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


1) ${X}\_{0}$ consists in two elements $\*\_{a}$, $\*\_{b}$,


2) ${X}\_{1}$ 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.


3) $X\_{\bullet}$ 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, which kind of condition should I put on $X\_{2}$ (I think thai in this case $X\_{\bullet}$ follows by applying the coskeleton)?



A groupoid which is homotopy equivalent to $BG$
Let $X = BG \times [0; 1]$, then $\ast\_a = (\ast; 0)$, $\ast\_b = (\ast; 1)$, $\gamma$ is the image of $[0;1]$, $f\_a$ is any choice of path in the class of $a$ and $f\_b$ is any choice of path in class $b$ conjugated by $\gamma$, i.e. $\gamma \bullet b \bullet \gamma^{-1}$.


Equivalently, this is the action groupoid for the trivial action of $G$ on the unit interval $[0;1]$.


