  The interplay between magnetism, band topology, and electronic correlation in
low dimensions has been a fascinating subject of research. Here, we propose
two-dimensional (2D) material systems which demonstrate such an interesting
interplay. Based on first-principles calculations and structural search
algorithms, we identify three lowest energy 2D CoSe structures, termed as the
$\alpha$-, $\beta$-, and $\gamma$-CoSe. {We show that $\alpha$- and
$\beta$-CoSe are two rare examples of 2D antiferromagnetic metals, which are
related to their Fermi surfaces nesting features, and meanwhile, $\gamma$-CoSe
is a ferromagnetic metal. They possess a range of interesting physical
properties, including anisotropic valleys connected by crystalline symmetries,
strain-tunable valley polarization, strain-induced metal-semiconductor and/or
magnetic phase transitions, as well as topological band features such as the
magnetic Weyl point and the magnetic Weyl loop. Remarkably, all the topological
features here are robust against spin-orbit coupling.} Some experimental
aspects of our predictions have been discussed.
