  We study the faint radio population using wide-field very long baseline
interferometry (VLBI) observations of 2865 known radio sources in the Cosmic
Evolution Survey (COSMOS) field. The main objective of the project was to
determine where active galactic nuclei (AGN) are present. The combination of
number of sources, sensitivity, angular resolution and area covered by this
project are unprecedented. We have detected 468 radio sources, expected to be
AGNs, with the Very Long Baseline Array (VLBA) at 1.4 GHz. This is, to date,
the largest sample assembled of VLBI detected sources in the sub-mJy regime.
The input sample was taken from previous observations with the Very Large Array
(VLA). We present the catalogue with additional multiwavelength information. We
find a detection fraction of 20%, considering only those sources from the input
catalogue which were in principle detectable with the VLBA (2361). As a
function of redshift, we see no evolution of the detection fraction over the
redshift range 0.5<z<3. In addition, we find that faint radio sources typically
have a greater fraction of their radio luminosity in a compact core: ~70% of
the sub-mJy sources detected with the VLBA have more than half of their total
radio luminosity in a VLBI-scale component, whereas this is true for only ~30%
of the sources that are brighter than 10 mJy. This suggests that fainter radio
sources differ intrinsically from brighter ones. Across our entire sample, we
find the predominant morphological classification of the host galaxies of the
VLBA detected sources to be early type (57%), although this varies with
redshift and at z>1.5 we find that spiral galaxies become the most prevalent
(48%). We demonstrate that wide-field VLBI observations, together with new
calibration methods such as multi-source self-calibration and mosaicing, result
in information which is difficult or impossible to obtain otherwise.
