In Bredon's Sheaf Theory he defines a sheaf this way
A Sheaf (of abelian groups) on $X$ is a pair $(\mathscr{A},\pi)$ where:
- $\mathscr{A}$ is a topological space
- $\pi:\mathscr{A}\to X$ is a local homeomorphism
- each $\mathscr{A}_x=\pi^{-1}(x)$, for $x\in X$ is an abelian group
- the group operations are continuous.
(In the sense that the map from $\{(a,b)\in\mathscr{A}\times\mathscr{A}:\pi(a)=\pi(b)\}\to\mathscr{A}$ defined by multiplication is continuous).
However most other sources (like wikipedia) I've looked at define a sheaf as a presheaf subject to some extra conditions. Clearly these are different objects. One is a topological space with a map to another and group structures on the fibers, and the other is a contravariant functor. How does one get from one definition to the other, and why did he choose to define a sheaf this way?