What is the extra property of this sheaf? - MathOverflow

What is the extra property of this sheaf?

2011-04-02T19:25:44Z

I have a particular mathematical structure, and I think it would be enlightening to try to place it in a categorical context.

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 not overlapping.

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.

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 $\mathcal{F}((r,t))$.

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.

Answer by Simon Rose for What is the extra property of this sheaf?

2011-04-02T20:05:26Z

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 $$ \mathcal{F}(I) \cong \bigoplus_{i \in \mathbb{N}} \mathcal{F}(I). $$

It seems to me that such sheaves would be very badly behaved, unless they are trivial.

Answer by Inna for What is the extra property of this sheaf?

2011-04-05T02:04:28Z

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 $\{U_i \rightarrow U\}_{i\in I}$ is a covering family if $\bigcup_{i\in I} U_i = U$. If instead we say that $\{U_i\rightarrow U\}_{i\in I}$ is a covering family if $\bigcup_{i\in I}U_i$ is dense in $U$ you'll get the site you're looking for.