TITLE: What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)
QUESTION [7 upvotes]: Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, then the fibre of $X_{+}$ at $\nu$ is defined as $\text{colim} \\ X_{+}(V)$ where the colimit is over all Nisnevich neighborhoods $V$ of $\nu$. 
What is this fiber now? Is this just the constant simplicial set associated to the set of all points of $X$ with residue field isomorphic to the one of $\nu$?
And as a further question; If I have a morphism $f:X\rightarrow Y$ between two smooths $k$-schemes, when is the associated morphism of simplicial Nisnevich sheaves $f_{+}:X_{+}\rightarrow Y_{+}$ a simplicial weak equivalence?  For example in the case where $Y$ is irreducible and $f$ an open immersion of a dense open subscheme $X$?

REPLY [2 votes]: Let $N:=\textrm{Neigh}\_{Nis}(U,\nu)$ denote the filtered category of Nisnevich neighborhoods of the point $\nu\in U$. Then $\textrm{colim}\_{V\in N^{op}} V\cong \textrm{spec}(\mathcal{O}^{h}\_{U,\nu})$, where $N^{op}$ denotes the opposite category of $N$ and $\mathcal{O}^{h}\_{U,\nu}$ is the henselization of the local ring $\mathcal{O}\_{U,\nu}$.
My guess now would be that the fiber is the constant simplicial set associated to the set      $\textrm{Hom}\_{k}(\textrm{spec}(\mathcal{O}^{h}\_{U,\nu}),X)$. 
Maybe someone can clarify this as well as your second question.