  We delved into the assembly pathways and environments of compact groups (CGs)
of galaxies using mock catalogues generated from semi-analytical models (SAMs)
on the Millennium simulation. We investigate the ability of SAMs to replicate
the observed CG environments and whether CGs with different assembly histories
tend to inhabit specific cosmic environments. We also analyse whether the
environment or the assembly history is more important in tailoring CG
properties. We find that about half of the CGs in SAMs are non-embedded
systems, 40% are inhabiting loose groups or nodes of filaments, while the rest
distribute evenly in filaments and voids, in agreement with observations. We
observe that early-assembled CGs preferentially inhabit large galaxy systems (~
60%), while around 30% remain non-embedded. Conversely, lately-formed CGs
exhibit the opposite trend. We also obtain that lately-formed CGs have lower
velocity dispersions and larger crossing times than early-formed CGs, but
mainly because they are preferentially non-embedded. Those lately-formed CGs
that inhabit large systems do not show the same features. Therefore, the
environment plays a strong role in these properties for lately-formed CGs.
Early-formed CGs are more evolved, displaying larger velocity dispersions,
shorter crossing times, and more dominant first-ranked galaxies, regardless of
the environment. Finally, the difference in brightness between the two
brightest members of CGs is dependent only on the assembly history and not on
the environment. CGs residing in diverse environments have undergone varied
assembly processes, making them suitable for studying their evolution and the
interplay of nature and nurture on their traits.
