Examples of classifying spaces of monoids

Question

Definition. The classifying space functor for monoids is the functor $|{-}|\colon\mathsf{Mon}\to\mathsf{Top}$ defined as the composition $$ \mathsf{Mon} \xrightarrow{\mathbf{B}} \mathsf{Cats} \xrightarrow{\mathrm{N}_{\bullet}} \mathsf{sSets} \xrightarrow{|-|} \mathsf{Top}, $$ where

• $\mathbf{B}$ is the delooping functor for monoids;
• $\mathrm{N}_{\bullet}$ is the nerve functor;
• $|{-}|$ is the geometric realisation functor.

---

Examples. So far, I've found the following:

1. The classifying space of the additive monoid of natural numbers is homotopic to the circle, i.e. $|\mathbb{N}|\simeq S^{1}$ (proof).
2. We have $|\mathbb{Z}|\simeq|\mathbb{N}|\simeq S^1$ also.
3. More generally, we have $|\mathbb{N}^{\oplus n}|\simeq|\mathbb{Z}^{\oplus n}|\simeq (S^1)^{\times n}$, the $n$-torus $\mathbf{T}^{n}$.
4. (For examples involving groups; see MO 56363.)

Question. What are other examples? In particular, what are the classifying spaces of the following monoids?

• The multiplicative monoid $(\mathbb{N},\cdot,1)$ of natural numbers;
• The multiplicative monoid $(\mathbb{Z},\cdot,1)$ of integers;
• The free monoid on a set $S$; by Proposition 4.4.1 of Lenz, if a monoid $A$ is free, then $|A|\simeq|A^\mathrm{grp}|$, and hence the classifying space of the free monoid $F_n$ on $n$ elements will be homotopy equivalent to the classifying space of the free group on $n$ elements, which is the wedge $S^1\vee\cdots\vee S^1$ of $n$ circles.