A motivation for derived field theory comes from the deformation of the algebraic input data. To study such one first needs to specify the ""ambient space"" of the deformation. We study the case of tensor categories and investigate retracts of tensor categories A in larger tensor categories A_z parametrized by objects z in the Drinfeld center of A. In case A is semisimple, the categories A_z correspond to (non-semisimple) categories of quiver modules. Thus the theory provides a construction of quivers with a rigid monoidal structure on their modules. For such fusion quivers, the moduli spaces carry interesting structures. As applications we consider relations on the quiver modules that are compatible with the rigid monoidal structures and study preprojective algebras.