Quotients of real algebraic sets and AS-sets equipped with a linearizable compact group action

We present the full-detailed construction of new geometric quotient structures for affine real algebraic varieties equipped with a linearizable action of a compact Lie group G. These quotients are functors from the category of arc-symmetric/AS-sets equipped with a linearizable action of G and equivariant continuous maps with AS-graph to the category of AS-sets. We furthermore provide a complete review of the results employed in the constructions, including properties of equivariant real algebraic geometry with respect to polynomial group actions, as well as an introduction to semialgebraic arc-symmetric sets and AS-sets of compact affine Nash manifolds.