Why do people care to define fine sheaves? What useful property do they have for which softness is not sufficient?
some observations (because I feel guilty about a the one-line question):
The point of fine sheaves is of course having partitions of unity. But soft sheaves already have something almost as good: If $\mathcal{F}$ on $X$ is soft then for every $s \in \mathcal{F}(X)$ and every cover $(U_α)$ of $X$ there exist $s_α \in \mathcal{F}(X)$ with $\operatorname{supp} s_α ⊆ U_α$ such that $\sum_α s_α = α$ and the sum is locally finite.
So the only advantage of fine sheaves is that we can decompose sections $s \in \mathcal{F}(X)$ "uniformly" in $s$. But I don't know of any case where this is beneficial...
Incidentally: Does anyone know an example of a soft but not fine sheaf? (ideally on a paracompact space)
