Final cooling for a high-energy high-luminosity lepton collider

A high-energy muon collider requires a "final cooling" system that reduces transverse emittance by a factor of similar to 10, while allowing the longitudinal emittance to increase. The baseline approach has low-energy transverse cooling within high-field solenoids, with strong longitudinal heating. This approach and its recent simulation are discussed. Alternative approaches, which more explicitly include emittance exchange are also presented. Round-to-flat beam transform, transverse slicing, and longitudinal bunch coalescence are possible components of an alternative approach. Wedge-based emittance exchange could provide much of the required transverse cooling with longitudinal heating. Li-lens and quadrupole focusing systems could also provide much of the required final cooling.