I'm looking at the fundamental group $\pi\_{1}(M)$ of the $n^{th}$ unordered configuration space $M$ of $\mathbb{R}^{d}$. In particular, it's well-known that $\pi\_{1}(M)\cong S\_{n}$ (symmetric group) for $d\geq3$ and $\pi\_{1}(M)\cong B\_{n}$ (braid group) for $d=2$.


Let $\Pi(a,b)$ denote the homotopy classes of paths from $a$ to $b$ in $M$ (for some $M$ as described above). We know $\Pi(a,a)\cong \pi\_{1}(M)$ for any $a\in M$ (since $M$ is connected).


**Question:**
For any $a,b\in M$, does there exist a bijection $h:\Pi(a,a)\to\Pi(a,b)$ that respects composition in the obvious way? That is, $h:[g]^{(a,a)}\mapsto[g]^{(a,b)}$ such that $[g\_{2}]^{(b,c)}\circ[g\_{1}]^{(a,b)}=[g\_2 g\_1]^{(a,c)}\in\Pi(a,c)$ ? (Have $g,g\_1,g\_2\in\pi\_{1}(M)$).


**Comments**: For a while I've thought the answer would be yes, but a concrete argument or reference would be much appreciated (or a refutation if that's the case!). I have a very rough sketch for why this might be true (involving the idea that it should suffice to consider moving one of $n$ points within a subregion of $\mathbb{R}^d$ (where adjacent points lie on the boundary), and that the homotopy classes from the fundamental group would be preserved under deformations within such a region).



Fundamental group to groupoid : bijection between homotopy classes?
Just fix a basepoint $a$ and choose a path $u\_x$ from $a$ to $x$ for each $x$. Then define $h\_{xy}\colon \Pi(a,a)\to\Pi(x,y)$ by $h\_{xy}(p)=u\_y\circ p\circ u\_x^{-1}$. These maps are bijections and satisfy $h\_{yz}(q)\circ h\_{xy}(p)=h\_{xz}(q\circ p)$. This is an instance of the general fact that any connected groupoid is equivalent to a one-object groupoid. In particular, this is nothing to do with the particular space $M$, apart from the fact that $M$ is path-connected.


