  We study the category $\mathcal{A}$ 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
$\mathcal{A}$, e.g., classification of injective objects, finiteness of
injective dimension, computation of the Grothendieck group, and so on. We also
prove that $\mathcal{A}$ is (essentially) equivalent to a simpler linear
algebraic category $\mathcal{B}$, which makes many properties of $\mathcal{A}$
transparent.
