  A hom-associative algebra is an algebra whose associativity is twisted by an
algebra homomorphism. In this paper, we introduce a strongly homotopy version
of hom-associative algebras ($HA_\infty$-algebras in short) on a graded vector
space. We describe $2$-term $HA_\infty$-algebras in details. In particular, we
study 'skeletal' and 'strict' $2$-term $HA_\infty$-algebras. We also introduce
hom-associative $2$-algebras as categorification of hom-associative algebras.
The category of $2$-term $HA_\infty$-algebras and the category of
hom-associative $2$-algebras are shown to be equivalent. An appropriate
skew-symmetrization of $HA_\infty$-algebras give rise to $HL_\infty$-algebras
introduced by Sheng and Chen. Finally, we define a suitable Hochschild
cohomology theory for $HA_\infty$-algebras which control the deformation of the
structures.
