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



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> 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](https://doi.org/10.1007/s00233-010-9209-2), arXiv:[math/0508617](https://arxiv.org/abs/math/0508617).
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proves that $\pi\_1(|\mathcal C'|)$ is isomorphic to Thompson's group $F$.



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> **Question:** Is $|\mathcal C'|$ a classifying space for Thompson's group $F$?
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Classifying space for Thompson's group F?
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](https://bookstore.ams.org/memo-130-620/)). Farley in 



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> "Finiteness and CAT(0) properties of diagram groups", Topology **42** (2003), no. 5, 1065–1082 doi:[10.1016/S0040-9383(02)00029-0](https://doi.org/10.1016/S0040-9383%2802%2900029-0), [author pdf](http://www.users.miamioh.edu/farleyds/Far1.pdf)
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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.


