Remarks on $2$–dimensional HQFTs

Abstract

We introduce and study algebraic structures underlying $2$–dimensional Homotopy Quantum Field Theories (HQFTs) with arbitrary target spaces. These algebraic structures are formalized in the notion of a twisted Frobenius algebra. Our work generalizes results of Brightwell, Turner and the second author on $2$–dimensional HQFTs with simply connected or aspherical targets.