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 $\mathcal C'$.

The article

Marcelo Fiore, Tom Leinster,

An abstract characterization of Thompson's group $F$, Semigroup Forum80(2010), 325-340, doi:10.1007/s00233-010-9209-2, arXiv:math/0508617.

proves that $\pi_1(|\mathcal C'|)$ is isomorphic to Thompson's group $F$.

Question:Is $|\mathcal C'|$ a classifying space for Thompson's group $F$?

isomorphism$X \otimes X \to X$. That category is a groupoid, so your question has a positive answer for their category. Now it easily follows from their result that for your non-groupoid category the fundamental group is $F$, and you are asking if you also get a $K(F,1)$ from your category. Right? $\endgroup$ – Omar Antolín-Camarena Jan 21 '20 at 23:54