C2-equivariant orthogonal calculus

For example, the functor BO(-), which sends a C_2 representation V to the classifying space of its orthogonal group BO(V). We obtain a bigraded sequence of approximations to F, called the strongly (p,q)-polynomial approximations T_{p,q}F. The bigrading arises from the bigrading on C_2 representations.

The homotopy fibre D_{p,q}F of the map from T_{p+1,q}T_{p,q+1}F to T_{p,q}F is such that the approximation T_{p+1,q}T_{p,q+1}D_{p,q}F is equivalent to the functor D_{p,q}F itself and the approximation T_{p,q}D_{p,q}F is trivial. A functor with these properties is called (p,q)-homogeneous. Via a zig-zag of Quillen equivalences, we prove that (p,q)-homogeneous functors are fully determined by orthogonal spectra with a genuine action of C_2 and a naive action of the orthogonal group O(p,q). <br><br>