  The dynamics of band-gap renormalization and gain build-up in monolayer
MoTe$_2$ is investigated by evaluating the non-equilibrium Dirac-Bloch
equations with the incoherent carrier-carrier and carrier-phonon scattering
treated via quantum-Boltzmann type scattering equations. For the case where an
approximately $300$ fs-long high intensity optical pulse generates
charge-carrier densities in the gain regime, the strong Coulomb coupling leads
to a relaxation of excited carriers on a few fs time scale. The pump-pulse
generation of excited carriers induces a large band-gap renormalization during
the time scale of the pulse. Efficient phonon coupling leads to a subsequent
carrier thermalization within a few ps, which defines the time scale for the
optical gain build-up energetically close to the low-density exciton resonance.
