# Category-theoretic characterization of locally constant sheaves

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