\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}.