ABSTRACT
A Grothendieck representation of an arbitrary category B has been introduced in [9], with an aim to explain the appearance of canonical noncommutative topologies in noncommutative geometry. This point of view also provided a natural framework for a theory of schemes for noncommutative algebras, including a general categorical version of Serre’s global sections theorem. Now the noncommutative theory of Proj may also be fit into the context of a Grothendieck representation but if one wants to relate the projective to the corresponding “affine” theory, two Grothendieck representations of the same category have to be compared. In this note we introduce the quotient of a Grothendieck representation with respect to a so-called topological nerve with respect to B. We introduce some restricting condition, i.e. a spectral representation, allowing to retranslate certain properties holding in the representing Grothendieck categories into properties related to a notion of spectrum in B.
