Algebraic structures and representation stability in configuration spaces

Let FI be the category of finite sets and injections. For a fixed space X, Church-Ellenberg-Farb demonstrated that packaging the sequence of ordered configuration spaces {PConf_n(X)} as a single object, namely a contravariant functor FI -> Top is very convenient in analyzing the "stable behavior" of the sequence. After reviewing the general framework, I will present general elementary conditions on X (which need not be a manifold) which allows the action of FI to be extended to a larger category. The extra structure yields several improvements in the onset of stabilization.