  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.
