KiDS-1000 catalogue: Redshift distributions and their calibration
  We present redshift distribution estimates of galaxies selected from the
fourth data release of the Kilo-Degree Survey over an area of $\sim1000$
deg$^2$ (KiDS-1000). These redshift distributions represent one of the crucial
ingredients for weak gravitational lensing measurements with the KiDS-1000
data. The primary estimate is based on deep spectroscopic reference catalogues
that are re-weighted with the help of a self-organising map (SOM) to closely
resemble the KiDS-1000 sources, split into five tomographic redshift bins in
the photometric redshift range $0.1<z_\mathrm{B}\le1.2$. Sources are selected
such that they only occupy that volume of nine-dimensional magnitude-space that
is also covered by the reference samples (`gold' selection). Residual biases in
the mean redshifts determined from this calibration are estimated from mock
catalogues to be $\lesssim0.01$ for all five bins with uncertainties of $\sim
0.01$. This primary SOM estimate of the KiDS-1000 redshift distributions is
complemented with an independent clustering redshift approach. After validation
of the clustering-$z$ on the same mock catalogues and a careful assessment of
systematic errors, we find no significant bias of the SOM redshift
distributions with respect to the clustering-$z$ measurements. The SOM redshift
distributions re-calibrated by the clustering-$z$ represent an alternative
calibration of the redshift distributions with only slightly larger
uncertainties in the mean redshifts of $\sim 0.01-0.02$ to be used in KiDS-1000
cosmological weak lensing analyses. As this includes the SOM uncertainty,
clustering-$z$ are shown to be fully competitive on KiDS-1000 data.
