  The notion of central stability was first formulated for sequences of
representations of the symmetric groups by Putman. A categorical reformulation
was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion
of FI-modules, where FI is the category of finite sets and injective maps. We
extend the notion of central stability from FI to a wide class of categories,
and prove that a module is presented in finite degrees if and only if it is
centrally stable. We also introduce the notion of $d$-step central stability,
and prove that if the ideal of relations of a category is generated in degrees
at most $d$, then every module presented in finite degrees is $d$-step
centrally stable.
