Bott and Tu exercise 10.7 is as follow (not an exact quote):
Let $\mathcal{F}$ be the presheaf on $S^1\}$ which associates to every open set the group $\mathbb{Z}$. They include a picture indicating three open sets but I will be explicit.
Let
$$ U_0 = \exp\left(( -\frac{\pi}{6},\frac{2\pi}{3} + \frac{\pi}{6})\right) $$
$$ U_1 = \exp\left(( \frac{2\pi}{3} -\frac{\pi}{6},\frac{4\pi}{3} + \frac{\pi}{6})\right)$$
$$ U_2 = \exp\left(( \frac{4\pi}{3} -\frac{\pi}{6},2\pi + \frac{\pi}{6})\right)$$$\mathcal{U} = \{U_0,U_1,U_2\}$ is a good cover of $S^1$.
Let $U_{ij} = U_i \cap U_j$.
Define restriction homomorphisms $\rho^t_{ij}: U_t \to U_{ij}$ to be the identity map except for $\rho^2_{02}$ which is $-\textrm{Id}$. Compute $H^*(\mathcal{U}, \mathcal{F})$.
I think I know what computation they want me to do here, but I do not think that $\mathcal{F}$ has actually been defined as a presheaf, nor do I think that there is an obvious presheaf which can be defined here which agrees with this on the given cover. As such, I do not think that $H^*(\mathcal{U}, \mathcal{F})$ has actually been defined at this point.
Am I being dense? Is there actually an obvious presheaf which is being defined by this data?
I can formalize my question a bit more. Say we have a good cover $\mathcal{U} = \{U_i\}$, then we can form a category with objects $U_i$, $U_{ij}$, $U_{ijk}, …$ with an arrow from $U$ to $V$ if $U \subset V$. Then define a psuedo-presheaf $\mathcal{F}$ of Abelian groups as a contravariant functor of this category to the category of abelian groups. Then define $H^*(\mathcal{U}, \mathcal{F})$ as the cohomology of the analogous complex to how presheaf cohomology is defined in the book.
The theorem, then, would be that presheaf cohomology can be computed from the psuedo-presheaf cohomology of a good cover. However, there may be psuedo-presheaves which do not arise from a presheaf. My question is whether the $\mathcal{F}$ defined in this exercise actually arises from a sheaf.
Best Answer
$\newcommand{\C}[1]{\mathcal{#1}}$Yes, $\C F$ does come from a sheaf $\C X$, in the sense that for each $U_*$, where $*$ is one of $1,2,3,12,13,23$, there is an inclusion $\C I_{U_*}: \C F(U_*)\to\C X(U_*)$, such that for every inclusion of open sets among $U_*$'s $$i^V_U: V\to U$$ we have $$\C I_{V_*}\circ \C F(i^V_U) = \C X(i^V_U)\circ \C I_{U_*}.$$ In fact, we can just let $X$ be the sheafification of the given "presheaf on the open cover $U_1,U_2, U_3$" and $\C I_{U_*}$ be the associated map.
Please note that the notion of "presheaf on an open cover" is only introduced later in chapter $13$, Monodromy of the same book, where you can find more explanation on this example of "cohomology with twisted coefficients".