Let $({\mathscr{C}},m)$ be a weighted integral ${\mathbb{R}}$-affine polyhedral complex of pure dimension $n$ on $N_{\mathbb{R}}$. Then the following conditions are equivalent:

• (a)

  $({\mathscr{C}},m)$ is a tropical cycle;

• (b)

  $\delta_{({\mathscr{C}},m)}$ is a $d^{\prime}$-closed supercurrent on $N_{\mathbb{R}}$;

• (c)

  $\delta_{({\mathscr{C}},m)}$ is a $d^{\prime\prime}$-closed supercurrent on $N_{\mathbb{R}}$.