Definition 1

Let G be a groupoid, H a quiver, $\varphi$ and $\varphi '$ morphisms of quivers from H into G. A homotopy connecting $\varphi$ to $\varphi '$ is a mapping $h$ of the set of vertices of H into the set of arrows of G having the following properties:

(i) For every vertex $a$ of H, the arrow $h(a)$ has origin $\varphi (a)$ and end $\varphi '(a)$;

(ii) For every arrow $f$ of H, with origin $a$ and end $b$, one has $\varphi (f)h(b) =h(a)\varphi '(f)$.

One says that $\varphi$ and $\varphi '$ are homotopic if there exists a homotopy connecting $\varphi$ to $\varphi '$.

Let G be a groupoid, let H be a quiver and let $\varphi ,\varphi ',\varphi ''$ be morphisms of quivers from H into G. The mapping $a\mapsto e_{\varphi(a)}$ is a homotopy connecting $\varphi$ to $\varphi$. If $h$ is a homotopy connecting $\varphi$ to $\varphi '$, the mapping $a\mapsto h(a)^{-1}$ is a homotopy connecting $\varphi '$ to $\varphi$. Let $h$ and $h'$ be homotopies connecting $\varphi$ to $\varphi '$ and $\varphi '$ to $\varphi ''$ respectively. For every vertex $a$ of H, the arrows $h(a)$ and $h'(a)$ are composable. The mapping $a\mapsto h(a)h'(a)$ is a homotopy connecting $\varphi$ to $\varphi ''$.

It follows that the relation “ $\varphi$ is homotopic to $\varphi '$ ” is an equivalence relation on the set of morphisms of quivers from H into G.

Let G be a groupoid, H a quiver and $\varphi ,\varphi '$ morphisms of quivers from H into G which are homotopic. By condition (i) of Definition 1, for every vertex $a$ of H, the vertices $\varphi (a)$ and $\varphi '(a)$ belong to one and the same orbit of G.

Suppose moreover that H is a groupoid, and let $h$ be a homotopy joining $\varphi$ to $\varphi '$. The maps Orb($\varphi$ ) and Orb($\varphi '$) deduced from $\varphi$ and $\varphi '$ by passing to orbits are therefore equal. For every vertex $a$ of H and every arrow $f\in H_a$, one has $\varphi (f) =h(a)\varphi '(f)h(a)^{-1}=$ Int($h(a)$)$(\varphi '(f))$, by condition (ii) of Definition 1. In other words, the homomorphism $\varphi_a$ is equal to Int($h(a)$)$\circ \varphi '_a$. In particular, if the homomorphism $\varphi_a$ is injective (resp. bijective, resp. surjective), the same is true of the homomorphism $\varphi '_a$.