Modular functors in homotopy quantum field theory and tortile structures

We study a version of a modular functor for Turaevʹs homotopy quantum field theories using 2-categories of surfaces. We define the homotopy surface 2-category of a space X and define an -structure to be a monoidal 2-functor from this to the 2-category of idempotent-complete additive k-linear categories. We initiate the study of the algebraic structure arising from these functors. In particular we show that a unitary -structure gives rise to a lax tortile π-category when the background space is an Eilenberg–Maclane space X=K(π,1), and to a tortile category with lax π2X-action when the background space is simply connected.