  Results of the first principles study on the electronic structure and
magnetism of the superconducting weak ferromagnet Y$_4$Co$_3$, are presented.
Using the full potential Korringa-Kohn-Rostoker (FP-KKR) method, densities of
states, dispersion curves and magnetic moments were calculated for
quasi-ordered structural model of the compound in the framework of the local
density approximation. Spin-polarized KKR calculations confirm that weak
ferromagnetic properties of Y$_4$Co$_3$ can be attributed to only one cobalt
atom located on (2b) site in the unit cell, while other twenty Co and Y atoms
acts as a diamagnetic environment. Moreover, the magnetic Co atoms form a
quasi-one-dimensional chains along $z$ direction. The magnitude of Co(2b)
magnetic moment ($0.55~\mu_B$) markedly overestimates the experimental value
(0.23~$\mu_B$), which suggests the importance of spin fluctuations in this
system. Calculated distribution of spin magnetization in the unit cell provides
a background for discussion of the coexistence of ferromagnetism and
superconductivity in Y$_4$Co$_3$. Finally, the effect of pressure on magnetism
is discussed and compared with experimental data, also supporting weak
ferromagnetic behaviors in the system.
