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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.

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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.

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To amplify Inna's answer, your sheaf $\mathcal{F}$ on $X$ is even a sheaf for the Lawvere–Tierney topology on $\mathrm{Sh}(X)$ given by the double-negation modality. In other words, your sheaf is the pushforward of some sheaf along the embedding $X^{\neg\neg} \hookrightarrow X$, where $X^{\neg\neg}$ is the smallest dense sublocale of $X$. (Recall that usually, there is no smallest dense sub_space_.)

So, the answer to your question "A sheaf on something other than a topological space?" is yes. It's a sheaf on a certain sublocale of your topological space.

I can provide details if wanted.

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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.

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    $\begingroup$ In one dimension the condition just means that removing one point doesn't change the sections. For example $U\mapsto L^p(U)$ seems to have this property, and it's hard to call $L^p(\mathbb{R})$ "badly behaved"... $\endgroup$
    – Tom Church
    Apr 2, 2011 at 20:24
  • $\begingroup$ Well, those aren't finitely generated... Point taken though. Not all such sheaves are horrible beasts. $\endgroup$
    – Simon Rose
    Apr 2, 2011 at 22:41
  • $\begingroup$ @Simon: yes my sheaves have this property. In fact one way of seeing these sheaves is that each $\mathcal{F}(I)$ is a probability space, and then your observation is (roughly) that the distributions are infinitely divisible (classically these are Levy processes, i.e. Brownian motions etc.). $\endgroup$
    – Tom Ellis
    Apr 2, 2011 at 22:43
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    $\begingroup$ @TomChurch: Perhaps you should take $U\mapsto L^p_{\mathrm{loc}}(U)$ instead, otherwise it's not a sheaf: $\exp | _{-\infty , x}$ are in $L^p(-\infty , x)$, and $\mathbb{R}$ is the union of the $(-\infty , x)$ but $\exp$ is not in $L^p(\mathbb{R})$. $\endgroup$
    – Qfwfq
    Apr 2, 2011 at 23:53
  • $\begingroup$ @unknowngoogle: good point; you're right. $\endgroup$
    – Tom Church
    Apr 5, 2011 at 14:50

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