Homotopy quantum field theories and tortile structures

Abstract

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 $π$-category when the background space is an Eilenberg-Maclane space $X=K(π,1)$, and to a tortile category with lax $π_2X$-action when the background space is simply-connected.