  We study Karoubian tensor categories which interpolate representation
categories of families of so-called easy quantum groups in the same sense in
which Deligne's interpolation categories $\mathrm{\underline{Rep}}(S_t)$
interpolate the representation categories of the symmetric groups. As such
categories can be described using a graphical calculus of partitions, we call
them interpolating partition categories. They include
$\mathrm{\underline{Rep}}(S_t)$ as a special case and can generally be viewed
as subcategories of the latter. Focusing on semisimplicity and descriptions of
the indecomposable objects, we prove uniform generalisations of results known
for special cases, including $\mathrm{\underline{Rep}}(S_t)$ or Temperley--Lieb
categories. In particular, we identify those values of the interpolation
parameter which correspond to semisimple and non-semisimple categories,
respectively, for all so-called group-theoretical easy quantum groups. A
crucial ingredient is an abstract analysis of certain subobject lattices
developed by Knop, which we adapt to categories of partitions. We go on to
prove a parametrisation of the indecomposable objects in all interpolating
partition categories for non-zero interpolation parameters via a system of
finite groups which we associate to any partition category, and which we also
use to describe the associated graded rings of the Grothendieck rings of these
interpolation categories.

Representation Theory;Category Theory;Operator Algebras;Quantum Algebra