A deformation of Borel equivariant homotopy

We describe a deformation of the $\mathrm{\infty }$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin–Tate motivic stable homotopy category when $G=C_2$ and gives a new interpretation of the $a$-completed $C_2$-effective slice spectral sequence. As a new computational tool, we present a modified Adams–Novikov spectral sequence which computes the $RO(G)$-graded Mackey functor valued homotopy of Borel $G$-spectra.