A few general questions about pre-sheaves and sheaves

By | June 13, 2018

I am no specialist in sheaf theory, so I would be glad to get some help regarding the following:

I have a pre-sheaf $F$ of abelian groups above a topological space $X$, and I have found an open cover $\{U_i\}$ of $X$ such that, for any $i$, $F(U_i)$ is a direct sum: $$F(U_i) = \underset{l}{\bigoplus} F^l (U_i),$$ where $F^l$ are pre-sheaves on $X$.

I have the two following questions:

  1. Is it true in general that a direct sum of pre-sheaves (resp. sheaves) of abelian groups is a pre-sheaf (resp. sheaf), or does it have to be a finite sum ? If it is true, how do we define sections and restriction morphisms for general direct sums of pre-sheaves ?
  2. How to prove that the sheafification $F^{\#}$ of $F$ is given by the direct sum: $$F^{\#} = \underset{l}{\bigoplus} F^{l \#},$$ where $F^{l \#}$ is the sheafification of $F^l$ ?

Thanks a lot for your help !