We consider now a compact analytic space $Z$ over $K$ of pure dimension $n$. Theorem 7.3 shows that the tropical variety ${\varphi_{\rm trop}}(Z)$ is the support of an integral $\Gamma$-affine polytopal complex in $N_{\mathbb{R}}$. Our next goal is to endow this complex with canonical tropical multiplicities. This will lead to the definition of a weighted polytopal complex $({\varphi_{\rm trop}})_{*}({\rm cyc}(Z))$ which is canonical up to subdivision.

If $\dim({\varphi_{\rm trop}}(Z))<n$, then we set $({\varphi_{\rm trop}})_{*}({\rm cyc}(Z))=0$ meaning that we choose all tropical weights equal to zero. It remains to consider the case $\dim({\varphi_{\rm trop}}(Z))=n$. As seen in Example 4.5, we may identify ${\mathbb{R}}^{r}$ with the skeleton $S({T^{\rm an}})$ of ${T^{\rm an}}$. We choose a generic surjective homomorphism $q:T\rightarrow T^{\prime}$ onto a split multiplicative torus $T^{\prime}={\rm Spec}(K[M^{\prime}])$ of rank $n=\dim(Z)$. Generic means that the corresponding linear map ${\rm Trop}(q)$ is injective on every polytope contained in ${\varphi_{\rm trop}}(Z)$. By Theorem 7.3, there is an integral $\Gamma$-affine polytopal complex ${\mathscr{C}}$ in $N_{\mathbb{R}}$ with $|{\mathscr{C}}|={\varphi_{\rm trop}}(Z)$ such that ${\rm Trop}(q)(\tau)$ is disjoint from $(q\circ\varphi)_{\rm trop}(\partial Z)$ for every $n$-dimensional face $\sigma$ of ${\mathscr{C}}$ and $\tau:={\rm relint}(\sigma)$. By passing to a subdivision, we may assume that ${\rm Trop}(q)_{*}({\mathscr{C}})$ is a polyhedral complex in $N_{\mathbb{R}}^{\prime}$ as in 3.9.

We identify ${\rm Trop}(q)(\tau)\subset{\mathbb{R}}^{n}$ with a subset of the skeleton $S((T^{\prime})^{\rm an})$ as in Remark 4.5. Then it is clear that $q\circ\varphi$ restricts to a map $\varphi_{\rm trop}^{-1}(\tau)\rightarrow{\rm Trop}(q)(\tau)$ which agrees with ${\rm Trop}(q)\circ{\varphi_{\rm trop}}$ on $\varphi_{\rm trop}^{-1}(\tau)$ using the identification $S((T^{\prime})^{\rm an})={\mathbb{R}}^{n}$. It is shown in [CD12], §2.4, that $\varphi_{\rm trop}^{-1}(\tau)\rightarrow{\rm Trop}(q)(\tau)$ is a finite flat and surjective morphism which means that every point $p$ of ${\rm Trop}(q)(\tau)$ has a neighbourhood $W^{\prime}$ in $(T^{\prime})^{\rm an}$ such that $(q\circ\varphi)^{-1}(W^{\prime})\rightarrow W^{\prime}$ has these properties. Since $\tau$ is connected, the corresponding degree depends only on $\tau$ and not on the choice of $p$. We denote this degree by $[\varphi_{\rm trop}^{-1}(\tau):{\rm Trop}(q)(\tau)]$.

Recall that $N_{\sigma}$ is the canonical lattice in the affine space generated by $\sigma$. Then the character lattice $M^{\prime}$ of $T^{\prime}$ is of finite index in $M_{\sigma}={\rm Hom}(N_{\sigma},{\mathbb{Z}})$.