We study the category A \mathcal {A} of smooth semilinear representations of the infinite symmetric group over the field of rational functions in infinitely many variables. We establish a number of results about the structure of A \mathcal {A} , e.g., classification of injective objects, finiteness of injective dimension, computation of the Grothendieck group, and so on. We also prove that A \mathcal {A} is (essentially) equivalent to a simpler linear algebraic category B \mathcal {B} , which makes many properties of A \mathcal {A} transparent.