  Building on To\"en's work on affine stacks, we develop a certain homotopy
theory for schemes, which we call "unipotent homotopy theory." Over a field of
characteristic $p>0$, we prove that the unipotent homotopy group schemes
$\pi_i^{\mathrm{U}}(\,\cdot\,)$ introduced in our paper recover the unipotent
Nori fundamental group scheme, the $p$-adic \'etale homotopy groups, as well as
certain formal groups introduced by Artin and Mazur. We prove a version of the
classical Freudenthal suspension theorem as well as a profiniteness theorem for
unipotent homotopy group schemes. We also introduce the notion of a formal
sphere and use it to show that for Calabi-Yau varieties of dimension $n$, the
group schemes $\pi_i^{\mathrm{U}}(\,\cdot\,)$ are derived invariants for all $i
\ge 0$; the case $i=n$ is related to recent work of Antieau and Bragg involving
topological Hochschild homology. Using the unipotent homotopy group schemes, we
establish a correspondence between formal Lie groups and certain higher
algebraic structures.
