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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 16th 2012
   • (edited Jan 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI would like to play the following game, eventually in higher toposes, but to warmup just for ordinary toposes and simple special cases there: _Suppose we have given the notion of [[covering space]] / [[locally constant sheaf]]. Express the more general notion of [[étale space]] / [[sheaf]] in terms of it._ What I am trying to get at should be standard, but I am not sure I have seen it. Maybe it's just too late at night for me. Anyway, consider the following kind of setup: let $Q \to X$ be an [[étale space]] over a manifold $X$ Write $\Gamma_X(Q)$ for its sheaf of sections. Let $\{U_i\}$ be a cover, maybe even the set of _all_ open subsets of $X$. For each $U_i$ we can form the trivial covering space $U_i \times \Gamma_X(Q)(U_i) \to U_i$ (any fiber is the discrete set of sections of $Q \to X$ over $U_i$). And this sits in a canonical diagram $$ \array{ U_i \times \Gamma_X(Q)(U_i) &\to& Q \\ \downarrow && \downarrow \\ U_i &\hookrightarrow& X } \,, $$ where the top morphism on $U_i \times \{\sigma\}$ for a given section $\sigma$ is the composite $$ \array{ U_i \times \{\sigma\} &\to& Q \\ \downarrow &{}^\sigma\nearrow& \downarrow \\ U_i &\hookrightarrow& X } \,. $$ Taking the coproduct over all the $U_i$s, we get a total diagram $$ \array{ \coprod_i ( U_i \times \Gamma_X(Q)(U_i)) &\to& Q \\ \downarrow && \downarrow \\ \coprod_i U_i &\to& X } \,. $$ This has the following properties: * the left morphism is a covering space / the étale morphism corresponding to a locally constant sheaf; * the two horizontal morphisms are epis (in the category of sheaves on manifolds). So we have "covered the étale map by a covering map". It seems that a converse to this holds: if a map $Q \to X$ can be covered in this fashion, then it must be étale.

   I would like to play the following game, eventually in higher toposes, but to warmup just for ordinary toposes and simple special cases there:

   Suppose we have given the notion of covering space / locally constant sheaf. Express the more general notion of étale space / sheaf in terms of it.

   What I am trying to get at should be standard, but I am not sure I have seen it. Maybe it’s just too late at night for me.

   Anyway, consider the following kind of setup: let $Q \to X$ be an étale space over a manifold $X$ Write $\Gamma_X(Q)$ for its sheaf of sections. Let $\{U_i\}$ be a cover, maybe even the set of all open subsets of $X$.

   For each $U_i$ we can form the trivial covering space $U_i \times \Gamma_X(Q)(U_i) \to U_i$ (any fiber is the discrete set of sections of $Q \to X$ over $U_i$). And this sits in a canonical diagram

   $$\array{ U_i \times \Gamma_X(Q)(U_i) &\to& Q \\ \downarrow && \downarrow \\ U_i &\hookrightarrow& X } \,,$$

   where the top morphism on $U_i \times \{\sigma\}$ for a given section $\sigma$ is the composite

   $$\array{ U_i \times \{\sigma\} &\to& Q \\ \downarrow &{}^\sigma\nearrow& \downarrow \\ U_i &\hookrightarrow& X } \,.$$

   Taking the coproduct over all the $U_i$s, we get a total diagram

   $$\array{ \coprod_i ( U_i \times \Gamma_X(Q)(U_i)) &\to& Q \\ \downarrow && \downarrow \\ \coprod_i U_i &\to& X } \,.$$

   This has the following properties:

   • the left morphism is a covering space / the étale morphism corresponding to a locally constant sheaf;

   • the two horizontal morphisms are epis (in the category of sheaves on manifolds).

   So we have “covered the étale map by a covering map”.

   It seems that a converse to this holds: if a map $Q \to X$ can be covered in this fashion, then it must be étale.