Ogus in his slides https://math.berkeley.edu/~ogus/preprints/colloqhandout.pdf presents the following picture of a monoidal space $\operatorname{Spec}(\mathbb{N} \longrightarrow \mathbb{C}[\mathbb{N}])$:
Picture of $\operatorname{Spec}(\mathbb{N})$ is clear, the generic point corresponds to the empty ideal. But what is the precise meaning of the bottom picture? Thanks in advance.
Ogus states that he draws a log scheme $(X,\mathcal{M})$ by first drawing $X$ and then adding a picture of $\operatorname{Spec}\mathcal{M}_x$ at each $x\in X.$ (He says this on page 21.)
In this case, $\operatorname{Spec}\Bbb{C}[\Bbb{N}] \simeq \operatorname{Spec}\Bbb{C}[t],$ so the pictured disk is the complex affine line, represented as a subset of the complex plane. We can also compute $\mathcal{M}_x$ for each $x\in\operatorname{Spec}\Bbb{C}[\Bbb{N}].$ At the points other than the origin, the log structure on $\operatorname{Spec}\Bbb{C}[\Bbb{N}]$ is trivial in the sense that $\mathcal{M}_x \simeq\mathcal{O}_{X,x}^\times.$ However, at the origin, we have $\mathcal{M}_x\simeq \mathcal{O}_{X,x}^\times\times\Bbb{N},$ and the origin gets embellished with a picture of $\operatorname{Spec}\Bbb{N}.$
It seems to me that the picture really decorates each point $x\in X$ with a picture of $\operatorname{Spec}(\overline{\mathcal{M}}_x),$ where $\overline{\mathcal{M}} = \mathcal{M}/\mathcal{O}_X^\times$ is the characteristic of the log structure.
