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We describe an inductive machinery to prove various properties ofrepresentations of a category equipped with a generic shift functor.Specifically, we show that if a property (P) of representations of the categorybehaves well under the generic shift functor, then all finitely generatedrepresentations of the category have the property (P). In this way, we obtainsimple criteria for properties such as Noetherianity, finiteness ofCastelnuovo-Mumford regularity, and polynomial growth of dimension to hold.This gives a systemetic and uniform proof of such properties forrepresentations of the categories $\FI_G$ and $\OI_G$ which appear inrepresentation stability theory.

An inductive machinery for representations of categories with shift functors