The yoga of schemic Grothendieck rings, a topos-theoretical approach

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring, in which we can formulate a form of integration resembling Kontsevich's motivic integration via arc schemes. Whereas the original construction was via definability, we have translated in this paper everything into a topos-theoretic framework.