Let $G$ be a finitely generated group, then its action groupoid $BG$ is a simplicial set. In fact $BG$ is the nerve of a groupoid where the set of objects is given by a point $*$ and the set of maps is given by $G$. Now assume that $G=F_{2}$, i.e. the free group on two generators $a$ and $b$. I would like to construct a simplicial set ${X}_{\bullet}$ such that
${X}_{0}$ consists in two elements $*_{a}$, $*_{b}$,
${X}_{1}$ is generated by the following elements:
- a $f_{a}\in\operatorname{Hom}_{X}(*_{a}, *_{a})$,
- a $f_{b}\in\operatorname{Hom}_{X}(*_{b}, *_{b})$,
- a $\gamma\in\operatorname{Hom}_{X}(*_{a}, *_{b})$,
by generated here I mean that each element of $X_{1}$ can be written (not. nec. in a unique way) as a concatenation of the elements listed above.
- $X_{\bullet}$ is a Kan complex and it is homotopy equivalent to $BF_{2}$.
Here my questions:
a) Is it possible to construct a groupoid $\mathcal{G}$ such that its nerve satisfies the above properties?
b) If not, wichwhich kind of condition should I put on $X_{2}$ (I think thai in this case $X_{\bullet}$ folowsfollows by applyngapplying the coskeleton)?
