We study a version of a modular functor for Turaev's homotopy quantum field theories using 2-categories of surfaces. We define the homotopy surface 2-category of a space X and define an SX-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 SX-structure gives rise to a lax tortile p-category when the background space is an Eilenberg-Maclane space X=K(p,1), and to a tortile category with lax p2X-action when the background space is simply connected. © 2003. Published by Elsevier B.V.