Rational and $p$-local Motivic Homotopy Theory

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.