Definition 7

Let G be a groupoid. One says that a subgroupoid H of G is normal if Som(H) = Som(G) and if for every pair $(a, b)$ of vertices of H and every arrow $f\in G_{a,b}$, one has Int($f$)$(H_b) = H_a$.

Let G be a groupoid and let H be a subgroupoid of G whose set of vertices is equal to Som(G). To verify that H is distinguished, it is enough to prove that $fH_bf^{-1}\subset H_a$ for every $a\in$ Som(G), every $b\in$ Som(G) and every $f\in G_{a,b}$. In fact, if this is so, one also has $f^{-1}H_af\subset H_b$ for every arrow $f\in G_{b,a}$, that is, $H_a\subset fH_bf^{-1}$, and consequently $fH_bf^{-1}= H_a$.

Let G be a groupoid and let H be a distinguished subgroupoid of G. For every vertex $a$ of G, the subgroup $H_a$ of $G_a$ is a distinguished subgroup of $G_a$.

Let $\varphi : G\rightarrow G'$ be a morphism of groupoids. The kernel of $\varphi ($II, p. 166, example 5) is a distinguished subgroupoid of G. In fact, let $f$ be an arrow in G joining $a$ to $b$ and let $g\in$ Ker($\varphi$ )$_b$; then one has $\varphi (f gf^{-1}) =\varphi (f)\varphi (g)\varphi (f^{-1}) =\varphi (f)e_{\varphi(b)}\varphi (f)^{-1}=e_{\varphi(a)}$, whence the inclusion $f$Ker($\varphi$ )$_bf^{-1}\subset$ Ker($\varphi$ )$_a$.

Let G be a groupoid. The groupoid G and the subgroupoid of G whose set of arrows is the set of identity elements of G are distinguished subgroupoids of G. The intersection of a family of distinguished subgroupoids of G is a distinguished subgroupoid of G. In particular, for every subset $F\subset$ Fl(G), there exists a smallest distinguished subgroupoid of G whose set of arrows contains F. It is called the distinguished subgroupoid generated by F.

Let H be a distinguished subgroupoid of G. Let $\mathscr{R}$ be the equivalence relation in Fl(G) defined by $\mathscr{R}\{f, g\}$ if and only if there exist arrows $x$ and $y$ in Fl(H) such that $f=xgy$. If $f$ and $g$ are arrows of G equivalent modulo $\mathscr{R}$, their origins (resp. their ends) belong to the same orbit of H. Put $F'=$ Fl(G)$/\mathscr{R}$ and denote by $o'$ and $t'$ the mappings of $F'$ into Orb(H) deduced from $o$ and $t$ by passing to the quotients. Denote by $G/H$ the quiver (Orb(H)$,F', o', t'$).