of the parallel transport homotopy quantum field theory\[secconstruchqft\]
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In this section we state our main result as Theorem \[mainthm\]: To a flat $n-1$-gerbe on a topological space $T$ represented by a singular cocycle $\theta \in Z^n(T;{\operatorname{U}}(1))$ we associate an extended homotopy quantum field theory $$\begin{aligned}
 T_\theta : {T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)}{\longrightarrow}{{2\mathbf{Vect}_{\mathbb{C}}}}\end{aligned}$$ which can be understood as the parallel transport operator of $\theta$. The evaluation of $T_\theta$ on an $n$-dimensional closed oriented manifold $M$ together with a map to $\psi : M {\longrightarrow}T$ yields an element in ${\operatorname{U}}(1)$, namely the holonomy of $\theta$ with respect to $\psi$. As explained in the introduction, the homotopy quantum field theory nature of the parallel transport operator reflects a high degree of locality of the Wess-Zumino term.

Definition on objects
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Consider an object $(S,\xi)$ in ${T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)}$, i.e. $S$ is a closed oriented $n-2$-dimensional manifold and $\xi : S {\longrightarrow}T$ a continuous map. The extended homotopy quantum field theory $T_\theta$ assigns to $(S,\xi)$ a 2-vector space $T_\theta(S,\xi)$ that we are going to define in this subsection.

To this end, denote by ${\operatorname{Fund}}(S)$ the groupoid of fundamental cycles of $S$. Its objects are fundamental cycles of $S$, i.e. those elements of $Z_{n-2}(S)$ representing the fundamental class in $H_{n-2}(S)$. 