  In the setting of compactly generated triangulated categories, we show that
the heart of a (co)silting t-structure is a Grothendieck category if and only
if the (co)silting object satisfies a purity assumption. Moreover, in the
cosilting case the previous conditions are related to the coaisle of the
t-structure being a definable subcategory. If we further assume our
triangulated category to be algebraic, it follows that the heart of any
nondegenerate compactly generated t-structure is a Grothendieck category.
