Relative FP-injective and FP-flat complexes and their model structures

The present article studies homological and homotopical aspects of FPn-injective and FPn-flat complexes, and describes homological dimensions associated to them. After establishing a relation between these dimensions via Pontrjagin duality, we construct (pre)covers and (pre)envelopes by complexes with finite FPn-injective and FPn-flat dimensions, thus setting up the bases for an approximation theory relative to complexes of finite type. We also construct on the category of complexes several model structures from modules and complexes with finite FPn-injective and FPn-flat dimensions, and analyze some situations where it is possible to connect these structures via Quillen functors.