A weighted integral ${\mathbb{R}}$-affine polyhedral complex $({\mathscr{C}},m)$ of pure dimension $n$ is called a tropical cycle if its weight $m$ satisfies the following balancing condition: For every $n-1$-dimensional $\rho\in{\mathscr{C}}$, we have

$$\sum_{\sigma\in{\mathscr{C}}_{n},\,\sigma\supset\rho}m_{\sigma}\omega_{\rho,\sigma}\in N_{\rho}.$$

Here, $N_{\rho}$ is the canonical lattice contained in the affine space generated by $\rho$ and $\omega_{\rho,\sigma}\in N_{\sigma}$ is the lattice vector pointing outwards of $\sigma$ (see 2.8). Tropical cycles are the basic objects in tropical geometry.