Let $(V_{i},\varphi_{U_{i}})_{i\in I}$ be finitely many tropical charts contained in $V_{i}$ and let $\Delta_{i}$ be a polytope contained in the open subset $\Omega_{i}:={\rm trop}_{U_{i}}(V_{i})$ of ${\rm Trop}(U_{i})$. We consider the space $A^{p,q}(V_{i},U_{i},\Delta_{i}:i\in I)$ of $(p,q)$-forms $\alpha$ on $W$ with support in $C:=\bigcup_{i\in I}{\rm trop}_{U_{i}}^{-1}(\Delta_{i})$ such that $\alpha$ is given on $V_{i}$ by a superform $\alpha_{i}\in A^{p,q}(\Omega_{i})$ for every $i\in I$. Since the tropicalization map is proper (see 4.4), the set $C$ is compact. Similarly as in the complex case, we endow $A^{p,q}(V_{i},U_{i},\Delta_{i}:i\in I)$ with the structure of a locally convex space such that a sequence $\alpha_{k}$ converges to $\alpha$ if and only if all derivatives of the superforms $\alpha_{k,i}$ converge uniformly to the derivatives of the superform $\alpha_{i}$ on $\Delta_{i}$. Here, $\alpha_{k,i}$ (resp. $\alpha_{i}$) is the superform on $\Omega_{i}$ which defines $\alpha_{k}$ (resp. $\alpha$) on $V_{i}$ and we mean more precisely the derivatives of the coefficients of $\alpha_{k,i}|_{\Delta_{i}}$ (resp. $\alpha_{i}|_{\Delta_{i}}$). It follows easily from Proposition 4.16 that $A_{c}^{p,q}(W)$ is the union of all spaces $A^{p,q}(V_{i},U_{i},\Delta_{i}:i\in I)$ with $(V_{i},U_{i},\Delta_{i}:i\in I)$ ranging over all possibilities as above.

A current on an open subset $W$ of ${X^{\rm an}}$ is a linear functional $T$ on $A_{c}^{p,q}(W)$ such that the restriction of $T$ to all subspaces $A^{p,q}(V_{i},U_{i},\Delta_{i}:i\in I)$ is continuous. The space of currents is a $C^{\infty}(W)$-module denoted by $D_{p,q}(W)$. As usual, we define the differential operators $d^{\prime}$, $d^{\prime\prime}$ and $d:=d^{\prime}+d^{\prime\prime}$ on the total space of currents $D(W):=\bigoplus_{p,q}D_{p,q}(W)$. Using partitions of unity from Proposition 5.10, it is easy to show that the currents form a sheaf on ${X^{\rm an}}$.

Let $T\in D_{p,q}(W)$ and $\omega\in A^{r,s}(W)$ for an open subset $W$ of ${X^{\rm an}}$. Then we define $T\wedge\omega\in D_{p-r,q-s}(W)$ by $(T\wedge\omega)(\alpha)=T(\omega\wedge\alpha)$ for $\alpha\in A_{c}^{p-r,q-s}(W)$. Since the wedge product with a given form is a continuous operation on $A_{c}(W)$, it is clear that $T\wedge\omega$ is really a current on $W$.

Let us again consider an open subset $W$ of ${X^{\rm an}}$. A function $f:W\rightarrow{\mathbb{R}}\cup\{\pm\infty\}$ is called locally integrable if $f$ is integrable with respect to the measure $\mu$ associated to any $\omega\in A_{c}^{2n}(W)$. Then we write $\int_{W}f\omega:=\int_{W}fd\mu$.

For a locally integrable function $f$ on $W$ and $\eta\in A^{p,q}(W)$, we define $[f\cdot\eta]\in D_{p,q}(W)$ by $[f\cdot\eta](\alpha):=\int_{W}f\eta\wedge\alpha$ for every $\alpha\in A_{c}^{n-p,n-q}(W)$.