First of all, I give my definition of weak sheaves:
By a weak sheaf on a topological space $ X $, we mean a presheaf $F$ such that for all open covering $\{ U_i\}_{i\in I} $ of $X$ sheaf condition holds; that is, the diagram
\begin{equation*} F(X)\stackrel{}{\rightarrow} \prod_{i\in I}F(U_i)\stackrel{}{\rightrightarrows}\displaystyle\prod_{(i,{i'})\in I\times I}F(U_i\times_XU_{i'}) \end{equation*} is an equalizer.
Notice that this condition for other open subsets of $X$ is not necessary. While for a sheaf this condition holds for every open subset $U$ of $X$ and all open coverings $\{ U_i\}_{i\in I} $ of $U$.
Update 1: Clearly, every sheaf is a weak sheaf, but not conversely. As an example, if $S=\{0,1\}$, then the constant presheaf with the value $S$ on a non-empty space is a weak sheaf but not a sheaf.
Consider weak sheaves as a full subcategory of the category of presheaves.
Now my questions are
(1) Does the category of "weak sheaves" constitute an (elementary) topos? or at least which properties of the category of sheaves are true for that of "weak sheaves"?
(2) Is the category of "weak sheaves" of abelian groups an abelian category?
In sheaf theory, these properties are obtained by a tool called sheafification. So I think we should have a reflection from presheaves to weak sheaves. But I don't know if there is such a reflection. Maybe another idea works as well!
Update 2: I don't know if it is possible to discuss a similar situation over a site, (say on a final object of the site, if it exists), or as David Roberts mentioned in comments, consider a fixed object $X$ in the site and only $X$ itself has nontrivial covering families (probably).
One motivation for this question is that you are still able to survey local problems by weak sheaves, mainly about the whole space. For example, local solutions of differential equations in de Rham cohomology by the Poincare lemma. But what is the need for these solutions to open submanifolds, as a sheaf does? I am aware that this gives rise to a resolution of the constant sheaf $\underline{\mathbb{R}}$, and then isomorphisms between de Rham cohomology and singular, Cech, Alexander-Spanier cohomologies on manifolds. But I don't believe these are admissible answers.
In fact, I want to know why we need to strengthen "weak sheaves" to the notion of sheaves.
Update 3:
Every sheaf is an assignment of a weak sheaf to each open subset of the space.
I am interested in a clear explanation.
