  We introduce a new topological invariant of a rigidly-compactly generated
tensor-triangulated category and two new notions of support. The first is based
on smashing subcategories: it is unknown whether the frame of smashing
subcategories is spatial in general, but supposing it is as in the examples we
understand, we show it is equipped with a morphism to the Balmer spectrum which
detects the failure of the telescope conjecture and we develop the
corresponding support theory. The new invariant, the big spectrum, results from
taking the entire collection of localizing ideals seriously and considering
prime localizing ideals. Although there are, in principle, a proper class of
localizing ideals, we are able to prove the existence of at least one big prime
lying over every Balmer prime. We conclude with a pair of examples illustrating
our constructions.
