I believe you would be correct on the first point, but have the second backwards. If V' was instead a chunk without the player, then res^V_V' Γ(V,F) = ∅, and the sections of sheaves are supported on {p} (as well as on any other entity with a valid chunk assignment). These sections would basically be functions providing the chunk containing a given entity, supported on the collection of all entities that actually exist on the map. Any other would give you 0 in the constructed top. space (no chunks if you don't exist).
