Categories of Dimension Zero

If D is a category and R is a commutative ring, the functors from D to R-Mod can be thought of as representations of D; we say D is dimension zero over R if its finitely generated representations are finite length. We show that this happens exactly when R is Artinian, the hom-sets of D are finite, and D admits a homological modulus over R (Definition 1.2). Even if R is not Artinian, we show that a homological modulus provides finite models of finitely generated D-representations that are amenable to homological computation. The main technical innovation is Definition 1.1 giving preorders refining the retract preorder on the objects of any category.