Fundamental group schemes for stratified sheaves

We study stratified sheaves in positive characteristic algebraic geometry using the technique of Tannakian categories. We show that the associated Tannakian group scheme Π is a perfect scheme. We also prove that in the proper case the largest unipotent quotient of Π is pro-etale. Special attention is paid to abelian varieties, where a description of Π is obtained. We finish with a discussion of the case where the ground field is valued.