Given an ane apartment with action of the ane Weyl group, we can look at compact convex subsets whose boundaries are re ection hyperplanes. The category of these \enclosures" has a natural Grothendieck topology, whose sheaves constitute a natural environment in which to consider spaces which look like buildings or subsets thereof. This provides a technical tool, of a rather combinatorial nature, in our ongoing work with Katzarkov, Noll and Pandit on the construction of pre-buildings and versal harmonic maps.