Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1+1

We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n,X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord}(n,X), including the fact that it only depends on the (n+1)-homotopy type of X. We also prove that an HQFT with target space X and in dimension n+1 implies the existence of geometrical structures in X; in particular, flat gerbes make their appearance. We give a complete characterization of HCobord(n,X) for n=1 (or the 1+1 case) and X the Eilenberg-Maclane space K(G,2). In the final section we derive state sum models for these HQFT's.