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.

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.