We extend the well-known construction of the Grothendieck ring of varieties to categories whose objects can be partitioned into predefined strata (i.e. stratifiable spaces). To this end, we introduce "Lego categories" which are subcategories of the category of spaces stratifiable by locally compact strata in Euclidean space. Restricting to strata that are cohomologically of finite type, as well as their closures, we construct a well-defined motivic morphism on the associated Grothendieck ring which coincides with the Euler characteristic with compact supports on locally compact spaces.
This "categorification" allows streamlined combinatorial derivations of Euler characteristics, be they topological or with compact supports. Main applications pertain to spaces stratified by configuration spaces, with new results including the computation of the Grothendieck class of graph configuration spaces, of orbit configuration spaces and of finite subset spaces.