Let $\mathcal{S}^{q}$ be a presheaf of singular cochain on a topological space $X$, that is the functor $\mathcal{S}^{q} \colon \mathcal{O}(X)^{op} \to \mathbb{Z}$-$\mathsf{mod}$ is given by $\mathcal{S}^{q}(U)= \operatorname{Hom}(S_{q}(U), \mathbb{Z})$, where $S_q(U)$ is a singular chain on X. This is a well known fact that these cochains are presheaves but not sheaves. However, I cannot check this fact by myself. I am reading Warner's Foundations of Differential Manifolds and Lie Groups.
Let $\{ U_{\alpha} \}_{\alpha \in A}$ be a covering of an open set $U \subset X$. I learned that a presheaf $\mathcal{F}\colon \mathcal{O}(X)^{op} \to \mathbb{Z}$-$\mathsf{mod}$ is called a sheaf if following conditions 1 and 2 are satisfied.
- If there are $s, t \in \mathcal{F}(U)$ satisfing $\ s|_{U_{\alpha}} = t|_{U_{\alpha}}$ for arbitrary $\alpha \in A$, then $s=t$.
- If there is a family $\{f_{\alpha} \in \mathcal{F}(U_{\alpha}) \}_{\alpha \in A}$ satisfing $s_{\alpha}|_{U_{\alpha} \cap U_{\beta}} = s_{\beta}|_{U_{\alpha} \cap U_{\beta}} \ $ for $\ \alpha, \beta \in A, \ $there exists a global section $f \in \mathcal{F}(U) $ such that $f|_{U_{\alpha}} = f_{\alpha}$ for $\alpha \in A$.
In this book of page 192 insist that the presheaf $\mathcal{S}^{q} \colon \mathcal{O}(X)^{op} \to \mathbb{Z}$-$\mathsf{mod}$ ( for $q \ge 1$) satisfies later condition 2 only. I can not check this. How should I do to confirm these fact?(It is satisfied with 2 and not 1.)
Let $U,V$ be an open cover of $X$, and let $\sigma$ be a singular simplex whose image si neither contained in $U$ nor in $V$. Let $f$ be the singular cochain that has value zero in all singular simplices and let $g$ be the singular cochain that has value zero in all simplices but $\sigma$, where it takes value 1.
Then the restriction of $f$ to $U$ is the zero cochain, and so is the restriction of $g$ to $U$. The same holds for $V$. But $f\neq g$.