Apologies for the blog silence! I’d like to get this thing back underway with a relatively brief post outlining one of the really weird things which happens with sheaves. After this post, I’d like to spend some time moving forward to talking about schemes and possibly a return to more concrete areas with a first look at elliptic curves.

First, suppose X is a topological space with a base of open sets \mathcal{B}. If \mathcal{C} is a target category (think sets or groups) and we are given objects F(U) \in \mathcal{C} and morphisms \rho^U_{V} :F(U) \rightarrow F(V) for each inclusion of basic open sets V\subseteq U such that these objects and morphisms obey the sheaf conditions with respect to inclusions and covering of basic opens by basic opens, we call F a \mathcal{B}-sheaf on X. A nice result is that every \mathcal{B}-sheaf extends uniquely (up to isomorphism!) to a sheaf on X; similarly each morphism of \mathcal{B}-sheaves (a collection of morphisms \phi_U : F(U) \rightarrow G(U) for each basic open set U \in \mathcal{B} commuting with restrictions of the basic open sets) extends uniquely to a morphism of sheaves \phi: F\rightarrow G on X. This is all obtained by setting, for an open set U\subseteq XF(U) = \lim_{\substack{V\subseteq U \\ V \in\mathcal{B}}} F(V) . The properties above are ensured by the UMP of the limit.

Now suppose \mathcal{U} is an open cover for a topological space X. Suppose for each U\in\mathcal{U} we have a sheaf (in whatever category you want – sets, say) F_U \in \text{Sh}(U). Furthermore suppose we are given isomorphisms F_U \vert_{U\cap V} \xrightarrow{\phi_{VU}} F_V \vert_{U\cap V} such that \phi_{WV} \circ \phi_{VU} = \phi_{WU} on the intersection U\cap V\cap W.

Then there is a unique sheaf F on X whose restriction F\vert_U to each U\in\mathcal{U} is isomorphic to F_U via isomorphisms \psi_U : F\vert_U \rightarrow F_U such that \phi_{VU} \circ \psi_U \vert_{U\cap V} = \psi_V \vert_{U\cap V} for all open sets U, V\in\mathcal{U}

To see this, define a base for the topology on X – set \mathcal{B} as collection of open sets contained in some open set in the cover \mathcal{U}. For each V\in\mathcal{B} arbitrarily choose some U\in\mathcal{U} containing V and set $F(V) = F_U (V)$. The conditions above allow us to make well-defined restriction maps. This gives us a \mathcal{B}-sheaf, which extends uniquely to a sheaf.

I didn’t really spell out how these maps compose correctly, but it isn’t too tricky to check everything works out. The important thing is what is actually said here. Starting with sheaves on each open set in a covering of a topological space which are isomorphic (not necessarily equal) on the overlaps between those open sets we obtain a unique sheaf on the space which, when restricted to each open set in the cover, is isomorphic to one of the original sheaves. This exactly mirrors what happens a level down internally to sheaves – given sections s_U \in F(U) for each open set U in a covering of a topological space X which are equal on overlaps, there is a global section s\in F(X) such that the restriction of s to U is equal to s_U.

This is quite freaky, because it looks like the sheaves defined on open covers of a space behave similarly (although not identically) to the sections internal to all those sheaves. In other words, the set of sheaves defined on open covers of a topological space is almost a sheaf itself! The technical difference is how at this level our sheaves and their restrictions are not equal to each other but rather isomorphic. This leads one already into a 2-categorical “stacky” world which I’m not quite ready to deal with at the moment. But it is an interesting point to which I would like to return soon…

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