Hom-associative algebras up to homotopy

Abstract

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