Mario Carrasco Aug 26 '11 at 21:02
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If we take the question as written then $\mathcal{P}(\emptyset)$ is a singleton and Ryan's answer is valid. However, with that definition I would not call $\mathcal{P}$ the constant presheaf. Instead we should have $\mathcal{P}(U)=\mathbb{R}$ for all $U$. With this definition the case $U=\emptyset$ means that we cannot think of $\mathcal{P}(U)$ as a set of functions. Now the problem is more subtle. Suppose that $U$ is the union of a family of open subsets $U_i$, and we have a family of elements $p_i\in\mathcal{P}(U_i)$ for all $i$, and that $p_i$ and $p_j$ always have the same image in $\mathcal{P}(U_i\cap U_j)$. This just means that the $p_i$ are real numbers, and that $p_i=p_j$ for all $i$ and $j$, so there is a single real number $p\in\mathcal{P}(U)$ that maps to $p_i$ for all $i$. This means that $\mathcal{P}$ has unique gluing and so is a sheaf ... except that we have missed one special case.

The empty set can be covered by the empty family of subsets. (It can also be covered by the family consisting of one subset, namely the empty set, but that is different.) For this family, the gluing condition says that $\mathcal{P}(\emptyset)$ should be the equaliser of two maps between two sets. Each of those sets is the product of no terms, which is a single point. Thus $\mathcal{P}(\emptyset)$ must also be a single point. The first stage of sheafification is to force $\mathcal{P}(\emptyset)$ to be a single point, and then we are back to Ryan's answer.

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Thanks Neil, yeah, I was completely missing that –  Mario Carrasco Aug 26 '11 at 20:58
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