Let $\mathcal C$ be the free monoidal category generated by an object $X$, and a morphism $X \otimes X \to X$.
This category contains exactly two connected components: that of the monoidal unit $1\in \mathcal C$, and that of $X\in \mathcal C$. (In general, two object $A$ and $B$ of a category are said to be in the same connected component if they are related by a zig-zag of arrows $$A\to Y_1\leftarrow Y_2\to Y_3\leftarrow Y_4\to Y_5\leftarrow\ldots \to B.$$ In the case of $\mathcal C$, any two non-unit objects are related by a single morphism.)
Let $\mathcal C'\subset \mathcal C$ be the connected component of $X$ and let $|\mathcal C'|$ denote the geometric realisation (of the simplicial nerve) of $\mathcal C'$.
proves that $\pi_1(|\mathcal C'|)$ is isomorphic to Thompson's group $F$.
Question: Is $|\mathcal C'|$ a classifying space for Thompson's group $F$?