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Let's consider $X$ a (locally connected) topological space and $\mathcal{Sh}(X)$ the topos of sheaves over $X$. If you see sheaves as étale spaces, locally constants sheaves correspond to covering spaces.

Is there an internal (topos-theoretic) characterization of the locally constant sheaves?

I've searched in Sheaves in Geometry and Logic, but I haven't found anything, and in the nLab there seems to be something but I do not understand it (and it does not really look like an internal characterization).

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Do you know an internal characterization of the representable sheaves? That seems like a simpler problem than yours. I know how to characterize the representables in a presheaf category, but not in a sheaf category. (Perhaps the first helps with the second.) – Tom Leinster Mar 13 '12 at 13:46
The nlab definition has some typos. Let $\mathcal{E}$ be a topos over a base $\mathcal{S}$ with structural morphism $\Gamma : \mathcal{E} \to \mathcal{S}$ and left adjoint $\Delta : \mathcal{S} \to \mathcal{E}$. Then $E \in \mathcal{E}$ is called locally constant if there is some epimorphism $i : U \to 1$ ("covering") such that there is an isomorphism $E \times U \cong \Delta(S) \times U$ in $\mathcal{E}/U$ for some $S \in \mathcal{S}$. – Martin Brandenburg Mar 13 '12 at 14:21
Now I wonder in how far this definition is not internal. Guillaume what are you looking for exactly? – Martin Brandenburg Mar 13 '12 at 14:26
The only sense I can think of in which that definition is "not internal" is that it refers to the geometric morphism $\mathcal{E}\to\mathcal{S}$ rather than merely to $\mathcal{E}$. But when $\mathcal{S}$ is Set, that geometric morphism is uniquely determined by $\mathcal{E}$. – Mike Shulman Mar 13 '12 at 20:02
Furthermore, saying that $mathcal{E}$ is a topos over a fixed base $\mathcal{S}$ means you have $mathcal{E}$ together with a geometric morphism $mathcal{E} \to \mathcal{S}$, so even for based topoi, this definition is internal. Anyhow, this level of generality is not needed to answer the question. Every topos can be regarded as a topos over $SET$ in a unique way, since $SET$ is the terminal topos, as Mike points out. – David Carchedi Mar 14 '12 at 11:43

One can talk about locally constant objects in a topos and these lead to covering spaces for sheaves on a space. Finite covers are already in SGA or the first Johnstone topos book. Barr and Diaconescu On locally simply connected toposes and their fundamental groups begins the modern theory. See papers by Bunge for more modern formulations.

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