  The Mu3e experiment searches for the lepton flavor violating decay
$\mu^+~\rightarrow~e^+~e^+~e^-$ with an ultimate aimed sensitivity of 1 event
in $10^{16}$ decays. This goal can only be achieved by reducing the material
budget per tracking layer to $X/X_0 \approx 0.1 \%$. High-Voltage Monolithic
Active Pixel Sensors (HV-MAPS) which are thinned to $50\ \mu m$ serve as
sensors. Gaseous helium is chosen as coolant.
  Results of recent studies related to the sensor prototypes, the helium
cooling, and module prototyping are presented. The recent chip submission
MuPix10 has proven its functionality regarding efficiency and time resolution.
The helium cooling system for the inner tracker could be verified using a
full-scale prototype. A complete prototype equipped with MuPix10 chips will be
tested inside the Mu3e magnet in summer 2021.
