$\theta$, the germs form an exhaustive increasing filtration of $\L_\theta$ with respect to the order defined by the equation $(\ref{order-c})$ above.

2.  Set $\L_{< c,\theta}:=\sum_{c'<c}\L_{\leqslant c', \theta}$. It defines subsheaf $\L_{<c}$ in $\L$. The second condition is that the sheaves ${\mathrm{gr}}_c \L:=\L_{\leqslant c}/\L_{<c}$ are in ${\sf Loc}_{\bm k}(S^1)$.

3.  Set ${\mathrm{gr}}\L:=\bigoplus_c {\mathrm{gr}}_c\L$. It has a natural filtration given by $({\mathrm{gr}}\L_{\leqslant c})_{\theta}:=\bigoplus_{c'\leqslant_\theta c}{\mathrm{gr}}_{c'}\L_\theta$. The last condition is that near any point $\theta\in S^1$, there are local isomorphisms $\eta:\L {\xrightarrow{\sim}}{\mathrm{gr}}\L$ such that $\eta (\L_{\leqslant c})\subset {\mathrm{gr}}\L_{\leqslant c}$ for all $c\in{\mathbb{C}}$, and that the induced graded morphism is identity.

We call the pair $(\L,\L_\bullet)$ a Stokes filtered local system. Let $(\L^i,\L^i_\bullet)$ be two Stokes filtered local systems $(i=0,1)$. A morphism of local systems $\lambda:\L^0\to\L^1$ is called a morphism of Stokes filtered local systems if it preserves Stokes filtrations, i.e. $\lambda{\L^0_{\leqslant c}} \subset \L^1_{\leqslant c}$ for all $c\in{\mathbb{C}}$.

For example, we define a Stokes filtration on ${\bm k}_{S^1}$ by ${\bm k}_{S^1<0,\theta}=0$ and ${\bm k}_{S^1,\leqslant 0,\theta}={\bm k}_{S^1}$ for all $\theta \in S^1$. For two Stokes structures $(\L^i,\L^i_\bullet)$ $(i=0,1)$, the tensor product $\L^0\otimes \L^1$ is equipped with a natural Stokes filtration by $$\begin{aligned}
\label{tensor stokes}
              (\L^0\otimes \L^1)_{\leqslant c,\theta}
              :=\sum_{c^0+c^1\leqslant_\theta c} \L^0_{\leqslant c^0,\theta}\otimes \L^1_{\leqslant c^1,\theta}.
           \end{aligned}$$

Let 