A groupoid which is homotopy equivalent to $BG$

Question

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)?

Answer 1

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 $[0;1]$.