Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell complex equipped with the Alexandrov topology and $R$ is a field (in the case of cellular sheaves of vector spaces). In this case one can define cosheaf homology as the left derived functor of the global section functor. I am wondering if one can always define cosheaf homology in this way ? I read that the problem with cosheaves is that cosheafification does not exist in general. Is this problem connected with the problem of defining cosheaf homology ?