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.