We give a definition of the action of a tensor triangulated category T on a
triangulated category K. In the case that T is rigidly-compactly generated and
K is compactly generated we show this gives rise to a notion of supports which
categorifies work of Benson, Iyengar, and Krause and extends work of Balmer and
Favi. We prove that a suitable version of the local-to-global principle holds
very generally. A relative version of the telescope conjecture is formulated
and we give a sufficient condition for it to hold.Comment: 33 pages, to appear in Journal f\"ur die reine und angewandte
Mathemati
