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Suppose I have a topological space $X$. Let $\mathcal{O}(X)$ denote the poset of open subsets. There is a canonical functor $\mathcal{O}(X) \to Top/X$ which sends an open $U \in \mathcal{O}(X)$ to $U \hookrightarrow X$. By left-Kan extension, this produces an adjunction between $Set^{\mathcal{O}(X)^{op}}$ and $Top/X$. The left-adjoint, $L$, is precisely the etale space construction, whereas the right adjoint, $\Gamma$, is the "sheaf of sections" functor. $\Gamma \circ L$ is sheafification. By abstract nonsense, this adjunction restricts to an equivalence between, one one hand, the subcategory where the unit is an iso, and the other hand, the subcategory where the counit is an iso. The former is easily seen to be the category of sheaves over $X$. I know the later is the category of etale spaces over $X$, i.e. maps $Y \to X$ which are local homeomorphisms. This is traditionally proven usually an explicit description of the etale space of a sheaf, topologizing the germs of local sections etc. However, is there a way to see this at a higher level of abstraction, only appealing to the abstract definition of this induced adjunction and abstract properties of local homeomorphisms?

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Look up "local homemorphism (of topoi)" in the Elephant. It gives the topos-theoretic approach to the construction of the espace etale. –  Harry Gindi Aug 18 '10 at 23:21
Page 651, by the by. –  Harry Gindi Aug 19 '10 at 2:10

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up vote 5 down vote accepted

Here is a sketch of why I think the condition that $Y$ is a local homeomorphism over $X$ should be sufficient for the counit to be a homeomorphism. I haven't worked out the converse yet.

For a presheaf $F \in Set^{\mathcal{O}(X)^{op}}$, the formula for the left Kan extension should be $$L(F) = \mathrm{colim}_{y(U) \rightarrow F} U$$ where $y : \mathcal{O}(X) \rightarrow Set^{\mathcal{O}(X)^{op}}$ is the Yoneda embedding. By Yoneda's lemma, the indexing category for the colimit is exactly the category of elements of $F$ which I will write as $\int F$.

Now, consider the case where $F = \Gamma_Y$ for some space $p: Y \rightarrow X$ and assume that $p$ is a local homeomorphism. We have $\Gamma_Y(U) = \{\sigma : U \rightarrow Y | p \circ \sigma = \mathrm{id}_U \}$. So the objects in the category $\int \Gamma_Y$ are exactly the sections over the various open sets of $X$, and the morphisms are given by restriction of sections. I'll write $d(\sigma)$ for the domain of a given section.

Our Kan extension formula becomes $$L(\Gamma_Y) = \mathrm{colim}_{\sigma \in \int \Gamma_Y} d(\sigma)$$ From here it's easy to see what the counit is: since our object is given by a colimit, it suffices to construct a map $d(\sigma) \rightarrow Y$ for each $\sigma \in \int \Gamma_Y$. But clearly $\sigma$ itself qualifies.

Now choose an open covering $\{V_\alpha\}$ of $Y$ such that $p$ restricts to a homeomorphism on each $V_\alpha$. We then have a collection $\{\sigma_\alpha : p(V_\alpha) \rightarrow V_\alpha\}$ of sections by choosing the inverse to each restriction. My claim would be that this collection is cofinal (or final? I can never remember which) in the category $\int \Gamma_Y$ so that we can restrict our colimit to just this subcategory. Notice that in this case, the components of the counit map above are homeomorphisms.

Moreover, this subcategory should also be cofinal in $\mathcal{O}(Y)$ by associating $\sigma_\alpha$ with the open set $V_\alpha$. Then the fact that the counit is a homeomorphism should be the statement that a topological space is the colimit of any of its open coverings.

Is this along the lines of what you were thinking?

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No! Please don't use "etale" to modify the word "morphism" in this context! I'm sure you're familiar with the confusion arising from that very bad mistranslation (cf. The Elephant). –  Harry Gindi Aug 18 '10 at 23:23
Sorry. I did not realize this was a point of confusion. I have changed it above. –  Eric Finster Aug 18 '10 at 23:45
Off topic: what is "The Elephant"? Google only turns up novelty stories of zoo animals. –  Ryan Reich Aug 18 '10 at 23:50
Sketches of an Elephant: A Topos Theory Compendium by Johnstone –  Matt Aug 19 '10 at 0:04
It's a book by Peter Johnstone called "Sketches of an elephant: a topos theory compendium" –  Eric Finster Aug 19 '10 at 0:05

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