  Let $F$ and $k$ be perfect fields. The main goal of this paper is to
investigate algebraic models for the Morel-Voevodsky unstable motivic homotopy
category $\mathrm{Ho}(F)$ after $\mathbf{H}^{\mathbb{A}^1}k$ localization. More
specifically, we extend results of Goerss to the $\mathbb{A}^1$-algebraic
topology setting: we study the homotopy theory of the category
$s\mathrm{coCAlg}_k(Sm_F)$ of presheaves of simplicial coalgebras over a field
$k$ and their $\tau$ and $\mathbb{A}^1$-localizations. For $k$ algebraically
closed, we show that the unit of the adjunction $k^{\delta}[-]\dashv(-)^{gp}$
determines the $\mathbf{H}^{\mathbb{A}^1}k$ homotopy type, where
$k^{\delta}[-]$ is the canonical coalgebra functor induced by the diagonal map
$\Delta:\mathcal{X}\rightarrow \mathcal{X}\times \mathcal{X}$. We extend this
result for the category of presheaves of coalgebras over a non-algebraically
closed field $k$ and the category of discrete $G$-motivic spaces, for
$G=Gal(\bar{k}/k)$.
  On the other hand, we show that the category of coalgebra objects in
$\mathrm{PST}(Sm_F,k)$ is locally presentable, where $\mathrm{PST}(Sm_F,k)$ is
the category of presheaves with Voevodsky transfers and the monoidal structure
is given by a Day convolution product.
