Sheaves are equivalent to étale spaces

Let $X$ be a fixed topological space. We prove that the category $\mathbf{Sh}(X)$ of sheaves of sets on $X$ is equivalent to the category $\mathbf{Et}/X$ of étale spaces over $X$ (objects are local homeomorphisms $p:E\to X$, morphisms are continuous maps over $X$). The equivalence is given by two functors that are quasi-inverse to each other:

$\Gamma:\mathbf{Et}/X\to\mathbf{Sh}(X)$ — the sheaf of (continuous) sections functor.
$\mathcal{E}:\mathbf{Sh}(X)\to\mathbf{Et}/X$ — the étalé (espace étalé) construction sending a sheaf to the disjoint union of its stalks with a natural topology.

We will define both functors carefully, show they are well defined and functorial, then construct natural isomorphisms

$$ \eta: \operatorname{id}_{\mathbf{Sh}(X)} \overset{\simeq}{\Longrightarrow} \Gamma\circ\mathcal{E} \qquad\text{and}\qquad \varepsilon: \mathcal{E}\circ\Gamma \overset{\simeq}{\Longrightarrow} \operatorname{id}_{\mathbf{Et}/X}, $$

which prove the equivalence.

$\mathbf{Sh}(X)$: Objects are sheaves of sets $\mathcal F$ on $X$. Morphisms are natural transformations of sheaves (i.e. morphisms of presheaves respecting the sheaf axioms).

$\mathbf{Et}/X$: Objects are pairs $(E,p)$ where $E$ is a topological space and $p:E\to X$ is a local homeomorphism (equivalently, every $e\in E$ has an open neighborhood $U\subset E$ such that $p|_U:U\to p(U)$ is a homeomorphism onto an open subset of $X$). Morphisms $(E,p)\to (E',p')$ are continuous maps $f:E\to E'$ with $p'=p\circ f$ (i.e. maps over $X$).

$\Gamma:\mathbf{Et}/X\to\mathbf{Sh}(X)$

Let $(E,p)\in\mathbf{Et}/X$. For each open $U\subset X$ define

$$ \Gamma(E,p)(U) := \{ s:U\to E \mid s\text{ is continuous and }p\circ s=\operatorname{id}_U \}. $$

For $V\subset U$ the restriction $\Gamma(E,p)(U)\to\Gamma(E,p)(V)$ is just function restriction. We must check this presheaf is a sheaf:

Locality (identity axiom). If $s,t\in\Gamma(E,p)(U)$ and $s|_{U_i}=t|_{U_i}$ for an open cover $U=\bigcup U_i$, then $s=t$ because equality holds pointwise.
Gluing (existence axiom). If $s_i\in\Gamma(E,p)(U_i)$ for a cover $U=\bigcup U_i$ and $s_i$ agree on overlaps, then the function $s$ defined pointwise by $s(x)=s_i(x)$ for $x\in U_i$ is continuous because each $s_i$ is continuous and the $s_i$ agree on overlaps; since $p\circ s_i=\operatorname{id}$, we have $p\circ s=\operatorname{id}$. Thus $s\in\Gamma(E,p)(U)$. (One can also check continuity by observing that continuity is a local property and each point has a neighborhood where a single $s_i$ determines $s$.)

Hence $\Gamma(E,p)$ is a sheaf.

If $f:(E,p)\to (E',p')$ is a morphism in $\mathbf{Et}/X$ (so $p'=p\circ f$), then for each open $U\subset X$ define

$$ \Gamma(f)_U : \Gamma(E,p)(U)\to\Gamma(E',p')(U),\qquad s\mapsto f\circ s. $$

These commute with restrictions and give a natural transformation of sheaves. Thus $\Gamma$ is a functor.

$\mathcal{E}:\mathbf{Sh}(X)\to\mathbf{Et}/X$

Let $\mathcal F$ be a sheaf on $X$. For each $x\in X$ let

$$ \mathcal F_x := \varinjlim_{x\in U} \mathcal F(U) $$

be the stalk at $x$ (germs of sections near $x$). Define the underlying set of the étalé space by the disjoint union of stalks:

$$ \mathcal E(\mathcal F) := \bigsqcup_{x\in X} \mathcal F_x. $$

Define the projection map $\pi:\mathcal E(\mathcal F)\to X$ by $\pi([s]_x)=x$ where $[s]_x$ denotes the germ at $x$ of a section $s$ defined on some neighborhood of $x$.

We now put a topology on $\mathcal E(\mathcal F)$ by giving a basis. For any open $U\subset X$ and any section $s\in\mathcal F(U)$, define the basic open (often called the image of the section $s$) as

$$ \widetilde{s} := \{ [s]_x \in \bigsqcup_{x\in U} \mathcal F_x \mid x\in U \}. $$

Declare the family of all such $\widetilde{s}$ (for all open $U$ and $s\in\mathcal F(U)$) to be a basis for a topology on $\mathcal E(\mathcal F)$. Equivalently, a set is open iff it is a union of such $\widetilde{s}$.

Basic properties

$\pi|_{\widetilde{s}} : \widetilde{s}\to U$ is bijective (take inverse $x\mapsto [s]_x$) and continuous; its inverse $U\to\widetilde{s}$ is continuous because $\widetilde{s}$ is given the final topology from these basis sets. Hence $\pi|_{\widetilde{s}}$ is a homeomorphism onto $U$. Thus $\pi$ is a local homeomorphism, so $(\mathcal E(\mathcal F),\pi)$ is indeed an object of $\mathbf{Et}/X$.
The topology is well-defined: if $[s]_x=[t]_x$ (two representatives define the same germ at $x$), then there exists a neighborhood $V\ni x$ with $s|_V=t|_V$; on $V$ the basic opens $\widetilde{s}$ and $\widetilde{t}$ agree, so germs are consistently placed in basic opens. Thus distinct representatives produce the same basic neighborhoods near the point.

Definition on morphisms

If $\varphi:\mathcal F\to\mathcal G$ is a morphism of sheaves, for each $x\in X$ we get a map on stalks $\varphi_x:\mathcal F_x\to\mathcal G_x$ (induced by the direct limit). These assemble into a map of sets

$$ \mathcal E(\varphi):\mathcal E(\mathcal F)\to\mathcal E(\mathcal G),\qquad [s]_x \mapsto [\varphi_U(s)]_x $$

for any representative $s\in\mathcal F(U)$ of the germ. This is well-defined, continuous, and satisfies $\pi_{\mathcal G}\circ\mathcal E(\varphi)=\pi_{\mathcal F}$. Continuity follows because the image of a basic open $\widetilde{s}\subset\mathcal E(\mathcal F)$ is the basic open $\widetilde{\varphi_U(s)}\subset\mathcal E(\mathcal G)$.

Thus $\mathcal E$ is a functor $\mathbf{Sh}(X)\to\mathbf{Et}/X$.

$\eta:\operatorname{id}_{\mathbf{Sh}(X)}\Rightarrow \Gamma\circ\mathcal{E}$

Let $\mathcal F$ be a sheaf. We produce a morphism of sheaves

$$ \eta_{\mathcal F}: \mathcal F \longrightarrow \Gamma(\mathcal E(\mathcal F)) $$

(i.e. for each open $U$ a map $\eta_{\mathcal F}(U):\mathcal F(U)\to\Gamma(\mathcal E(\mathcal F))(U)$) as follows:

For $s\in\mathcal F(U)$ define $\eta_{\mathcal F}(U)(s):U\to\mathcal E(\mathcal F)$ by

$$ \eta_{\mathcal F}(U)(s)(x) := [s]_x \in \mathcal F_x. $$

This is well defined, the map $x\mapsto [s]_x$ is continuous (because for each $x$ there is a neighborhood $V$ where $s$ is represented and $\eta_{\mathcal F}(U)(s)$ maps $V$ homeomorphically onto the basic open $\widetilde{ s|_V}$). Also $\pi\circ \eta_{\mathcal F}(U)(s)=\operatorname{id}_U$, so this is a section of $\pi$. The assignments $\eta_{\mathcal F}(U)$ commute with restriction, so $\eta_{\mathcal F}$ is a morphism of sheaves.

We claim each $\eta_{\mathcal F}$ is an isomorphism of sheaves.

Injectivity

Suppose $s,t\in\mathcal F(U)$ and $\eta_{\mathcal F}(U)(s) = \eta_{\mathcal F}(U)(t)$ as sections $U\to\mathcal E(\mathcal F)$. Then for every $x\in U$ we have $[s]_x=[t]_x$ in the stalk $\mathcal F_x$. Thus for every $x$ there exists a neighborhood $V_x$ of $x$ with $s|_{V_x}=t|_{V_x}$. The family $(V_x)$ covers $U$, and by the sheaf identity axiom, $s=t$ on $U$. So $\eta_{\mathcal F}(U)$ is injective.

Surjectivity

Let $\sigma\in\Gamma(\mathcal E(\mathcal F))(U)$ be a continuous section $\sigma:U\to\mathcal E(\mathcal F)$ with $\pi\circ\sigma=\operatorname{id}_U$. For each point $x\in U$ the value $\sigma(x)$ is a germ $[s_x]_x$ for some representative $s_x$ defined on a neighborhood $V_x\ni x$. By continuity of $\sigma$, for each $x$ there exists a neighborhood $W_x\subset V_x$ such that $\sigma(W_x)\subset\widetilde{s_x}$ (because the $\widetilde{s_x}$ form a basis). On $W_x$ we then have $\sigma(y)=[s_x]_y$ for all $y\in W_x$, so $\sigma|_{W_x}$ is exactly the section associated to $s_x|_{W_x}$. The $W_x$ cover $U$. On overlaps $W_x\cap W_y$ the sections $s_x|_{W_x\cap W_y}$ and $s_y|_{W_x\cap W_y}$ have the same germs at every point, hence are equal there; by the sheaf gluing axiom they glue to a global section $s\in\mathcal F(U)$. By construction $\eta_{\mathcal F}(U)(s)=\sigma$. Thus $\eta_{\mathcal F}(U)$ is surjective.

So $\eta_{\mathcal F}$ is an isomorphism of sheaves. Naturalness in $\mathcal F$ is immediate from the definitions: for any morphism $\varphi:\mathcal F\to\mathcal G$ the square

$$ \begin{CD} \mathcal F @>{\eta_{\mathcal F}}>> \Gamma(\mathcal E(\mathcal F))\\ @V{\varphi}VV @VV{\Gamma(\mathcal E(\varphi))}V\\ \mathcal G @>{\eta_{\mathcal G}}>> \Gamma(\mathcal E(\mathcal G)) \end{CD} $$

commutes because both ways send a section $s$ to the section $x\mapsto [\varphi(s)]_x$. Thus $\eta:\operatorname{id}\Rightarrow\Gamma\circ\mathcal E$ is a natural isomorphism.

$\varepsilon:\mathcal E\circ\Gamma\Rightarrow \operatorname{id}_{\mathbf{Et}/X}$

Let $(E,p)$ be an étale space over $X$. Form the sheaf $\Gamma(E,p)$ of sections and then the étalé space $\mathcal E(\Gamma(E,p))$, whose points are germs $[s]_x$ of local continuous sections $s$ of $p$. Define

$$ \varepsilon_{(E,p)} : \mathcal E(\Gamma(E,p)) \longrightarrow E $$

on germs by

$$ \varepsilon_{(E,p)}\big([s]_x\big) := s(x)\in E. $$

We check this is well-defined, continuous, a bijection, and a homeomorphism over $X$.

Well-definedness

If $[s]_x=[t]_x$ as germs, there exists a neighborhood $V\ni x$ with $s|_V=t|_V$. Evaluating at $x$ gives $s(x)=t(x)$, so the map is well-defined.

Map over $X$

Clearly $p(\varepsilon_{(E,p)}([s]_x)) = p(s(x)) = x = \pi([s]_x)$ where $\pi:\mathcal E(\Gamma(E,p))\to X$ is the projection. So $\varepsilon_{(E,p)}$ is a map over $X$.

Continuity

Let $\widetilde{s}\subset\mathcal E(\Gamma(E,p))$ be a basic open (coming from a section $s$ defined on $U$). Then $\varepsilon_{(E,p)}(\widetilde{s}) = s(U)$, the image of $s$ in $E$. But $s:U\to E$ is continuous, and $\widetilde{s}$ is homeomorphic via $\pi$ to $U$. Concretely, $\varepsilon_{(E,p)}$ restricted to $\widetilde{s}$ equals $s\circ (\pi|_{\widetilde{s}})^{-1}$, which is continuous. Thus $\varepsilon_{(E,p)}$ is continuous (it is continuous on each basis open, hence continuous).

Bijectivity

Surjective. For any $e\in E$ with $x=p(e)$, because $p$ is a local homeomorphism there exists a neighborhood $V\ni e$ with $p|_V:V\to U$ a homeomorphism onto $U:=p(V)$. The inverse $s := (p|_V)^{-1}:U\to V\subset E$ is a section with $s(x)=e$. Then $[s]_x$ maps to $e$. So $\varepsilon_{(E,p)}$ is surjective.
Injective. Suppose $\varepsilon([s]_x)=\varepsilon([t]_y)$. Then $x=p(s(x))=p(t(y))=y$, so $x=y$. Let $e:=s(x)=t(x)$. Because $p$ is a local homeomorphism there exists some neighborhood $V$ of $e$ such that $p|_V$ is a homeomorphism onto an open neighborhood $U$ of $x$. Consider the sections $s$ and $t$. For points of $U$ we may view both $s$ and $t$ as maps into $E$; but $s(x)=e\in V$ and $t(x)=e\in V$, so by shrinking domains we can assume both sections are defined on $U$ and take values in $V$. Because $p|_V$ is injective, the equality $p\circ s = \operatorname{id}$ and $p\circ t=\operatorname{id}$ implies $s|_U=t|_U$. Hence $[s]_x=[t]_x$. So $\varepsilon$ is injective.

Thus $\varepsilon_{(E,p)}$ is a bijection.

Homeomorphism (open map / inverse continuous)

We have shown $\varepsilon_{(E,p)}$ is continuous and bijective. To see it is a homeomorphism, it suffices to show it is open (or equivalently its inverse is continuous). Let $W\subset\mathcal E(\Gamma(E,p))$ be a basic open $\widetilde{s}$ for a local section $s:U\to E$. Then $\varepsilon_{(E,p)}(\widetilde{s})=s(U)$, which is open in $E$ because $s$ is a local inverse of $p$ on $U$ (indeed, $s$ is continuous but need not be an embedding globally; however $s(U)$ is open because $p|_{s(U)}$ is a homeomorphism onto $U$). More directly: since $p|_{s(U)}$ is the inverse of the continuous map $s$, $s(U)$ is homeomorphic to $U$, hence open in $E$ (because $E$ admits a basis consisting of images of local sections). Therefore $\varepsilon_{(E,p)}$ maps basis opens to open sets; hence it is an open map. Therefore $\varepsilon_{(E,p)}$ is a homeomorphism.

Thus $\varepsilon_{(E,p)}$ is an isomorphism in $\mathbf{Et}/X$.

Naturality

Given a morphism $f:(E,p)\to (E',p')$ of étale spaces (i.e. a map over $X$), there is an induced map $\Gamma(f)$ on sheaves of sections and therefore $\mathcal E(\Gamma(f))$ on the étalé constructions. One checks directly that the square

$$ \begin{CD} \mathcal E(\Gamma(E,p)) @>{\varepsilon_{(E,p)}}>> E\\ @V{\mathcal E(\Gamma(f))}VV @VV{f}V\\ \mathcal E(\Gamma(E',p')) @>{\varepsilon_{(E',p')}}>> E' \end{CD} $$

commutes: both compositions send the germ $[s]_x$ to $f(s(x))$. So $\varepsilon$ is a natural transformation and each component is an isomorphism.

We have constructed two functors $\Gamma:\mathbf{Et}/X\to\mathbf{Sh}(X)$ and $\mathcal E:\mathbf{Sh}(X)\to\mathbf{Et}/X$, and natural isomorphisms

$$ \eta:\operatorname{id}_{\mathbf{Sh}(X)}\overset{\simeq}{\Longrightarrow}\Gamma\circ\mathcal E,\qquad \varepsilon:\mathcal E\circ\Gamma\overset{\simeq}{\Longrightarrow}\operatorname{id}_{\mathbf{Et}/X}. $$

Therefore $\Gamma$ and $\mathcal E$ are quasi-inverse equivalences of categories; equivalently,

$$ \mathbf{Sh}(X) \simeq \mathbf{Et}/X. $$

Remarks (clarifying technical points)

Topology on $\mathcal E(\mathcal F)$: the basis $\widetilde{s}$ is well defined and compatible with refinement of sections: if $s$ and $t$ have the same germ at $x$ then in a neighborhood they coincide, so the corresponding basic open neighborhoods coincide near that point. That ensures the topology is Hausdorff or not depending on the sheaf; one does not need Hausdorffness for the construction.
Local homeomorphism property: $\pi:\mathcal E(\mathcal F)\to X$ is by construction locally modeled on the homeomorphisms $\pi|_{\widetilde{s}}:\widetilde{s}\simeq U$. This is the essential geometric fact that makes $\mathcal E(\mathcal F)$ an étale space.
Germs and continuity: two key uses of the sheaf axioms were (i) to deduce injectivity of $\eta$ (identity axiom) and (ii) to perform the gluing necessary to prove surjectivity of $\eta$. The étale space topology was specifically tailored so that the maps $\eta$ and $\varepsilon$ are continuous and inverse to each other.
Set vs. other categories: this equivalence is for sheaves of sets. There are analogous constructions and equivalences for sheaves of other algebraic structures (abelian groups, rings, modules) where one requires the stalks to carry the corresponding algebraic structure and morphisms respect that structure; the étalé space approach requires a bit more structure (local group / local ring objects) if one wants a space representing the sheaf as a bundle with fibres carrying algebraic structure. The classical statement, though, is the sheaves-of-sets ↔ étale-spaces equivalence given above.

Sheaves are equivalent to étale spaces

\(\Gamma:\mathbf{Et}/X\to\mathbf{Sh}(X)\)

\(\mathcal{E}:\mathbf{Sh}(X)\to\mathbf{Et}/X\)