# 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 !