# Classifying space for Thompson's group F?

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 $$F$$?

• Just to make sure I understand: Fiore and Leinster talk about a different category, namely, the monoidal category freely generated by an object $X$ and an 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? – Omar Antolín-Camarena Jan 21 at 23:54
• @Omar Antolin-Camarera: Yes, your understanding is correct. – André Henriques Jan 22 at 7:58

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
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.