Classifying space for Thompson's group F?

Question

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 Forum 80 (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 $F$?

Answer 1

What you describe is the so called Squier complex of the semigroup presentation $\langle x \mid x^2=x\rangle$ (you did not describe the 2-cells, but it is straightforward). The fact that its fundamental group is $F$ was proved by Guba and myself in 1997, "Diagram groups", Memoirs of the AMS, November, 1997 (link). Farley in

"Finiteness and CAT(0) properties of diagram groups", Topology 42 (2003), no. 5, 1065–1082 doi:10.1016/S0040-9383(02)00029-0, author pdf

proved that its universal cover is a $CAT(0)$ cube complex. So indeed the Squier complex is a classifying space for $F$. The proof of all these from the category theory point of view can be found in Guba, V. S.; Sapir, M. V. "Diagram groups and directed 2-complexes: homotopy and homology. " J. Pure Appl. Algebra 205 (2006), no. 1, 1–47.