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$\def\et{\operatorname{\acute Et}} \def\sh{\operatorname{Sh}} \def\set{\mathsf{Set}} \def\top{\mathsf{Top}} \def\psh{\operatorname{PSh}} $From a google search, it appears to be a well-known fact that there is an equivalence of categories $\et(X)\simeq\sh_\set(X)$, where $X$ is a topological space. This equivalence of categories is induced by the pair of anti-parallel functors $\top/X\to\psh_\set(X)$ that sends a topological space over $X$ to its sheaf of continuous sections, and $\psh_\set(X)\to\top/X$, that sends a presheaf of sets over $X$ to the étale space of the presheaf. The essential image of the former are sheaves over $X$, whereas the essential image of the latter are étale spaces (local homeomorphisms) over $X$.

However, I have been unable to find a complete proof of the equivalence $\et(X)\simeq\sh_\set(X)$. I know that it is not hard to define the action on morphisms of these two functors, to show that all the constructions are well-defined and to work out the rest of the details. This can be done as an exercise by anyone learning about sheaves. Nevertheless, I was curious to know about some source that actually writes the full proof, making explicit all the details involved. I haven't found anything on the Stacks Project, nor on Kashiwara-Schapira Categories and Sheaves, or on the nLab.

I was thinking that maybe it isn't written anywhere because it may follow from more general results. I know nothing about abstract nonsense like sites, Grothendieck topologies or toposes. I don't know if the equivalence is just a corollary or a particular case from a more general idea coming out from there.

So my questions are:

  • Do you know about some source where the full proof of the equivalence is worked out?
  • Do you know if the equivalence follows naturally from a more general or more abstract result?
Elías Guisado Villalgordo
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  • The arguments are essentially worked out in Szamuely's Galois Groups and Fundamental Groups, Chapter 2. The caveat is that he assumes all spaces locally connected, though this doesn't actually make a difference for this part of the result (the point being that he also wants the equivalence to restrict the equivalence to one between the category of locally constant sheaves of $X$ and covering spaces of $X$, and this specification does require $X$ locally connected). – Thorgott Nov 05 '22 at 15:22
  • In Mac Lane/Moerdijk "Sheaves in Geometry and Logic" they mention that the adjointness between $Top/X$ and $PShSet(X)$ can be obtained from the left Kan extension of $Open(X) \to Top/X$ along the Yoneda embedding. – Marc Olschok Nov 06 '22 at 20:51

2 Answers2

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This is Corollary II.6.3 on p. 90 of MacLane and Moerdijk's "Sheaves in Geometry and Logic". Chapter II does not yet require the theory of Grothendieck topoi.

Elías Guisado Villalgordo
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Qi Zhu
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I just stumbled upon another proof of the result in T. Wedhorn, Manifolds, Sheaves, and Cohomology, Proposition 3.39.

Elías Guisado Villalgordo
  • 4,758