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.