  We define the cohomological Hall algebra ${AHA}_{Higgs(X)}$ of the
($2$-dimensional) Calabi-Yau category of Higgs sheaves on a smooth projective
curve $X$, as well as its nilpotent and semistable variants, in the context of
an arbitrary oriented Borel-Moore homology theory. In the case of usual
Borel-Moore homology, ${AHA}_{Higgs(X)}$ is a module over the (universal)
cohomology ring $\mathbb{H}$ of the stacks of coherent sheaves on $X$ . We show
that it is a torsion-free $\mathbb{H}$-module, and we provide an explicit
collection of generators (the collection of fundamental classes $[Coh_{r,d}]$
of the zero-sections of the map $Higgs_{r,d} \to Coh_{r,d}$, for $r \geq 0, d
\in Z$).
