Hom-associative algebras up to homotopy

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.