  We present constraints on the formation and evolution of early-type galaxies
(ETGs) with the empirical model EMERGE. The parameters of this model are
adjusted so that it reproduces the evolution of stellar mass functions,
specific star formation rates, and cosmic star formation rates since
$z\approx10$ as well as 'quenched' galaxy fractions and correlation functions.
We find that at fixed halo mass present-day ETGs are more massive than
late-type galaxies, whereas at fixed stellar mass ETGs populate more massive
halos in agreement with lensing results. This effect naturally results from the
shape and scatter of the stellar-to-halo mass relation and the galaxy formation
histories. The ETG stellar mass assembly is dominated by 'in-situ' star
formation below a stellar mass of $3\times10^{11}\mathrm{M}_\odot$ and by
merging and accretion of 'ex-situ' formed stars at higher mass. The mass
dependence is in tension with current cosmological simulations. Lower mass ETGs
show extended star formation towards low redshift in agreement with recent
estimates from IFU surveys. All ETGs have main progenitors on the 'main
sequence of star formation' with the 'red sequence' appearing at $z \approx 2$.
Above this redshift, over 95 per cent of the ETG progenitors are star-forming.
More than 90 per cent of $z \approx 2$ 'main sequence' galaxies with $m_* >
10^{10}\mathrm{M}_\odot$ evolve into present-day ETGs. Above redshift 6, more
than 80 per cent of the observed stellar mass functions above
$10^{9}\mathrm{M}_\odot$ can be accounted for by ETG progenitors with $m_* >
10^{10}\mathrm{M}_\odot$. This implies that current and future high redshift
observations mainly probe the birth of present-day ETGs. The source code and
documentation of EMERGE are available at github.com/bmoster/emerge.
