A polyhedral complex ${\mathscr{C}}$ in $N_{\mathbb{R}}$ is a finite set of polyhedra with the following two properties: Every polyhedron in ${\mathscr{C}}$ has all its closed faces in ${\mathscr{C}}$. If $\Delta,\sigma\in{{\mathscr{C}}}$, then $\Delta\cap\sigma$ is a closed face of $\Delta$ and $\sigma$. Note here that the empty set and also $\sigma$ are allowed as closed faces of a polyhedron $\sigma$ (see [Gu12], Appendix A, for details).

A polyhedral complex ${\mathscr{C}}$ is called integral $G$-affine for a subgroup $G$ of ${\mathbb{R}}$ if every polyhedron of ${\mathscr{C}}$ is integral $G$-affine. The support $|{\mathscr{C}}|$ of ${\mathscr{C}}$ is the union of all polyhedra in ${\mathscr{C}}$. The polyhedral complex ${\mathscr{C}}$ is called pure dimensional of dimension $n$ if every maximal polyhedron in ${\mathscr{C}}$ has dimension $n$. We will often use the notation ${\mathscr{C}}_{k}:=\{\sigma\in{\mathscr{C}}\mid\dim(\sigma)=k\}$ for $k\in{\mathbb{N}}$.

Let ${\mathscr{C}}$ be a polyhedral complex in $N_{\mathbb{R}}$. A superform on ${\mathscr{C}}$ is the restriction of a superform on (an open subset of) $N_{\mathbb{R}}$ to $|{\mathscr{C}}|$. This means that two superforms agree if their restrictions to any polyhedron of $|{\mathscr{C}}|$ agree. Let $A({\mathscr{C}})$ be the space of superforms on ${\mathscr{C}}$. It is an alternating algebra with respect to the induced wedge product. We have also differential operators $d$, $d^{\prime}$ and $d^{\prime\prime}$ on $A({\mathscr{C}})$ given by restriction of the corresponding operators on $A(N_{\mathbb{R}})$. Let $A^{p,q}({\mathscr{C}})$ be the space of $(p,q)$-superforms on ${\mathscr{C}}$. The support of $\alpha\in A({\mathscr{C}})$ is the complement of $\{\omega\in|{\mathscr{C}}|\mid\text{$\alpha$ vanishes identically in a neighbourhood of $\omega$}\}$ in $|{\mathscr{C}}|$. We denote by $A_{c}^{p,q}({\mathscr{C}})$ the subspace of $A^{p,q}({\mathscr{C}})$ of superforms of compact support.

Let $N^{\prime}$ be a free abelian group of rank $r^{\prime}$ and let $F:N^{\prime}_{\mathbb{R}}\rightarrow N_{\mathbb{R}}$ be an affine map. Suppose that ${\mathscr{C}}^{\prime}$ is a polyhedral complex of $N^{\prime}_{\mathbb{R}}$ with $F(|{\mathscr{C}}^{\prime}|)\subset|{\mathscr{C}}|$, then the pull-back in 2.3 induces a pull-back $F^{*}:A^{p,q}({\mathscr{C}})\rightarrow A^{p,q}({\mathscr{C}}^{\prime})$.

A polyhedral complex ${\mathscr{D}}$ subdivides the polyhedral complex ${\mathscr{C}}$ if they have the same support and if every polyhedron $\Delta$ of ${\mathscr{D}}$ is contained in a polyhedron of ${\mathscr{C}}$. In this case, we say that ${\mathscr{D}}$ is a subdivision of ${\mathscr{C}}$. All our constructions here will be compatible with subdivisions. This is no problem for the definition of superforms on ${\mathscr{C}}$ as they depend only on the support $|{\mathscr{C}}|$.

A weight on a pure dimensional polyhedral complex ${\mathscr{C}}$ is a function $m$ which assigns to every maximal polyhedron $\sigma\in{\mathscr{C}}$ a number $m_{\sigma}\in{\mathbb{Z}}$. Then we get a canonical weight on every subdivision of ${\mathscr{C}}$. For a weighted polyhedral complex $({\mathscr{C}},m)$, only the polyhedra $\Delta\in{\mathscr{C}}$ which are contained in a maximal dimensional $\sigma\in{\mathscr{C}}$ with $m_{\sigma}\neq 0$ are of interest. They form a subcomplex ${\mathscr{D}}$ of ${\mathscr{C}}$ and we define the support of $({\mathscr{C}},m)$ as the support of ${\mathscr{D}}$. The polyhedra of ${\mathscr{C}}\setminus{\mathscr{D}}$ will usually be neglected.

Let $({\mathscr{C}},m)$ be a weighted integral ${\mathbb{R}}$-affine polyhedral complex of pure dimension $n$. For $\alpha\in A_{c}^{n,n}({\mathscr{C}})$, we set

$$\int_{({\mathscr{C}},m)}\alpha:=\sum_{\sigma\in{\mathscr{C}}_{n}}m_{\sigma}\int_{\sigma}\alpha,$$

where we use integration from 2.4 on the right. We define integrals over the boundary of ${\mathscr{C}}$ for a superform $\beta$ in $A_{c}^{n-1,n}({\mathscr{C}})$ or in $A_{c}^{n,n-1}({\mathscr{C}})$ by

$$\int_{\partial({\mathscr{C}},m)}\beta=\sum_{\sigma\in{\mathscr{C}}_{n}}m_{\sigma}\int_{\partial\sigma}\beta,$$

where we use the boundary integrals from 2.8 on the right. Note that the boundary $\partial{\mathscr{C}}$ may be defined as the subcomplex consisting of the polyhedra of dimension at most $n-1$, but there is no canonical weight on $\partial{\mathscr{C}}$. Indeed, the boundary integral $\int_{\partial({\mathscr{C}},m)}\beta$ depends on the relative situation $\partial{\mathscr{C}}\subset{\mathscr{C}}$ because of the weight $m_{\sigma}$ and the contraction with respect to the vectors $\omega_{\rho,\sigma}$ used in the definitions. This is similar to the situation in real analysis where boundary integrals depend on the relative orientation. These classical boundary integrals do depend only on the restriction of the differential form to the boundary which is clearly wrong for our boundary integrals. However, it is still true that $\int_{\partial({\mathscr{C}},m)}\beta=0$ if the support of $\beta$ is disjoint from $\partial{\mathscr{C}}$.

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.

Now let $F:N_{\mathbb{R}}^{\prime}\rightarrow N_{\mathbb{R}}$ be an affine map whose underlying linear map is integral, i.e. induced by a homomorphism $N^{\prime}\rightarrow N$. We will define the push-forward of a weighted integral ${\mathbb{R}}$-affine polyhedral complex $({\mathscr{C}}^{\prime},m)$ of pure dimension $n$ on $N_{\mathbb{R}}^{\prime}$. For details, we refer to [AR10], §7. After a subdivision of ${\mathscr{C}}^{\prime}$, we may assume that

$$F_{*}({\mathscr{C}}^{\prime}):=\{F(\sigma^{\prime})\mid\text{$\sigma^{\prime}$ is a face of $\nu^{\prime}\in{\mathscr{C}}^{\prime}$ with $\dim(F(\nu^{\prime}))=n$}\}$$

is a polyhedral complex in $N_{\mathbb{R}}$. We define the multiplicity of an $n$-dimensional $F(\sigma^{\prime})\in F_{*}({\mathscr{C}}^{\prime})$ by

$$m_{F(\sigma^{\prime})}:=\sum_{\nu^{\prime}\in{\mathscr{C}}^{\prime}_{n},\,\nu^{\prime}\subset F^{-1}(F(\sigma^{\prime}))}[N_{\nu^{\prime}}^{\prime}:N_{F(\sigma^{\prime})}]m_{\nu^{\prime}}.$$

Endowed with these multiplicities, we get a weighted integral ${\mathbb{R}}$-affine polyhedral complex $F_{*}({\mathscr{C}}^{\prime},m)$ of $N_{\mathbb{R}}$. If $({\mathscr{C}}^{\prime},m)$ is a tropical cycle, then $F_{*}({\mathscr{C}}^{\prime},m)$ is also a tropical cycle. It might happen that $F_{*}({\mathscr{C}}^{\prime},m)$ is empty, then we get the tropical zero cycle.