What is the extra property of this sheaf? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:17:59Z http://mathoverflow.net/feeds/question/60380 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60380/what-is-the-extra-property-of-this-sheaf What is the extra property of this sheaf? Tom Ellis 2011-04-02T19:25:44Z 2011-04-05T02:04:28Z <p>I have a particular mathematical structure, and I think it would be enlightening to try to place it in a categorical context. </p> <p>The structure is a sheaf on a topological space, and the extra property is that not only can we patch together data from overlapping open sets, we can also do it sometimes when the open sets are <em>not</em> overlapping. </p> <p>For example, I can take the data assigned to the open intervals $(r,s)$ and $(s,t)$ and combine them to recover uniquely the data assigned to the interval $(r,t)$. In one dimension this all seems quite simple, but in higher dimensions the class of disjoint sets that you can patch together can be quite complicated. </p> <p>I suppose in more categorical language we would say that the sheaf $\mathcal{F}$ satisfies $\mathcal{F}((r,s) \cup (s,t))$ is canonically isomorphic to $\mathcal{F}((r,t))$.</p> <p>Is this kind of thing a known specialisation of a sheaf? A sheaf on something other than a topological space? A different sheaf-like object? I'm trying to work out what's the morally correct'' framework in which to study these objects that I have.</p> http://mathoverflow.net/questions/60380/what-is-the-extra-property-of-this-sheaf/60384#60384 Answer by Simon Rose for What is the extra property of this sheaf? Simon Rose 2011-04-02T20:05:26Z 2011-04-02T20:05:26Z <p>Such a sheaf is certainly not finitely generated, at least over $\mathbb{R}$. Since $\mathcal{F}(I_1 \cup I_2) \cong \mathcal{F}(I_1) \oplus \mathcal{F}(I_2)$, we can split up an interval, say $(0,1)$ into $(0,1/2), (1/2, 1)$ and continually subdivide this up; as $\mathcal{F}(I) \cong \mathcal{F}(J)$ for any bounded open intervals, we necessarily have $$\mathcal{F}(I) \cong \bigoplus_{i \in \mathbb{N}} \mathcal{F}(I).$$</p> <p>It seems to me that such sheaves would be very badly behaved, unless they are trivial.</p> http://mathoverflow.net/questions/60380/what-is-the-extra-property-of-this-sheaf/60634#60634 Answer by Inna for What is the extra property of this sheaf? Inna 2011-04-05T02:04:28Z 2011-04-05T02:04:28Z <p>This seems like a perfectly reasonable sheaf, just not on the normal site associated to the topological space. When we look at the site associated to a topological space $X$ we take the poset of open subsets of $X$ under inclusion. To define the topology, we say that <code>$\{U_i \rightarrow U\}_{i\in I}$</code> is a covering family if $\bigcup_{i\in I} U_i = U$. If instead we say that <code>$\{U_i\rightarrow U\}_{i\in I}$</code> is a covering family if $\bigcup_{i\in I}U_i$ is dense in $U$ you'll get the site you're looking for.</p>