\textsc{CompaSO}: A new halo finder for competitive assignment to
  spherical overdensities
  We describe a new method (\textsc{CompaSO}) for identifying groups of
particles in cosmological $N$-body simulations. \textsc{CompaSO} builds upon
existing spherical overdensity (SO) algorithms by taking into consideration the
tidal radius around a smaller halo before competitively assigning halo
membership to the particles. In this way, the \textsc{CompaSO} finder allows
for more effective deblending of haloes in close proximity as well as the
formation of new haloes on the outskirts of larger ones. This halo-finding
algorithm is used in the \textsc{AbacusSummit} suite of $N$-body simulations,
designed to meet the cosmological simulation requirements of the Dark Energy
Spectroscopic Instrument (DESI) survey. \textsc{CompaSO} is developed as a
highly efficient on-the-fly group finder, which is crucial for enabling good
load-balancing between the GPU and CPU and the creation of high-resolution
merger trees. In this paper, we describe the halo-finding procedure and its
particular implementation in \Abacus{Abacus}, accompanying it with a
qualitative analysis of the finder. {We test the robustness of the
\textsc{CompaSO} catalogues before and after applying the cleaning method
described in an accompanying paper and demonstrate its effectiveness by
comparing it with other validation techniques.} We then visualise the haloes
and their density profiles, finding that they are well fit by the NFW
formalism. Finally, we compare other properties such as radius-mass
relationships and two-point correlation functions with that of another widely
used halo finder, \textsc{ROCKSTAR}.
