To begin, we must first specify what a “local” ring is. Let $\StrTop{\ETop}$ be a topos ringed in $k$-algebras. It [trans.: $\Str{\ETop}$] can be identified with a left exact functor $M\to \ETop$ that sends an affine variety $V$ to the object $V\prn{\Str{\ETop}}$ of $\Str{\ETop}$-points of $V$. $\StrTop{\ETop}$ will be “local” if the functor is continuous for the topology on $M$, i.e. if whenever $\prn{U_i\to V}\Sub{i\in I}$ is a covering, the family

is surjective. In the case of the Zariski topology, this gives the local rings (without quotation marks) and in the case of the étale topology it gives the strict local rings (or Henselian local rings with a separably closed residue field).