A morphism $\sigma {\longrightarrow}\sigma'$ between two fundamental cycles is an $n-1$-chain $\tau$ such that $\partial \tau = \sigma'-\sigma$. Composition is given by addition of $n-1$-chains.

The objects in $T_\theta(S,\xi)$ are generated via biproduct from the object set of ${\operatorname{Fund}}(S)$, i.e. formal finite sums $\bigoplus_{i=1}^n V_i* \sigma_i$, where the $V_i$ are finite-dimensional complex vector spaces and the $\sigma_i$ are objects in ${\operatorname{Fund}}(S)$. Sometimes we write $\sigma$ for ${\mathbb{C}}* \sigma$. The space of morphisms between $\sigma,\sigma' \in {\operatorname{Fund}}(S)$ seen as objects of $T_\theta(S,\xi)$ is given by $$\begin{aligned}
{\operatorname{Hom}}_{T_\theta(S,\xi)}(\sigma,\sigma') := \frac{\mathbb{C}[{\operatorname{Hom}}_{{\operatorname{Fund}}(S)}(\sigma,\sigma')]}{\sim},      \label{defmorphismon0cells}\end{aligned}$$ where $\mathbb{C}[{\operatorname{Hom}}_{{\operatorname{Fund}}(S)}(\sigma,\sigma')]$ is the free complex vector space on the set ${\operatorname{Hom}}_{{\operatorname{Fund}}(S)}(\sigma,\sigma')$, and for two morphisms $\tau,\widetilde \tau : \sigma {\longrightarrow}\sigma'$ we make the identification $$\begin{aligned}
 \widetilde \tau \sim \langle \xi^* \theta,\lambda\rangle \tau,  \label{defmorphismon0cells2} \end{aligned}$$ whenever $\widetilde \tau-\tau=\partial \lambda$ for some $\lambda \in C_n(S)$. By angular brackets we denote the evaluation of cochains on chains. Note that in the choice of $\lambda$ does not matter. In order to obtain the morphism spaces between all objects in $T_\theta(S,\xi)$, has to be extended bilinearly, i.e. $$\begin{aligned}
{\operatorname{Hom}}_{T_\theta(S,\xi)} \left( \bigoplus_{i=1}^n V_i * \sigma_i\ ,\ \bigoplus_{j=1}^m V_j * \sigma_j \right) = \bigoplus_{i,j} {\operatorname{Hom}}(V_i,V_j) \otimes {\operatorname{Hom}}_{T_\theta(S,\xi)} (\sigma_i, \sigma_j)\end{aligned}$$ for all formal finite sums. Composition is defined 