the fundamental group $\pi_1(S^1)\simeq {\mathbb{Z}}$ over ${\bm k}$. This category is considered as the category of pairs $(V,T)$ of a finite dimensional $\bm{k}$-vector space $V$ and an automorphism $T$ on $V$. Let ${C}$ be a finite set and $\tau:{C}{\xrightarrow{\sim}}\{1,2,\dots, m\}$ be a bijection.

\[stokes data\] Let $(V,T)$ be an object in ${{\sf Rep}}_{\mathbb{Z}}({\bm k})$. Stokes data on $(V,T)$ of type $({C},\tau)$ are a family of objects $(V_{c},T_{c})_{{c}\in{C}}$ in ${{\sf Rep}}_{\mathbb{Z}}({\bm {k}})$, isomorphisms $f:\bigoplus_{{c}\in {C}}V_{c}\to V$, and $f^*:V{\xrightarrow{\sim}}\bigoplus_{{c}\in {C}}V_{c}$ of [*vector spaces*]{} with the following properties$:$

1.  Let $f_{c}:V_{c}\to V$ be the composition of $f$ and the natural inclusion $V_{c}{\hookrightarrow}\bigoplus_{{{c'}}\in{C}}V_{{{c'}}}$. Let $f^*_{c}:V \to V_{c}$ be the composition of the projection $\bigoplus_{{{c'}}} V_{{{c'}}}{\twoheadrightarrow}V_{c}$ and $f^*$. Then, $f^*_{{{c'}}} \circ f_{c}=0$ if $\tau({{c'}}) < \tau({c})$, and $f^*_{c}\circ f_{c}= {\mathrm{id}}_{V_{c}}$ for all ${c}\in {C}$.

2.  Set $f^!_{c}:=T_{c}^{-1}\circ f^*_{c}\circ T$ for ${c}\in {C}$. Then $f^!_{{{c'}}}\circ f_{{c}}=0$ for $\tau({c})<\tau({{c'}})$, and $f^!_{{c}} \circ f_{c}={\mathrm{id}}_{V_{c}}$ for all ${c}\in {C}$.

Let $\big{(}(V^{(a)},T^{(a)}),(V^{(a)}_{c},T^{(a)}_{c})_{{c}\in{C}},f^{(a)},f^{*(a)}\big{)}$ $(a=0,1)$ be two objects in ${{\sf Rep}}_{\mathbb{Z}}({\bm k})$ with Stokes data of type $({C},\tau)$. A morphism $g:(V^{0},T^{(0)})\to (V^{(1)},T^{(1)})$ is compatible with the Stokes data when the induced maps $(f^{(1)})^{-1}\circ g \circ f^{(0)}$ and $f^{*(1)}\circ g\circ (f^{*(0)})^{-1}$ are graded.

\[2.4\] We do not need to assume that $f^*$ is 