Rigid rational homotopy types

In this paper we define a rigid rational homotopy type, associated to any variety $X$ over a perfect field $k$ of positive characteristic. We prove comparison theorems with previous definitions in the smooth and proper, and log-smooth and proper case. Using these, we can show that if $k$ is a finite field, then the Frobenius structure on the higher rational homotopy groups is mixed. We also define a relative rigid rational homotopy type, and use it to define a homotopy obstruction for the existence of sections.Comment: 30 pages. Final version, published in Proceedings of the LM

Moduli stacks of algebraic structures and deformation theory

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial moduli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there is an associated obstruction theory for infinitesimal, higher order and formal deformations. The methods are general enough to be adapted for more general kinds of algebraic structures.Comment: several corrections, especially in sections 6 and 7. Final version, to appear in the J. Noncommut. Geo