  We study a variation of 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 $\cS_X$-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, under certain conditions, an $\cS_X$-structure gives rise to a
lax tortile $\pi$-category when the background space is an Eilenberg-Maclane
space $X=K(\pi,1)$, and to a tortile category with lax $\pi_2X$-action when the
background space is simply-connected.
