  A subunit in a monoidal category is a subobject of the monoidal unit for
which a canonical morphism is invertible. They correspond to open subsets of a
base topological space in categories such as those of sheaves or Hilbert
modules. We show that under mild conditions subunits endow any monoidal
category with a kind of topological intuition: there are well-behaved notions
of restriction, localisation, and support, even though the subunits in general
only form a semilattice. We develop universal constructions completing any
monoidal category to one whose subunits universally form a lattice, preframe,
or frame.
