Let $M$ be some connected topological space, assume it has the local system $\Bbb{Z}_M$ then we can associate a monodromy representation $\rho: \pi_1(M,x) \to \text{Aut}(\Bbb{Z})$.
Why this representation is trivial? I know isomorphism between $\Bbb{Z}$ should be $\text{Id},-\text{Id}$ it has two elements why it's trivial?
The question coming from this wonderful article by Fouad EI Zein and Jawad Snoussi linked in the comment. The theorem goes as follows:
Corollary 1.9. The group of global sections of the local system $\mathcal{L}$ is isomorphic to the invariant subspace of the fiber $L$ at the reference point $v$ under the action of the representation $\rho$ ( where $\rho$ is the monodromy representation of $\mathcal{L}$ here) $$ H^0(M, \mathcal{L}) \simeq L^\rho:=\left\{a \in L \mid \rho(\alpha)(a)=a, \forall \alpha \in \pi_1(M, v)\right\} $$
Proof.
(Step 1) Applied the equivalence between the local system and representation of fundamental group $\pi_1(M,v)$ to the constant sheaf $\mathbb{Z}_M$.
States that the space of morphisms $\varphi \in \operatorname{Hom}(\mathbb{Z}, \mathcal{L})$ is isomorphic to the space of morphisms of the trivial representation $\mathbb{Z}$ into $\rho$.
(Step 2) On one side $\operatorname{Hom}(\mathbb{Z}, \mathcal{L})$ is isomorphic to the space of global section $H^0(M, \mathcal{L})$
(Step 3) On the other side the morphisms from the trivial representation $\mathbb{Z}$ to $\mathcal{L}$ are defined by elements $a \in L$ satisfying the above formula where $\rho(\alpha)=I d$ since it is the image by $\varphi$ of the trivial action on $\mathbb{Z}$.