old ones by restriction to $[0,\varepsilon)$ in a way and allow us to glue them along the boundary and then rescale the interval. By disjoint union the structure of a symmetric monoidal bicategory with duals on $T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)$ is obtained.

\[defhqftext\] An *$n$-dimensional extended homotopy quantum field theory with target space $T$ and values in a symmetric monoidal bicategory $\mathcal{S}$* is defined to be a symmetric monoidal functor $Z: T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2) {\longrightarrow}\mathcal{S}$ fulfilling the property of *homotopy invariance*: Given 2-morphisms $(M,\psi), (M,\psi') : (\Sigmait_a,\varphi_a) \Longrightarrow (\Sigmait_b,\varphi_b)$ between 1-morphisms $(\Sigmait_a,\varphi_a) , (\Sigmait_b,\varphi_b) : (S_0,\xi_0) {\longrightarrow}(S_1,\xi_1)$ such that $\psi$ is homotopic to $\psi'$ relative the boundary of $M$ we require that $$\begin{tikzcd}
\, & \ \ \ \ \ \  \ar[Rightarrow, swap, shorten <= 8,shorten >= 8]{dd}{Z(M,\psi)} &  \\
Z(S_0,\xi_0) \ar[bend left=50]{rr}{Z(\Sigmait_a,\varphi_a)} \ar[bend right=50, swap]{rr}{Z(\Sigmait_b,\varphi_b)}  & & Z(S_1,\xi_1) \\
\, & \, &
\end{tikzcd} 
=
\begin{tikzcd}
\, & \ \ \ \ \ \  \ar[Rightarrow, swap, shorten <= 8,shorten >= 8]{dd}{Z(M,\psi')} &  \\
Z(S_0,\xi_0) \ar[bend left=50]{rr}{Z(\Sigmait_a,\varphi_a)} \ar[bend right=50, swap]{rr}{Z(\Sigmait_b,\varphi_b)}  & & Z(S_1,\xi_1) \\
\, & \, &
\end{tikzcd} \ .$$

An extended homotopy quantum field theory $Z: T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2) {\longrightarrow}\mathcal{S}$ is called *invertible* if it takes values in the sub-2-groupoid of $\mathcal{S}$ containing the objects, 