  The stellar mass assembly of galaxies can be affected by both secular and
environmental processes. In this study, for the first time, we investigate the
stellar mass assembly of $\sim90,000$ low redshift, central galaxies selected
from SDSS group catalogues (M$_{\rm Stellar}\gtrsim10^{9.5}$M$_{\odot}$,
M$_{\rm Halo}\gtrsim10^{12}$M$_{\odot}$) as a function of both stellar and halo
mass. We use estimates of the times at which 10, 50 and 90 per cent of the
stellar mass was assembled from photometric spectral energy distribution
fitting, allowing a more complete investigation than single stellar ages alone.
We consider trends in both stellar and halo mass simultaneously, finding
dependencies of all assembly times on both. We find that galaxies with higher
stellar masses (at constant halo mass) have on average older lookback times,
similar to previous studies of galaxy assembly. We also find that galaxies at
higher halo mass (at constant stellar mass) have younger lookback times,
possibly due to a larger reservoir of gas for star formation. An exception to
this is a sub sample with high stellar-to-halo mass ratios, which are likely
massive, field spirals. We compare these observed trends to those predicted by
the TNG300 simulation, finding good agreement overall as a function of either
stellar or halo mass. However, some differences in the assembly times (of up to
$\sim 3$ Gyr) appear when considering both stellar and halo mass
simultaneously, noticeably at intermediate stellar masses (M$_{\rm Stellar}
\sim 10^{11}$ M$_{\odot}$). These discrepancies are possibly linked to the
quenched fraction of galaxies and the kinetic mode AGN feedback implemented in
TNG300.
