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 (H A ∞ -algebras in short) on a graded vector space. We describe 2-term H A ∞ -algebras in detail. In particular, we study 'skeletal' and 'strict' 2-term H A ∞ -algebras. We also introduce hom-associative 2-algebras as categorification of hom-associative algebras. The category of 2-term H A ∞ -algebras and the category of hom-associative 2-algebras are shown to be equivalent. Next, we define a suitable Hochschild cohomology theory for H A ∞ -algebras which control the deformation of the structures. Finally, we visit H L ∞ -algebras introduced by Sheng and Chen and show that an appropriate skew-symmetrization of H A ∞ -algebras give rise to H L ∞ -algebras. [ABSTRACT FROM AUTHOR]