Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed C-algebras

摘要

Homotopy Quantum Field Theories (HQFTs) were introduced by the second authorto extend the ideas and methods of Topological Quantum Field Theories to closed$d$-manifolds endowed with extra structure in the form of homotopy classes ofmaps into a given `target' space, $B$. For $d = 1$, classifications of HQFTs interms of algebraic structures are known when $B$ is a $K(G,1)$ and also when itis simply connected. Here we study general HQFTs with $d = 1$ and target ageneral 2-type, giving a common generalisation of the classifying algebraicstructures for the two cases previously known. The algebraic models for 2-typesthat we use are crossed modules, $\mathcal{C}$, and we introduce a notion offormal $\mathcal{C}$-map, which extends the usual lattice-type constructions tothis setting. This leads to a classification of `formal' 2-dimensional HQFTswith target $\mathcal{C}$, in terms of crossed $\mathcal{C}$-algebras.