Let $X$ be a toric variety over $K$ with torus $\mathbb{T}_{K}$ and let ${\mathcal{X}}$ be a toric scheme over $S$ with torus $\mathbb{T}_{S}$. We say that ${\mathcal{X}}$ is a toric model of $X$ over $S$ if the identity of $\mathbb{T}_{K}$ can be extended to an isomorphism from $X$ to ${\mathcal{X}}_{\eta}$.

If ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ are toric models of $X$ and $\alpha\colon{\mathcal{X}}\to{\mathcal{X}}^{\prime}$ is an $S$-morphism, we say that $\alpha$ is a morphism of toric models if its restriction to $\mathbb{T}_{K}$ is the identity.