  By considering two {\it ab initio}-based complementary approaches, we analyze
the electronic structure and extract effective spin models of
BaCo$_2$(AsO$_4$)$_2$, a honeycomb material which has been proposed as a
candidate for Kitaev physics. Both methods show that the dominant direct
hopping makes the bond-dependent Kitaev term negligible averting the material
away from the sought-after spin-liquid regime. As a result, we present a simple
three-parameter exchange model to describe the interactions of the lowest
doublet of the honeycomb cobaltate BaCo$_2$(AsO$_4$)$_2$. Remarkably, it is the
third-neighbor interactions, both isotropic and anisotropic, that are
responsible for the standout double-zigzag ground state of
BaCo$_2$(AsO$_4$)$_2$, stabilized by quantum fluctuations. A significantly
large third-nearest neighbor hopping, observed in {\it ab initio}, supports the
importance of the third-neighbor interactions in the stabilization of the
unique ground state of BaCo$_2$(AsO$_4$)$_2$.
