 A \to A$ a map for all $i\in I$, and $C^a=(c^a_{jk})_{j,k \in I}$ a generalized Cartan matrix in ${\mathbb{Z}}^{I \times I}$ for all $a\in A$. The quadruple $${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$$ is called a *Cartan scheme* if

1.  ${r}_i^2 = {\operatorname{id}}$ for all $i \in I$,

2.  $c^a_{ij} = c^{{r}_i(a)}_{ij}$ for all $a\in A$ and $i,j\in I$.

Let $A=\{a\}$ be a set with a single element, and let $C$ be a generalized Cartan matrix. Then $r_i={\operatorname{id}}$ for all $i\in I$, and ${\mathcal{C}}$ becomes a Cartan scheme.

One says that a Cartan scheme ${\mathcal{C}}$ is *connected*, if the group $\langle {r}_i\,|\,i\in I\rangle \subset {\mathrm{Aut}}(A)$ acts transitively on $A$, that is, if for all $a,b\in A$ with $a\not=b$ there exist $n\in {\mathbb{N}}$ and $i_1,i_2,\ldots ,i_n\in I$ such that $b=r_{i_n}\cdots r_{i_2}
  r_{i_1}(a)$. Two Cartan schemes ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ and ${\mathcal{C}}'={\mathcal{C}}'(I',A',$ $({r}'_i)_{i\in I'},({C'}^a)_{a\in A'})$ are called *equivalent*, if there are bijections $\varphi _0:I\to I'$ and $\varphi _1:A\to A'$ such that $$\begin{aligned}
\label{eq:equivCS}
    \varphi _1({r}_i(a))={r}'_{\varphi _0(i)}(\varphi _1(a)),
    \qquad
    c^{\varphi _1(a)}_{\varphi _0(i) \varphi _0(j)}=c^a_{i j}
  \end{aligned}$$ for all $i,j\in I$ and $a\in A$.

Let ${\mathcal{C}}= {\mathcal{C}}(I,A,({r}_i)_{i \in I}, (C^a)_{a \in A})$ be a Cartan scheme. For all $i \in I$ and $a \in A$ define ${\sigma }_i^a \in
  \Aut({\mathbb{Z}}^I)$ by $$\begin{aligned}
    {\sigma }_i^a ({\alpha }_j) = {\alpha }_j - c_{ij}^a{\alpha }_i \qquad
    \text{for all $j \in I$.}
    \label{eq:sia}
  \end{aligned}$$ This map is a reflection. The *Weyl groupoid of* ${\mathcal{C}}$ is the category ${\mathcal{W}}({\mathcal{C}})$ such that ${\mathrm{Ob}}({\mathcal{W}}({\mathcal{C}}))=A$ and the morphisms are generated by the maps ${\sigma }_i^a\in {\mathrm{Hom}}(a,{r}_i(a))$ with $i\in I$, $a\in A$. Formally, for $a,b\in A$ the se