We consider TTF-classes which are not triangulated but just
cosuspended,
i.e., subcategories V of T for which there is a (nondegenerate)
t-structure
(U,V) and a co-t-structure (V,W). Examples of such TTF-triples
(U,V,W) arise
from compactly generated t-structures or from certain cosilting
t-structures
in derived module categories. In this talk, we show that in a
compactly
generated triangulated category, a large class of these TTF-triples
can be
parametrised by pure-injective silting objects. Moreover, it turns
out that
the heart of such a cosilting t-structure is a Grothendieck category
and, as
a consequence, nondegenerate compactly generated t-structures have
Grothendieck hearts. This talk is based on joint work with Lidia
Angeleri Hügel and Frederik Marks.