by matrix multiplication and composition in ${\operatorname{Fund}}(S)$.

The ${{\mathbf{FinVect}_{\mathbb{C}}}}$-module structure is given by $$\begin{aligned}
* \colon {{\mathbf{FinVect}_{\mathbb{C}}}}\times T_{\theta}(S,\xi) &\longrightarrow T_{\theta}(S,\xi) \\
 V\times \left(\bigoplus_{i=1}^n V_i * \sigma_i\right) & {\longmapsto}\left(\bigoplus_{i=1}^n (V\otimes V_i) * \sigma_i\right) \ .  \end{aligned}$$ It is now easy to see that $T_\theta(S,\xi)$ is a 2-vector space with one simple object, i.e. a 2-line.

Definition on 1-morphisms
-------------------------

Let $(\Sigmait,\varphi) : (S_0,\xi_0) {\longrightarrow}(S_1,\xi_1)$ be a 1-morphism in ${T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)}$. Again, we denote by ${\operatorname{Fund}}(\Sigmait)$ the groupoid of fundamental cycles of $\Sigmait$, i.e. the groupoid of relative cycles in $C_{n-1}(\Sigmait)$ representing the fundamental class of $\Sigmait$ in $H_{n-1}(\Sigmait,\partial \Sigmait)$. For fundamental cycles $\sigma_0$ and $\sigma_1$ of $S_0$ and $S_1$, respectively, we denote by ${\operatorname{Fund}}_{\sigma_0}^{\sigma_1}(\Sigmait)$ the subgroupoid of ${\operatorname{Fund}}(\Sigmait)$ spanned by all fundamental cycles $\mu$ of $\Sigmait$ with $\partial \mu = \sigma_1-\sigma_0$. Here we suppress the inclusion of the ingoing and outgoing boundary into $\Sigmait$ in the notation. By [@bredon VI., Lemma 9.1] the groupoid ${\operatorname{Fund}}_{\sigma_0}^{\sigma_1}(\Sigmait)$ is non-empty and connected.

In order to define the 2-linear map $T_\theta(\Sigmait,\varphi) : T_\theta (S_0,\xi_0) {\longrightarrow}T_\theta(S_1,\xi_1)$ we define on the free vector space $\mathbb{C}[{\operatorname{Fund}}_{\sigma_0}^{\sigma_1}(\Sigmait)]$ the equivalence relation $$\begin{aligned}
\mu' \sim \langle\varphi^* \theta ,\nu\rangle \mu \end{aligned}$$ for any $\nu \in C_n(\Sigmait)$ such that $\partial \nu =\mu'-\mu$. We use 