  We initiate a systematic study of lattices of thick subcategories for
arbitrary essentially small triangulated categories. To this end we give
several examples illustrating the various properties these lattices may, or may
not, have and show that as soon as a lattice of thick subcategories is
distributive it is automatically a spatial frame. We then construct two
non-commutative spectra, one functorial and one with restricted functoriality,
that give universal approximations of these lattices by spaces.
