There are (at least) two definitions for rigidity of a local system on $X = \mathbb{P}^1\setminus\{p\_1,\dots,p\_n\}$:


1. A local system $L$ on $X$ is rigid if any other local system with conjugate monodromy around each of the $p\_i$ is isomorphic to $L$
2. A local system $L$ of rank $n$ on $X$ is rigid its preimage under the map $Hom(\pi\_1(X),GL(n))\to M(X,n)$ is open.


In the literature people seem to treat these definitions as being equivalent, but I don't see how they could be. For example rank $1$ local systems are clearly always rigid in the first sense. But the map $Hom(\pi\_1(X),GL(1))\to M(X,1)$ is an isomorphism, so the preimage of a (closed) point will never be open. 


So what is the relation between these two notions of ridigity?



Equivalent definitions of rigid local systems
To avoid notational overlap, let $X = \mathbb P^1 \setminus \{p\_1,\dots,p\_m\}$.


The two definitions are equivalent for irreducible local systems if we restrict 2 to the subset of $M(X,n)$ with fixed local monodromy at the points $p\_1,\dots,p\_m$. (This uses the fact that we are working in $GL\_n$, otherwise the first condition may be stronger.)


As far as I know, anyone dropping the condition on the local monodromy in the second definition is either working with proper varieties (e.g. Carlos Simpson in his paper [Higgs bundles and local systems](http://www.numdam.org/item/PMIHES_1992__75__5_0)) or is making a mistake, most likely caused by copying from a paper working in the proper case and not making the appropriate adjustments.


The second definition is absurd in the setting of $\mathbb P^1 - \{p\_1,\dots,p\_m\}$ as it is easy to see there are no such representations except when $m=1$, in which case all representations are trivial.


