Complexes of stable birational invariants

We introduce a new stable birational invariant, which takes the form of a functor sending a degenerating variety to the homotopy type of a chain complex. Our invariant is a categorification of the motivic volume of Nicaise and Shinder. From the class of the chain complex in a Grothendieck group, we obtain a motivic obstruction to retract rationality, valued in a quotient of the Grothendieck ring of varieties. In addition, we construct a general class of stable birational invariants, with the invariant above as the universal example, given by applying any chosen stable birational invariant (e.g., unramified cohomology) to the strata of a semistable degeneration.