I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu.

Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open covers $\{U_i\rightarrow X\}$.

The result I am trying to understand is the following.

Let $F$ be a sheaf over $\mathfrak{S}$. Let $X$ be a manifold and $F\rightarrow X$ be a morphism. Suppose that $\{U_i\rightarrow X\}$ is a cover for $X$ and that for every $i$, the sheaf $F_i=U_i\times_XF$ is representable. Then $F$ is representable.

A sheaf $F:\mathfrak{S}^{op}\rightarrow (Set)$ defines a groupoid fibration (category fibered in groupoids $\mathcal{D}$) over $\mathfrak{S}$ as follows. Objects of $\mathcal{D}$ are $(U,x)$ such that $U$ is an object of $\mathfrak{S}$ and $x\in F(U)$. Given objects $(U,x)$ and $(V,y)$, a morphism $(U,x)\rightarrow (V,y)$ consists of a smooth map $f:U\rightarrow V$ such that $F(f):F(V)\rightarrow F(U)$ takes $y$ to $x$ i.e., $F(f)(y)=x$.

As $F_i$ is representable, there exists a manifold $Y_i$ representing $F_i$. I am trying to use that $F$ is a sheaf to glue these manifolds $Y_i$ to produce a manifold $Y$ and an isomorphism $F\rightarrow Y$ without success.

Any hints are welcome.

  • $\begingroup$ Did you consider the fact that multiple intersections are also representable (since open embeddings are representable maps) to construct the gluing? $\endgroup$ – Denis Nardin Jan 2 at 8:53
  • $\begingroup$ By multiple intersections you mean $U_{ij}\times_X F$?? @DenisNardin $\endgroup$ – Praphulla Koushik Jan 2 at 9:20
  • $\begingroup$ Indeed, precisely. $\endgroup$ – Denis Nardin Jan 2 at 10:01
  • $\begingroup$ @DenisNardin Yes. Even Kai Behrend paper tries in that way but I am not very happy in proceeding that way.. So, trying to come up with something on my own (with help from this site as I already posted the question :D) $\endgroup$ – Praphulla Koushik Jan 2 at 10:16

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